Research ArticleStudy on the Valuation Method for Overseas Oil and GasExtraction Based on the Modified Trinomial Tree OptionPricing Model

Jianye Liu 1 Zuxin Li 12 Dongkun Luo 1 and Ruolei Liu1

1School of Business Administration China University of Petroleum (Beijing) 18 Fuxue Road Changping DistrictBeijing 102200 China2Institute of Petroleum Exploration and Development of China National Petroleum Corporation Beijing 100083 China

Correspondence should be addressed to Jianye Liu jianyeliu2015163com Zuxin Li leezx-69hotmailcom andDongkun Luolizx609163com

Received 25 December 2019 Accepted 20 April 2020 Published 12 May 2020

Academic Editor Jian G Zhou

Copyright copy 2020 Jianye Liu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Wandering of oil prices at lower values and the bitter reality have forced people to look for a more accurate valuation method foroverseas oil and gas extraction of China However the currently available resource classification method discount cash flow(DCF) method and real option method all suffer from their own disadvantages is paper identifies multiple uncertainty factorssuch as oil prices and reserves It then investigates the transmission mechanism of how each uncertainty factor impacts the oil andgas extraction value and quantifies the transmission efficiency e probability distribution patterns of each uncertainty factorhave been determined the trinomial tree option pricing model is modified with consideration upon the nonstandardness of theprobability distribution Decision points and strategies space are designed in accordance with the practical oil and gas productionand the Bermuda option is adopted to replace the conventional decision-based tree model with the probability-based tree Finallya backward algorithm is developed to calculate the probability at each decision point which avoids difficulties in determining theasset volatility ratio and a case study is presented to demonstrate application of the proposed method Results show that decisionrights for overseas investment are valuable e value of extraction does not yet necessarily grow with higher uncertainty andinstead it is under joint effects of the cash flow and strategy space So valuation should incorporate the composite value of futurecash flow and decision rights Volatility of the value of extraction is not solely dependent on the oil price but affected by multiplefactors Similar to the Bermuda option the decision-making behavior for oil and gas extraction occurs only at finite decisionpoints to which the trinomial tree option pricingmodel is applicablee adoption of probability distribution can to a great extentexploit the uncertain information Replacement of the decision-based tree with the probability-based tree provides more accurateprobability distribution of the calculated value of extraction and moreover the disperse degree of the probability can reflect howhigh risks are which is conducive to decision-making for investment

1 Introduction

Since the 21th century the foreign dependency of Chinarsquos crudeoil has been evergrowing due to the increasing oil demand [1]In this regard major oil and gas companies in China all developtheir overseas development strategies for ldquogoing outrdquo [2 3] Afternearly 20 years of international development oil and gascompanies of China have possessed some oil and gas blocks inthe central Asia Africa etc However these blocks are often seen

with inferior opulence in resources and in many cases they arelocated in high-risk countries [4]

In recent years China has further increased its invest-ment in overseas oil and gas assets resulting in a significantincrease in overseas oil and gas production As shown inFigure 1 the overseas oil and gas production of Chinarsquos threemajor oil companies CNPC SINOPEC and CNOOC in-creased by 90 416 and 177 in 2018 compared with2011

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 4803909 15 pageshttpsdoiorg10115520204803909

Hence the investment evaluation for oil and gas ex-traction is of extreme importance With the prolongeddownturn of oil prices there will be more opportunities toacquire overseas oil and gas assets however higher re-quirements are also raised up upon decision-making inacquisition and extraction For oil and gas blocks that havebeen through deepened exploration and already put intodevelopment valuation is relatively simple e availableelaborate geological and production data are sufficient tobuild the geological model and gas production forecastmodel which are able to offer relatively precise valuation ofassets [5] Nonetheless as for oil and gas blocks with uncleargeological setting and incomplete data the existing con-ventional method fails to accomplish asset valuation due toits high uncertainty [6] Chinarsquos overseas oil and gas de-velopment business starts relatively late and correspond-ingly the current assets mostly belong to the latter case [4]Accordingly a valuation method specific to oil and gasextraction with higher uncertainty is required

Based on the Modified Trinomial Tree Option PricingModel and considering the fluctuation of asset uncertaintythis research explores the value of decision rights at eachdecision point in overseas oil and gas asset evaluation andproposes an asset evaluation method Firstly this paperpresents an overview of existing evaluation methods foroverseas oil and gas assets and reveals that the existingmethods have some defects in calculating the value of de-cision rights under uncertain conditions Secondly itidentifies some uncertainty factors of overseas oil and gasassets analyzes how these factors exert impacts on the assetvalue and constructs a formula to estimate the cash flow ofoil and gas asset development irdly it holds that there are

five main decision points and three strategy types in the oiland gas investment process By improving the trinomial treeoption pricing model and combining with the distributionand types of uncertainty factors this paper finally presentsan evaluation method that is based on the probability tree ofoil and gas exploration and development and that can beused to calculate the value of decision rights at each decisionpoint in inverse order and the overall oil and gas asset valueand also it gives some examples to illustrate the applicationof this method

2 Literature Review

e evaluation of oil and gas extraction refers to the eco-nomic benefit evaluation of the future exploration devel-opment and sales process of oil and gas assets Besides thedirect extraction value the value of overseas oil and gasassets also includes strategic value social value and politicalvalue which are not within the scope of this paper In termsof valuation of oil and gas extraction with higher uncer-tainty the current research approach can be divided into twogroups One is the quantitative analysis based on modellingand computing including the discounted cash flow and realoptions valuation e other is the qualitative or semi-quantitative research based on decision-makersrsquo percep-tions such as the Delphi method and resource classificationmethod

For the qualitative analysis the Delphi method is sus-ceptible to personal preferences and knowledge limitationsof surveyed experts and has been discarded as the corevaluation method for investment decision-makers e re-source classification method [7 8] is a semiquantitative

88599528

11287 12575

13709 13868 15298

16006

51690

29823

7079

52377

35104

7800

59202

35676

17994

65201

40695

19852

72032

44101

20954

76016

42697

19964

89119

43483

20376

98219

42235

19605

2011 2012 2013 2014 2015 2016 2017 2018

CNPC oilCNPC gasSINOPEC oilSINOPEC gas

CNOOC oilCNOOC gasTotal

0

2000

4000

6000

8000

10000

12000

14000

16000

18000Pr

oduc

tion

(104 to

n)

Years

Figure 1 e overseas oil and gas production of CNPC SINOPEC and CNOOC (1255m3 of natural gas production converted to 1 ton ofcrude oil production)

2 Mathematical Problems in Engineering

method for comprehensive valuation of oil and gas assetsinvolving grading hydrocarbon resources in accordancewith reserve quantities recovery difficulties oil and gasquality external risks etc is method is relatively accurateand moreover simple and can easily determine at whichgrade a certain oil and gas asset stays However compre-hensive grading based on this method is somewhat sub-jective and has difficulties in deciding which has highervalues a block with a smaller scale and yet better quality orthe one with secondary quality but expanded reservequantities Under such circumstances weights shall beassigned to each grading indicator by the decision makersand therefore this semiquantitative method is applicable tofuzzy comparison and preliminary screening out of multipleblocks

Valuation of overseas oil and gas extraction demandsmore precise quantitative research to support investmentdecision-making especially during the current downturn ofoil prices With respect to quantitative research themainstream method should be the discounted cash flow(DCF) method based on input and outpute DCFmethod[9] is widely used in the oil and gas industry and is able topresent relatively high precision in the case of oil and gasblocks with lower certainty Taking the time value of moneyinto consideration it precisely calculates the value of ex-traction by estimating the exploration and developmentinvestment operation expenditure and mortgages and taxesand predicting the sales revenue of crude

Nevertheless the estimation of investment cost andsales revenue in this method requires low uncertainty of theevaluated object otherwise the calculated NPV (net presentvalue) will not have credibility [10] Moreover the DCFmethod only considers the value of future cash flow andneglects the value of decision-making Clearly for overseasoil and gas extraction the value of decision-making isembodied as the ability to give up exercising rights whichmeans stopping exploitation in the case of money-losing oiland gas investment [11] is is a limitation of the DCFmethod

Some scholars introduce the option method to deal withuncertainty of overseas oil and gas extraction calculate thecomprehensive value and determine the investment timinge real options method [12 13] as an evaluation approachestimates the financial value of oil and gas extraction throughthe DCF process then calculates the option value using thevolatility ratio of the asset value and at last concludes thecomprehensive value e real option method is able to bettermimic the decision-making behavior and measure the value ofthe decision right by considering and calculating values ofoptions to defer investment and to abandon

Yet some issues still exist in terms of the basic as-sumption and actual implementation and have not been wellhandled Firstly it is hard to measure the volatility ratio ofthe value of extraction Transactions of oil and gas assets arecharacterized by their small quantity and discontinuationand thus the value volatility ratio cannot be directly cal-culated Generally the real option method [12 14 15] usesthe oil price fluctuation to represent the undulation of the oiland gas asset value between which the consistency has not

been confirmed yet In fact the value of oil and gas assets isnot only related to oil prices but also related to many otherfactors such as the level of risks associated with the re-sources the local political and economic status and lawsis is considered a major flaw of this method Secondlythere is no complete market for oil and gas assets like thatfor option transactions e excessively limited quantity ofbuyers and sellers decides that the transaction is not real-time and various options cannot be exercised in a timelyfashion Furthermore the transaction value of assets islargely determined through the game between the two sidesand the transaction value of options is hard to be estimatedIn addition some studies [16] reject two stylized facts of realoptions on oil one is that the correlation of the returns on oiland the stock market is positive the other is that it is in-variant to changes in oil price volatility ey state that thewidespread idea that higher volatility leads to increasedvalue and postponed investment is not necessarily valid

To sum up regarding valuation of overseas oil and gasextraction the qualitative method suffers from insufficientprecision the DCF method fails to capture the value ofdecision rights and the basic assumption of the real optionmethod is questionable erefore it is required to thinkabout it further to overcome these defects

3 Analyzing Uncertainties of Overseas Oil andGas Extraction

e value of the decision rights for overseas oil and assetsroots in uncertainty and thus we should first clarify in whichaspects uncertainty is embodied then analyze how theseuncertain factors impact the assessed value of assets and atlast characterize variations of these uncertainty factors

31 Identification ofUncertainty Factors Overseas oil and gasextraction are subject to various uncertainty factors of whichextensive identification investigation and subsequent riskquantification and asset valuation have been carried out Mostscholars [17ndash20] focus on uncertainty in the geology whichmainly include the reserves quality depth utilization rateproduction rate and decline rate of production Some scholars[21 22] also consider the external environment including thepolitical and economic environment sovereignty credit andglobal oil price Besides the geographic location topographicalsetting hydrogeological background and technical proficiencyof operators which are concerned with the discovery devel-opment construction and operating costs during hydrocarbonrecovery also have effects upon the value of assets In summarythe uncertainty factors can be classified into the followingcategories as shown in Table 1

Geological factors of resources refer to the geologicalconditions of oil and gas resources and their uncertainty ismainly caused by the error of measurement results of ex-ploration experimental wells ese uncertainty factors aremostly derived from the low level of exploration andfuzziness in geological data and will gradually decline withthe progressing geological understanding e uncertaintyfactors of production parameters mainly refer to the

Mathematical Problems in Engineering 3

parameters in the production process which are influencedby the geological conditions technical level of producersand the preference of decision makers and have certainsubjectivity External environmental factors mainly refer tothe political and economic environment of the target areawhich are objective macrofactors Other factors howeverdue to the relatively low level of uncertainty are not con-sidered in this research

32 Uncertainty Transmission Uncertainty of each factor isultimately transmitted onto the extraction value which isembodied as the uncertainty of the value of overseas oil andgas extraction e transmission mechanism of uncertaintyfactors is illustrated in Figure 2

e reserves utilization rate production rate anddecline rate impact the ultimately recovered saleable re-sources e larger the reserves the higher the utilizationratio the faster the production rate the slower the declinerate then the larger the saleable resources However theincrease of production rate will also generally lead to theincrease of decline rate e production rate and declinerate together affect the production curve of the whole lifecycle which further results in influence upon the time valueof capital Even if it has the same total production the netpresent value of different production curves is different Itis obvious that earlier recovery of hydrocarbons is favorableto amortizing the capital and repaying the loan e re-source quality and the global oil price have effects upon thefinal sales price and heavy oil and light oil are seen withdifferent global prices Generally quality compensation isused to represent the quality of resources e worse theresource quality is the higher the amount of compensationis needed and the lower the value of extraction is Fur-thermore the burial depth of resources is one of the de-cisive factors for drilling cost Higher capitalized costscompromise the value of extraction

e above work clarifies the route of each uncertaintyfactor to influence the value of oil and gas extraction and yethow much the influence is cannot be calculated through thepresented figure us we need to further investigate theefficiency of such a transmissionmechanisme value of oiland gas extraction is dependent on the cash flow generatedby future oil and gas production and the cash flow com-ponent is shown in Figure 3

e roles played by uncertainty factors in terms of cash flowgeneration are illustrated in Figure 3 For cash outflow theamount of the predicted resources is related to the explorationarea which will determine the exploration investment andreserves discovery cost is generally used to quantify the in-vestment e initial annual production will affect the facilities

construction investment and the total output will affect variousCAPEXs (capital expenditure) and OPEXs (operating expense)and then indirectly affect the expenditure of taxes of course allexpenditures are influenced by the investment environment taxlaws and local communities such as the level of local priceswhich will affect the workersrsquo wage For cash inflows global oilprices and the resource quality will affect sales prices whileproduction and sales prices determine the sales revenue Effortscould bemade to formulize these routes of cash flow generationwhich is in accordance with financial appraisal of oil and gasproduction

Many studies on oil and gas investment appraisal [19 23]have discussed those formulas and here a complete for-mulized description of cash inflow and outflow for oil andgas development is presented

In terms of exploration investment it is related topredicted recoverable reserves and also the discovery costper barrel oil erefore it can be expressed as

Iexp Iexpbbl times Rrec Rrec Rpre times rrec rrec 1113944

n

t1rpro 1 minus rdel( 1113857

tminus 1

(1)

where Iexp is the exploration investment Iexpbbl is the discoverycost per barrel oil Rrec are the predicted recoverable reservesRpre are the predicted development pending resources rrec is therecovery factor rpro stands for the production rate rdel repre-sents the decline rate n is the estimated recovery lifecycle

As for development investment it is primarily depen-dent on the predicted reserves and utilization and pro-duction rates and can be calculated as follows

Idev Idevbbl times Ptotal

Ptotal Rpre times ruti times rrec(2)

where Idev is the development investment including in-vestment on drilling fracturing and completion Idevbbl isthe development investment per barrel oil Ptotal is the totalproduction ruti is the utilization rate

en for construction investment it mainly involvesconstruction of surface oil and gas processing facilitieswhich is correlated to the maximum annual hydrocarbonoutput instead of total production Higher annual pro-duction results in higher construction investment which canbe computed using the following equation

Icon Iconbbl times Pini

Pini Rpre times ruti times rpro(3)

where Icon is the construction investment Iconbbl is theconstruction investment per barrel oil Pini stands for theinitial annual production

Table 1 Uncertainty factors of overseas oil and gas extraction

Categories Uncertainty factorsResource geology Oil and gas resource quantity quality depthProduction parameters Rate of utilization production declineExternal environment Politics economics sovereignty credit global oil priceOthers Geography topography hydrology operational proficiency

4 Mathematical Problems in Engineering

e sales revenue is determined by the global oil pricequality of produced oil and gas and produced amount of oiland gas Calculation can be done using the following equation

Rsal Psminusprice times Ptotal

Psminusprice Pgminusprice + iqua(4)

where Rsal is the sales revenue Ps-price is the sale price Pg-price isthe global oil price iqua is the price variation dependent onquality of hydrocarbon resources which may be negative

Calculating the operating cost and tax is relatively simpleand can be shown via the following equation

Ttax Ttaxbbl times Ptotal

Copex Copexbbl times Ptotal(5)

where Ttax stands for taxesCopex is the operating cost Ttaxbblis the average tax per barrel oil Copexbbl is the operating costper barrel oil

To sum up the composition of future cash flow of oil andgas assets development is shown in Table 2

33 Distribution Patterns of Uncertainty Factors Havingformulized the path of influence of each uncertainty factor

for future case flow of oil and gas extraction and clarified thetransmission mechanism of the uncertainty factor for thevalue of extraction we are still facing undefined degrees ofuncertainty for each factor itself In other words investi-gation of the distribution pattern of each factor should becarried out eg volatility of oil prices

In reference to volatility of each uncertainty factor manyfactors have available literatures for references Regardingreserves extensive research in the petroleum engineeringindustry [24 25] suggests that the oil reserve is undoubtedlyfound with the logarithmic normal distribution instead ofthe normal distribution assumed by some researchers in the

Recovery factorPredicted resources Predicted recoverable reserves

Utilization resources

Production rate

Utilization rate

Initial production Decline Rate Total production

Exploration investment

Facilities construction investment Investment environment amp society

Development investment Deep

Operating cost

Global oil prices Sale pricequality

Government Local tax

Sale revenue

Cash

flow

Figure 3 Components of the future cash flow of overseas oil and gas extraction

Table 2 Cash inflow and outflow of oil and gas assets

Categories Cash flows Notation Workflow stage

Outflow

Exploration investment Iexp ExplorationDevelopmentinvestment Idev Development

Construction investment Icon DevelopmentOperating cost Copex Sale

Taxes Ttax SaleInflow Sales revenue Rsal Sale

DepthRecoveryrate

Declinerate Reserves Utilization

rate Quality Global oilprices

Production Sales priceTime value

Overseas oil amp gas value

Sovereign credit

Investmentenvironment

Risk discount rate Investment

Resource conditions

Uncertainty factors

Local conditions Economic condition

Figure 2 Transmission of uncertainty with respect to the value of overseas oil and gas extraction

Mathematical Problems in Engineering 5

real option field For the distribution of oil prices anagreement in understanding among extensive scholars hasnot been reached yet Some [15 26] believe that oil pricevariation should be a type of geometric Brownian motionwhile others [20 27ndash30] conduct oil price forecast using thesupport vector machine Bayesian model system simulationor a combination of multiple approaches with variouscorresponding results is paper tends to believe that oilprice complies with the Mean-Reversion with Jumps [26]which means that oil price is endowed with a mean-re-version nature and the mean oil price will gradually growwith time in case of no unexpected outburst events

In terms of quality and depth of resources both aredetermined according to the results of exploration experi-mental Wells e uncertainty of resource depth comes fromthe measurement error of experimental well depth and theuncertainty of resource quality comes from the measure-ment error of sulfur content and other indicators ereforethey can be considered as normal distribution and can beexpressed by quality compensation amount and drilling cost

Utilization rate production rate decline rate and otherfactors are to some extent subject to the subjective influence ofthe developer after consulting relevant experts we make thefollowing assumptions For the utilization rate it should bewithin [0 1] and we assume that it obeys the trapezoidaldistribution having probability within a certain subintervalmuch higher than the averagee production rate of resourcesis somewhat susceptible to subjectivity and meanwhile is alsoconstrained by geological conditions It should be within (0 1)with the existence of an optimal value and is therefore assumedto follow the triangular distribution e decline rate of hy-drocarbon recovery with a supposed range of (0 1) is related tothe production rate and also under the constraints of geologicalconditions Consequently it is also assumed to follow the tri-angular distributionWhen it comes to the discount rate of risksit is dependent on the local investment and financing envi-ronment sovereignty credit and politics Its distribution patternis still unclear For projects with low risks it may present theT-shaped distribution while for projects with higher risks itmay follow the normal distribution and we have not reached anagreement yet In addition in most cases the risk-free rate ofreturn is replaced with the long-term treasury bond rate (LTBR)of theUS and the risk-free rate plus the risk discount rate shouldbe the discount rate i used in calculating net present values

To sum up the distribution patterns of uncertaintyfactors are summarized in Table 3

4 AModified ApproachBased on the TrinomialTree Option Pricing Model

Upon accomplishment of identification of uncertainty fac-tors and investigation of transmission routes and distribu-tion patterns of probability we are able to calculate the netpresent value distribution on the basis of the establishedprobability density function and transmission route formulaof uncertainty factors e calculation is simple as is shownin equation (6) and the expectation value and variance of netpresent values can be obtained which is similar to theappraisal concept based on the DCF method Nonetheless

such practice still neglects the value of decision rights andthus we need to modify the trinomial tree option pricingmodel in a way inspired by the real options method

NPV 1113944n

t1Rsalminust minus Iexpminust minus Idevminust minus Iconminust minus Copexminust minus Ttaxminust1113872 1113873

times(1 + i)minus t

(6)

where NPV is the net present value of overseas oil and gasextraction i is the discount rate Rsal-t Iexp-t Idev-t Icon-t Copex-tand Ttax-t are the sales revenue exploration investment de-velopment investment construction investment operatingcost and taxes at the t-th year respectively (in case of noincome or expense under a specific term it should be zero)

41 Decision Points and Strategies during Appraisal (DeferredDevelopment Immediate Development and Sale of Assets)e real optionmethod can deal with the asset volatility ratiovia an approach combing the uncertainty factor and thetransmission route formula since the calculated net presentvalue presents itself as a distribution It should be noted thatthe resultant distribution does not necessarily follow thenormal or logarithmic normal distribution and thereforesome currently available option calculation models may beinapplicable e volatility ratio of the calculated distribu-tion does not solely depend on the oil price this singleuncertainty factor instead should be computed usingmultiple factors through the transmission route formulas

Another major disadvantage of the real option method isthat exercising rights cannot be done in a real-time manner Itis not like that one can immediately exercise the right at anymoment and there is no such thing as a simple switch forturning on and off lights to allow for immediate startupsuspension and termination of petroleum exploration anddevelopment For example the option to defer cannot beexercised in themiddle of drilling to instantaneously shut downthe development Oil and gas fields cannot be sold out duringexploration and development to exercise the option to aban-don Consequently the timing at which it is feasible to exercisethe option should be analyzed which is referred to as thedecision point in this paper At non-decision points optionscannot be exercised or partially exercised to defer or abandon

e general extraction workflow of oil and gas is illus-trated in Figure 4 At each decision point occurs a decision-making behavior which may have various strategy spacese corresponding decision space is concluded in Table 4

Here is a brief statement of the strategy space of eachdecision point listed in Table 4 For each major stage wehave three strategies namely starting investment deferringand waiting and abandoning investment right before ini-tiation of exploration development and sale In this regardthis paper is consistent with Tang et al [13] yet two ad-ditional intermediate decision points are considered in thispaper After accomplishment of regional exploration apreliminary appraisal is carried out before trap explorationwhich is consistent with the practice of oil companies Ifregional exploration presents favorable results exploration

6 Mathematical Problems in Engineering

goes on otherwise it will be abandoned Moreover afterfinishing the drilling engineering some oil and gas com-panies may decide not to perforate the payzone for the timebeing and wait for the right moment in accordance of theirown status and estimation of future oil price and supplytendencies It is based on this very fact that the decisionpoint is designed

In addition to the five main decision points mentionedabove in fact there are many possible accidental decisionpoints in the process of oil and gas extraction For examplelarge fluctuations of the oil price may delay or bring forwardthe exploration safety incidents may lead to the suspensionof someWells and oil and gas productionmay be suspendedfor political reasons However it is difficult to predictwhether these decision points will occur when they willoccur and how long a project may be suspended Since thispaper is only a method study there is no obvious differencein the application of the method whether it is 5 6 or moredecision points this model simplifies the actual situation andonly considers 5 main decision points that inevitably exist

42 Modification to the Trinomial Tree Option Pricing ModelOn the basis of the analysis on the decision-makingworkflow presented above it is found that investment onoverseas oil and gas extraction is characterized by limited

decision points and should be a type of Bermudan optionsto which the tree option model is applicable instead of theAmerican-style or European-style options

e conventional binomial modal develops the decision-making tree with respect to the probabilities of upward anddownward movements Magnitudes of upward and down-ward movements are dependent on the volatility ratio of thetotal asset erefore there are infinite decision points andthe resultant ultimate value of extraction follows theprobability distribution However decision points foroverseas oil and gas extraction are finite which means thatone is incapable of simulating the asset volatility throughinfinite decision points Moreover the NPV at each decisionpoint calculated using Table 3 and equations (1)ndash(6) presentsitself as a distribution Under such circumstances we are notable to plot and handle an N-ary tree with infinite upwardand downward points Given this some modification has tobe made upon the tree option pricing model e tree is notplotted in accordance with upward and downward move-ments instead it is developed in reference to the strategyspace Consequently we are able to calculate the probabilitydistribution of the pre-decision value of extraction in abackward manner as is shown in Figure 5

e probability tree-based method to estimate value stilladopts the concept of calculating the initial value of ex-traction in a backward manner although it is slightly dif-ferent with that based on the decision-making tree First thevalue at the last decision point ldquobefore productionrdquo iscomputed Upon accomplishment of exploration and de-velopment there are three available strategies namelyabandonment deferring and immediate production In thecase of the value at production higher than zero decisionmakers will choose immediate development in the case ofthe value at production lower than zero decisionmakers willchoose to abandon development and recover residual valuesor straightforward sale of assets with a production value ofabout zero decision makes will choose to defer developmentand wait for growing back of oil prices e specific boundlimit is dependent on preferences for utilities of investors

Table 3 Distribution patterns of uncertainty factors for overseas oil and gas extraction

Uncertainty factor Distribution pattern Source Range Additional remarksResource reserves Logarithmic normal distribution Reliable literature gt0 mdashSale price of oil Mean-reversion with jumps Available literature gt0 Mean value growing with timeResource depth Normal distribution mdash gt0 e measurement error is normally distributedResource quality Normal distribution mdash mdashUtilization rate Trapezoidal distribution Assumed [0 1] Consulting relevant expertsamp consider the realityRate of production Triangular distribution Assumed (0 1)Decline rate Triangular distribution Assumed (0 1)Discount rate T-shaped or normal distribution Assumed mdash Not in an agreement

Regional exploration Trap exploration Drilling engineering Completion engineering

Ground facility construction engineering

Production

Exploration Development Sale

Figure 4 Workflow of overseas oil and gas exploration and development

Table 4 Decision points and their strategy space for overseas oiland gas extraction

Stage Decision point Strategy space

ExplorationBefore regionalexploration Begin defer abandon

Before trap exploration Continue abandon

Development Before drilling Continue deferabandon

Before completion Continue defer

Sale Before production Continue deferabandon

Mathematical Problems in Engineering 7

e value at this decision point can be expressed using thefollowing equation

C5

V5 minus I5 V5 gt 0 V5 1113936n

tt5

Rsalminust minus Copexminust minus Ttaxminust1113872 1113873 times(1 + i)minus t

0 V5 asymp 0

Rrecminus5 minus I5 V5 lt 0 I5 1113936t5

tt4

Idev2minust + Icon2minust1113872 1113873 times(1 + i)minus t

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

where V5 is the present value of future earnings in case the oiland gas field is being developed I5 stands for the present valueof investment made in the previous stage which will becomethe sunk cost if the development is abandoned and the holdingasset if the development is deferred t5 is the time corre-sponding to the decision point ldquobefore productionrdquo C5 is thevalue at the decision point Rrecminus5 refers to the residual value ofprevious investment that can be recovered if the development isabandoned or the value that the asset can realize in case it issold out generally far less than I5 It should be noted that V5and I5 calculated in accordance with Table 3 are probabilitydistributions instead of specific numbers

Similarly value distributions at other decision points canbe computed as is shown in the following equation

C1

C2 C2 gt 0

0 C2 asymp 0

Rrecminus1 minus I1 C2 lt 0

⎧⎪⎨

⎪⎩

C2 C3 minus I2 C3 gt 0

Rrecminus2 minus I2 C3 le 01113896

C3

C4 minus I3 C4 gt 0

0 C4 asymp 0

Rrecminus3 minus I3 C4 lt 0

⎧⎪⎨

⎪⎩

C4 C5 minus I4 C5 gt 0

0 C5 le 01113896

I2 1113944

t1

t1Iexp1minust1113872 1113873 times(1 + i)

minus t

I3 1113944

t2

tt1

Iexp2minust1113872 1113873 times(1 + i)minus t

I4 1113944

t3

tt2

Idev1minust + Icon1minust1113872 1113873 times(1 + i)minus t

(8)

where I1 I2 I3 and I4 are present values of existing in-vestment at each decision point respectively Rrecminus1 Rrecminus2 andRrecminus3 are the investment residual values that can be paid back ifthe development is abandoned In most cases one will notabandon completion after finishing drilling and the decision islimited to either immediate or deferred completion

After obtaining distributions of the value at each deci-sion point through backward calculation based on theprobability tree we can further compute the expectedeconomic value with respect to the value distribution Forinstance at the first decision point that is before decision-making upon regional exploration the incurred cost is theacquisition cost of the oil and gas asset I1 If the developmentis implemented the obtained revenue is C2 In the case ofdeferred development the investment changes into the assetof which the value is assumed to be equal to the acquisitioncost Under the circumstance that the development isabandoned part of previous investment will be lost Giventhe aforementioned information the expected value at theldquobefore regional explorationrdquo decision point is shown in thefollowing equation

E C1( 1113857 1113946P10

0Rrecminus1 minus I1 + 1113946

P10+P12

P10

0+ 11139461

P10+P12

C2

P10 P C2 lt 0( 1113857

P12 P C2 asymp 0( 1113857

P11 P C2 gt 0( 1113857

(9)

e expectation of the value of extraction at otherintermediate decision points can be calculated in asimilar approach and thus detailed description is dis-carded here

Regional exploration Trap exploration Drilling engineering Completion engineering Production

P0

P10

P11

P12

P20

P21

P30

P31

P32

P41

P42

P52

P51

P50

Yes

No

Wait

Yes Yes

No

Yes

No

Yes

No

Wait Wait Wait

Figure 5 Probability tree of overseas oil and gas extraction (there are three possibilities of whether to proceed to the next stage Yes meansimmediate development No means immediate abandonment and Wait means waiting for opportunity)

8 Mathematical Problems in Engineering

43 Application of the Modified Method In this paper theconventional decision-based binomial model is replaced bythe probability-based one For purposes of understandinghow this model can be applied this paper takes an overseasoil and gas extraction project as an example to brieflydemonstrate the application of the modified method

Sudan X is an oil and gas block located in Africa thatCNPC plans to invest inis block is seen with no elaborategeological data and only part of parameters required forasset appraisal can be determined on the basis of analogousblocks Acquisition of the exploration and developmentinterest of this block costs three million USD and thediscovery cost per barrel oil Iexpbbl is 12 $bbl e con-struction investment per barrel oil is 50 $bbl while theoperating cost per barrel oil is 17 $bbl e composite taxrate is about 20 e estimated period of exploration lastsfor two years and production capacity construction needsthree years followed by a sustained recovery of 17 yearsOther factors are all uncertain with dispersed probabilitydistribution and high uncertainty (Table 5)

It is easy to compute the probability distribution ofannual production of each year on the basis of the distri-butions of the resource reserves utilization rate productionrate and decline rate equation (1) and the assumption thatproduction capacity establishment can be completely fin-ished within three yearse calculation process is illustratedin Figure 6

With respect to oil price volatility discount rate anddistribution of sale price variation induced resource qualitywe can compute the distribution of the present value ofearnings V5 at each future year using equation (7) Resultsare presented in Figure 7

It is seen that V5gt 0 is highly likely and thus at thedecision point ldquoproductionrdquo strategies of deferring orabandoning are seldom chosen is is consistent with ourcommon observation Once all investments of explorationand development are down oil and gas operators barelydecide to abandon production

With equations (1)ndash(3) and distributions of relevantparameters the probability distributions of I4 and I5 can becomputed which subsequently leads to the distributions ofC4 and C5

Figures 7 and 8 indicate high odds of the case of C4gt 0and C5gt 0 is means that once oil and gas resources arefound during exploration probability of the proceedingdevelopment is very high Given this the values of the lattertwo decision points are overridden which is inconsistentwith the view held in the conventional real option law thathigher uncertainty results in the higher asset value

Similarly the probability distribution of C3 can be de-termined with the help of I3 and C4 and is shown in Figure 9

Unlike other decision points the future value at thedecision point ldquotrap explorationrdquo presents negative valueswhich is decisive in whether or not to execute trap

exploration If the future cash value is below zero it iscertain that investment stops A recovery ratio of 20 isset for the total previous investment residual value that isRrecminus2 20 I A range of [minus20 I2 +20 I2] for C3 is alsoset to define the case in which cash value is about zerois range is man-made and in fact will not be usedgiven equation (8) since no deferring strategy exists at thetrap exploration decision point At last the probabilitydistribution of C2 can be computed on the basis of I2 andC3 (Figure 10)

From Figure 10 it is seen that the probability of C2 belowzero is significantly reduced with decision-making beforetrap exploration However negative values still exist becauseloss can be induced by volatility of uncertainty factors eventhough rational decision-making has been executed edecision-making behavior is able to only reduce such lossesand yet it is incapable of entirely avoiding such losses Fi-nally the probability distribution of C1 can be computedusing equation (8) as is shown in Figure 11

Figure 11 demonstrates that before initiation of regionalexploration thanks to the availability of a waiting strategythe calculated value distribution presents high probabilityfor the value of about zeroe resultant expectation value ofC1 is 1148 million USD

In the case of the DCF method with no consideration ofdecision rights the probability distribution of C1 calculateddirectly using equation (6) is plotted in Figure 12 e ex-pectation value of C1-DCF calculated in this way is 1058million USD

Table 5 Distribution of uncertainty factors in Block Sudan X

Uncertainty factor Distribution of affected parametersResource reserves Ln (Rpre)simN(9 0422)

Global oil price3 dLn Ps-price 08(4-Ln Ps-price)dt + 002dztdztsim(0 dt)

Resource depth IdevbblsimN (16 322)Resource quality iquasimN (4 0782)Utilization rate rutisimTra (08 085 094 1)Production rate rprosimTri (001 002 003)Decline rate RdelsimTri (015 02 022)Discount rate4 isim001T(4) + 012ere are three possibilities of whether to proceed to the next stage Yesmeans immediate development No means immediate abandonment andWait means waiting for opportunity 3Ps-price is the sales price of crude oilLn Ps-price refers to the logarithm of crude oil price and 4 is themean valueof the logarithm of Brent oil price during 1999-2019 08 is the reversionspeed of the mean value the larger the value is the faster the logarithm ofprice approaches the long-term mean value 002 is the logarithm volatilityobtained by calculating the logarithms of Brent oil price in recent 20 yearsdzt refers to the standard Brownian motion dztsim (0 dt) 4e design ofdiscount rate distribution is based on the discount rate currently used byCNPC data from Evaluation parameters of investment projects of CNPC(2019)

Mathematical Problems in Engineering 9

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

method for comprehensive valuation of oil and gas assetsinvolving grading hydrocarbon resources in accordancewith reserve quantities recovery difficulties oil and gasquality external risks etc is method is relatively accurateand moreover simple and can easily determine at whichgrade a certain oil and gas asset stays However compre-hensive grading based on this method is somewhat sub-jective and has difficulties in deciding which has highervalues a block with a smaller scale and yet better quality orthe one with secondary quality but expanded reservequantities Under such circumstances weights shall beassigned to each grading indicator by the decision makersand therefore this semiquantitative method is applicable tofuzzy comparison and preliminary screening out of multipleblocks

Valuation of overseas oil and gas extraction demandsmore precise quantitative research to support investmentdecision-making especially during the current downturn ofoil prices With respect to quantitative research themainstream method should be the discounted cash flow(DCF) method based on input and outpute DCFmethod[9] is widely used in the oil and gas industry and is able topresent relatively high precision in the case of oil and gasblocks with lower certainty Taking the time value of moneyinto consideration it precisely calculates the value of ex-traction by estimating the exploration and developmentinvestment operation expenditure and mortgages and taxesand predicting the sales revenue of crude

Nevertheless the estimation of investment cost andsales revenue in this method requires low uncertainty of theevaluated object otherwise the calculated NPV (net presentvalue) will not have credibility [10] Moreover the DCFmethod only considers the value of future cash flow andneglects the value of decision-making Clearly for overseasoil and gas extraction the value of decision-making isembodied as the ability to give up exercising rights whichmeans stopping exploitation in the case of money-losing oiland gas investment [11] is is a limitation of the DCFmethod

Some scholars introduce the option method to deal withuncertainty of overseas oil and gas extraction calculate thecomprehensive value and determine the investment timinge real options method [12 13] as an evaluation approachestimates the financial value of oil and gas extraction throughthe DCF process then calculates the option value using thevolatility ratio of the asset value and at last concludes thecomprehensive value e real option method is able to bettermimic the decision-making behavior and measure the value ofthe decision right by considering and calculating values ofoptions to defer investment and to abandon

Yet some issues still exist in terms of the basic as-sumption and actual implementation and have not been wellhandled Firstly it is hard to measure the volatility ratio ofthe value of extraction Transactions of oil and gas assets arecharacterized by their small quantity and discontinuationand thus the value volatility ratio cannot be directly cal-culated Generally the real option method [12 14 15] usesthe oil price fluctuation to represent the undulation of the oiland gas asset value between which the consistency has not

been confirmed yet In fact the value of oil and gas assets isnot only related to oil prices but also related to many otherfactors such as the level of risks associated with the re-sources the local political and economic status and lawsis is considered a major flaw of this method Secondlythere is no complete market for oil and gas assets like thatfor option transactions e excessively limited quantity ofbuyers and sellers decides that the transaction is not real-time and various options cannot be exercised in a timelyfashion Furthermore the transaction value of assets islargely determined through the game between the two sidesand the transaction value of options is hard to be estimatedIn addition some studies [16] reject two stylized facts of realoptions on oil one is that the correlation of the returns on oiland the stock market is positive the other is that it is in-variant to changes in oil price volatility ey state that thewidespread idea that higher volatility leads to increasedvalue and postponed investment is not necessarily valid

To sum up regarding valuation of overseas oil and gasextraction the qualitative method suffers from insufficientprecision the DCF method fails to capture the value ofdecision rights and the basic assumption of the real optionmethod is questionable erefore it is required to thinkabout it further to overcome these defects

3 Analyzing Uncertainties of Overseas Oil andGas Extraction

e value of the decision rights for overseas oil and assetsroots in uncertainty and thus we should first clarify in whichaspects uncertainty is embodied then analyze how theseuncertain factors impact the assessed value of assets and atlast characterize variations of these uncertainty factors

31 Identification ofUncertainty Factors Overseas oil and gasextraction are subject to various uncertainty factors of whichextensive identification investigation and subsequent riskquantification and asset valuation have been carried out Mostscholars [17ndash20] focus on uncertainty in the geology whichmainly include the reserves quality depth utilization rateproduction rate and decline rate of production Some scholars[21 22] also consider the external environment including thepolitical and economic environment sovereignty credit andglobal oil price Besides the geographic location topographicalsetting hydrogeological background and technical proficiencyof operators which are concerned with the discovery devel-opment construction and operating costs during hydrocarbonrecovery also have effects upon the value of assets In summarythe uncertainty factors can be classified into the followingcategories as shown in Table 1

Geological factors of resources refer to the geologicalconditions of oil and gas resources and their uncertainty ismainly caused by the error of measurement results of ex-ploration experimental wells ese uncertainty factors aremostly derived from the low level of exploration andfuzziness in geological data and will gradually decline withthe progressing geological understanding e uncertaintyfactors of production parameters mainly refer to the

Mathematical Problems in Engineering 3

parameters in the production process which are influencedby the geological conditions technical level of producersand the preference of decision makers and have certainsubjectivity External environmental factors mainly refer tothe political and economic environment of the target areawhich are objective macrofactors Other factors howeverdue to the relatively low level of uncertainty are not con-sidered in this research

32 Uncertainty Transmission Uncertainty of each factor isultimately transmitted onto the extraction value which isembodied as the uncertainty of the value of overseas oil andgas extraction e transmission mechanism of uncertaintyfactors is illustrated in Figure 2

e reserves utilization rate production rate anddecline rate impact the ultimately recovered saleable re-sources e larger the reserves the higher the utilizationratio the faster the production rate the slower the declinerate then the larger the saleable resources However theincrease of production rate will also generally lead to theincrease of decline rate e production rate and declinerate together affect the production curve of the whole lifecycle which further results in influence upon the time valueof capital Even if it has the same total production the netpresent value of different production curves is different Itis obvious that earlier recovery of hydrocarbons is favorableto amortizing the capital and repaying the loan e re-source quality and the global oil price have effects upon thefinal sales price and heavy oil and light oil are seen withdifferent global prices Generally quality compensation isused to represent the quality of resources e worse theresource quality is the higher the amount of compensationis needed and the lower the value of extraction is Fur-thermore the burial depth of resources is one of the de-cisive factors for drilling cost Higher capitalized costscompromise the value of extraction

e above work clarifies the route of each uncertaintyfactor to influence the value of oil and gas extraction and yethow much the influence is cannot be calculated through thepresented figure us we need to further investigate theefficiency of such a transmissionmechanisme value of oiland gas extraction is dependent on the cash flow generatedby future oil and gas production and the cash flow com-ponent is shown in Figure 3

e roles played by uncertainty factors in terms of cash flowgeneration are illustrated in Figure 3 For cash outflow theamount of the predicted resources is related to the explorationarea which will determine the exploration investment andreserves discovery cost is generally used to quantify the in-vestment e initial annual production will affect the facilities

construction investment and the total output will affect variousCAPEXs (capital expenditure) and OPEXs (operating expense)and then indirectly affect the expenditure of taxes of course allexpenditures are influenced by the investment environment taxlaws and local communities such as the level of local priceswhich will affect the workersrsquo wage For cash inflows global oilprices and the resource quality will affect sales prices whileproduction and sales prices determine the sales revenue Effortscould bemade to formulize these routes of cash flow generationwhich is in accordance with financial appraisal of oil and gasproduction

Many studies on oil and gas investment appraisal [19 23]have discussed those formulas and here a complete for-mulized description of cash inflow and outflow for oil andgas development is presented

In terms of exploration investment it is related topredicted recoverable reserves and also the discovery costper barrel oil erefore it can be expressed as

Iexp Iexpbbl times Rrec Rrec Rpre times rrec rrec 1113944

n

t1rpro 1 minus rdel( 1113857

tminus 1

(1)

where Iexp is the exploration investment Iexpbbl is the discoverycost per barrel oil Rrec are the predicted recoverable reservesRpre are the predicted development pending resources rrec is therecovery factor rpro stands for the production rate rdel repre-sents the decline rate n is the estimated recovery lifecycle

As for development investment it is primarily depen-dent on the predicted reserves and utilization and pro-duction rates and can be calculated as follows

Idev Idevbbl times Ptotal

Ptotal Rpre times ruti times rrec(2)

where Idev is the development investment including in-vestment on drilling fracturing and completion Idevbbl isthe development investment per barrel oil Ptotal is the totalproduction ruti is the utilization rate

en for construction investment it mainly involvesconstruction of surface oil and gas processing facilitieswhich is correlated to the maximum annual hydrocarbonoutput instead of total production Higher annual pro-duction results in higher construction investment which canbe computed using the following equation

Icon Iconbbl times Pini

Pini Rpre times ruti times rpro(3)

where Icon is the construction investment Iconbbl is theconstruction investment per barrel oil Pini stands for theinitial annual production

Table 1 Uncertainty factors of overseas oil and gas extraction

Categories Uncertainty factorsResource geology Oil and gas resource quantity quality depthProduction parameters Rate of utilization production declineExternal environment Politics economics sovereignty credit global oil priceOthers Geography topography hydrology operational proficiency

4 Mathematical Problems in Engineering

e sales revenue is determined by the global oil pricequality of produced oil and gas and produced amount of oiland gas Calculation can be done using the following equation

Rsal Psminusprice times Ptotal

Psminusprice Pgminusprice + iqua(4)

where Rsal is the sales revenue Ps-price is the sale price Pg-price isthe global oil price iqua is the price variation dependent onquality of hydrocarbon resources which may be negative

Calculating the operating cost and tax is relatively simpleand can be shown via the following equation

Ttax Ttaxbbl times Ptotal

Copex Copexbbl times Ptotal(5)

where Ttax stands for taxesCopex is the operating cost Ttaxbblis the average tax per barrel oil Copexbbl is the operating costper barrel oil

To sum up the composition of future cash flow of oil andgas assets development is shown in Table 2

33 Distribution Patterns of Uncertainty Factors Havingformulized the path of influence of each uncertainty factor

for future case flow of oil and gas extraction and clarified thetransmission mechanism of the uncertainty factor for thevalue of extraction we are still facing undefined degrees ofuncertainty for each factor itself In other words investi-gation of the distribution pattern of each factor should becarried out eg volatility of oil prices

In reference to volatility of each uncertainty factor manyfactors have available literatures for references Regardingreserves extensive research in the petroleum engineeringindustry [24 25] suggests that the oil reserve is undoubtedlyfound with the logarithmic normal distribution instead ofthe normal distribution assumed by some researchers in the

Recovery factorPredicted resources Predicted recoverable reserves

Utilization resources

Production rate

Utilization rate

Initial production Decline Rate Total production

Exploration investment

Facilities construction investment Investment environment amp society

Development investment Deep

Operating cost

Global oil prices Sale pricequality

Government Local tax

Sale revenue

Cash

flow

Figure 3 Components of the future cash flow of overseas oil and gas extraction

Table 2 Cash inflow and outflow of oil and gas assets

Categories Cash flows Notation Workflow stage

Outflow

Exploration investment Iexp ExplorationDevelopmentinvestment Idev Development

Construction investment Icon DevelopmentOperating cost Copex Sale

Taxes Ttax SaleInflow Sales revenue Rsal Sale

DepthRecoveryrate

Declinerate Reserves Utilization

rate Quality Global oilprices

Production Sales priceTime value

Overseas oil amp gas value

Sovereign credit

Investmentenvironment

Risk discount rate Investment

Resource conditions

Uncertainty factors

Local conditions Economic condition

Figure 2 Transmission of uncertainty with respect to the value of overseas oil and gas extraction

Mathematical Problems in Engineering 5

real option field For the distribution of oil prices anagreement in understanding among extensive scholars hasnot been reached yet Some [15 26] believe that oil pricevariation should be a type of geometric Brownian motionwhile others [20 27ndash30] conduct oil price forecast using thesupport vector machine Bayesian model system simulationor a combination of multiple approaches with variouscorresponding results is paper tends to believe that oilprice complies with the Mean-Reversion with Jumps [26]which means that oil price is endowed with a mean-re-version nature and the mean oil price will gradually growwith time in case of no unexpected outburst events

In terms of quality and depth of resources both aredetermined according to the results of exploration experi-mental Wells e uncertainty of resource depth comes fromthe measurement error of experimental well depth and theuncertainty of resource quality comes from the measure-ment error of sulfur content and other indicators ereforethey can be considered as normal distribution and can beexpressed by quality compensation amount and drilling cost

Utilization rate production rate decline rate and otherfactors are to some extent subject to the subjective influence ofthe developer after consulting relevant experts we make thefollowing assumptions For the utilization rate it should bewithin [0 1] and we assume that it obeys the trapezoidaldistribution having probability within a certain subintervalmuch higher than the averagee production rate of resourcesis somewhat susceptible to subjectivity and meanwhile is alsoconstrained by geological conditions It should be within (0 1)with the existence of an optimal value and is therefore assumedto follow the triangular distribution e decline rate of hy-drocarbon recovery with a supposed range of (0 1) is related tothe production rate and also under the constraints of geologicalconditions Consequently it is also assumed to follow the tri-angular distributionWhen it comes to the discount rate of risksit is dependent on the local investment and financing envi-ronment sovereignty credit and politics Its distribution patternis still unclear For projects with low risks it may present theT-shaped distribution while for projects with higher risks itmay follow the normal distribution and we have not reached anagreement yet In addition in most cases the risk-free rate ofreturn is replaced with the long-term treasury bond rate (LTBR)of theUS and the risk-free rate plus the risk discount rate shouldbe the discount rate i used in calculating net present values

To sum up the distribution patterns of uncertaintyfactors are summarized in Table 3

4 AModified ApproachBased on the TrinomialTree Option Pricing Model

Upon accomplishment of identification of uncertainty fac-tors and investigation of transmission routes and distribu-tion patterns of probability we are able to calculate the netpresent value distribution on the basis of the establishedprobability density function and transmission route formulaof uncertainty factors e calculation is simple as is shownin equation (6) and the expectation value and variance of netpresent values can be obtained which is similar to theappraisal concept based on the DCF method Nonetheless

such practice still neglects the value of decision rights andthus we need to modify the trinomial tree option pricingmodel in a way inspired by the real options method

NPV 1113944n

t1Rsalminust minus Iexpminust minus Idevminust minus Iconminust minus Copexminust minus Ttaxminust1113872 1113873

times(1 + i)minus t

(6)

where NPV is the net present value of overseas oil and gasextraction i is the discount rate Rsal-t Iexp-t Idev-t Icon-t Copex-tand Ttax-t are the sales revenue exploration investment de-velopment investment construction investment operatingcost and taxes at the t-th year respectively (in case of noincome or expense under a specific term it should be zero)

41 Decision Points and Strategies during Appraisal (DeferredDevelopment Immediate Development and Sale of Assets)e real optionmethod can deal with the asset volatility ratiovia an approach combing the uncertainty factor and thetransmission route formula since the calculated net presentvalue presents itself as a distribution It should be noted thatthe resultant distribution does not necessarily follow thenormal or logarithmic normal distribution and thereforesome currently available option calculation models may beinapplicable e volatility ratio of the calculated distribu-tion does not solely depend on the oil price this singleuncertainty factor instead should be computed usingmultiple factors through the transmission route formulas

Another major disadvantage of the real option method isthat exercising rights cannot be done in a real-time manner Itis not like that one can immediately exercise the right at anymoment and there is no such thing as a simple switch forturning on and off lights to allow for immediate startupsuspension and termination of petroleum exploration anddevelopment For example the option to defer cannot beexercised in themiddle of drilling to instantaneously shut downthe development Oil and gas fields cannot be sold out duringexploration and development to exercise the option to aban-don Consequently the timing at which it is feasible to exercisethe option should be analyzed which is referred to as thedecision point in this paper At non-decision points optionscannot be exercised or partially exercised to defer or abandon

e general extraction workflow of oil and gas is illus-trated in Figure 4 At each decision point occurs a decision-making behavior which may have various strategy spacese corresponding decision space is concluded in Table 4

Here is a brief statement of the strategy space of eachdecision point listed in Table 4 For each major stage wehave three strategies namely starting investment deferringand waiting and abandoning investment right before ini-tiation of exploration development and sale In this regardthis paper is consistent with Tang et al [13] yet two ad-ditional intermediate decision points are considered in thispaper After accomplishment of regional exploration apreliminary appraisal is carried out before trap explorationwhich is consistent with the practice of oil companies Ifregional exploration presents favorable results exploration

6 Mathematical Problems in Engineering

goes on otherwise it will be abandoned Moreover afterfinishing the drilling engineering some oil and gas com-panies may decide not to perforate the payzone for the timebeing and wait for the right moment in accordance of theirown status and estimation of future oil price and supplytendencies It is based on this very fact that the decisionpoint is designed

In addition to the five main decision points mentionedabove in fact there are many possible accidental decisionpoints in the process of oil and gas extraction For examplelarge fluctuations of the oil price may delay or bring forwardthe exploration safety incidents may lead to the suspensionof someWells and oil and gas productionmay be suspendedfor political reasons However it is difficult to predictwhether these decision points will occur when they willoccur and how long a project may be suspended Since thispaper is only a method study there is no obvious differencein the application of the method whether it is 5 6 or moredecision points this model simplifies the actual situation andonly considers 5 main decision points that inevitably exist

42 Modification to the Trinomial Tree Option Pricing ModelOn the basis of the analysis on the decision-makingworkflow presented above it is found that investment onoverseas oil and gas extraction is characterized by limited

decision points and should be a type of Bermudan optionsto which the tree option model is applicable instead of theAmerican-style or European-style options

e conventional binomial modal develops the decision-making tree with respect to the probabilities of upward anddownward movements Magnitudes of upward and down-ward movements are dependent on the volatility ratio of thetotal asset erefore there are infinite decision points andthe resultant ultimate value of extraction follows theprobability distribution However decision points foroverseas oil and gas extraction are finite which means thatone is incapable of simulating the asset volatility throughinfinite decision points Moreover the NPV at each decisionpoint calculated using Table 3 and equations (1)ndash(6) presentsitself as a distribution Under such circumstances we are notable to plot and handle an N-ary tree with infinite upwardand downward points Given this some modification has tobe made upon the tree option pricing model e tree is notplotted in accordance with upward and downward move-ments instead it is developed in reference to the strategyspace Consequently we are able to calculate the probabilitydistribution of the pre-decision value of extraction in abackward manner as is shown in Figure 5

e probability tree-based method to estimate value stilladopts the concept of calculating the initial value of ex-traction in a backward manner although it is slightly dif-ferent with that based on the decision-making tree First thevalue at the last decision point ldquobefore productionrdquo iscomputed Upon accomplishment of exploration and de-velopment there are three available strategies namelyabandonment deferring and immediate production In thecase of the value at production higher than zero decisionmakers will choose immediate development in the case ofthe value at production lower than zero decisionmakers willchoose to abandon development and recover residual valuesor straightforward sale of assets with a production value ofabout zero decision makes will choose to defer developmentand wait for growing back of oil prices e specific boundlimit is dependent on preferences for utilities of investors

Table 3 Distribution patterns of uncertainty factors for overseas oil and gas extraction

Uncertainty factor Distribution pattern Source Range Additional remarksResource reserves Logarithmic normal distribution Reliable literature gt0 mdashSale price of oil Mean-reversion with jumps Available literature gt0 Mean value growing with timeResource depth Normal distribution mdash gt0 e measurement error is normally distributedResource quality Normal distribution mdash mdashUtilization rate Trapezoidal distribution Assumed [0 1] Consulting relevant expertsamp consider the realityRate of production Triangular distribution Assumed (0 1)Decline rate Triangular distribution Assumed (0 1)Discount rate T-shaped or normal distribution Assumed mdash Not in an agreement

Regional exploration Trap exploration Drilling engineering Completion engineering

Ground facility construction engineering

Production

Exploration Development Sale

Figure 4 Workflow of overseas oil and gas exploration and development

Table 4 Decision points and their strategy space for overseas oiland gas extraction

Stage Decision point Strategy space

ExplorationBefore regionalexploration Begin defer abandon

Before trap exploration Continue abandon

Development Before drilling Continue deferabandon

Before completion Continue defer

Sale Before production Continue deferabandon

Mathematical Problems in Engineering 7

e value at this decision point can be expressed using thefollowing equation

C5

V5 minus I5 V5 gt 0 V5 1113936n

tt5

Rsalminust minus Copexminust minus Ttaxminust1113872 1113873 times(1 + i)minus t

0 V5 asymp 0

Rrecminus5 minus I5 V5 lt 0 I5 1113936t5

tt4

Idev2minust + Icon2minust1113872 1113873 times(1 + i)minus t

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

where V5 is the present value of future earnings in case the oiland gas field is being developed I5 stands for the present valueof investment made in the previous stage which will becomethe sunk cost if the development is abandoned and the holdingasset if the development is deferred t5 is the time corre-sponding to the decision point ldquobefore productionrdquo C5 is thevalue at the decision point Rrecminus5 refers to the residual value ofprevious investment that can be recovered if the development isabandoned or the value that the asset can realize in case it issold out generally far less than I5 It should be noted that V5and I5 calculated in accordance with Table 3 are probabilitydistributions instead of specific numbers

Similarly value distributions at other decision points canbe computed as is shown in the following equation

C1

C2 C2 gt 0

0 C2 asymp 0

Rrecminus1 minus I1 C2 lt 0

⎧⎪⎨

⎪⎩

C2 C3 minus I2 C3 gt 0

Rrecminus2 minus I2 C3 le 01113896

C3

C4 minus I3 C4 gt 0

0 C4 asymp 0

Rrecminus3 minus I3 C4 lt 0

⎧⎪⎨

⎪⎩

C4 C5 minus I4 C5 gt 0

0 C5 le 01113896

I2 1113944

t1

t1Iexp1minust1113872 1113873 times(1 + i)

minus t

I3 1113944

t2

tt1

Iexp2minust1113872 1113873 times(1 + i)minus t

I4 1113944

t3

tt2

Idev1minust + Icon1minust1113872 1113873 times(1 + i)minus t

(8)

where I1 I2 I3 and I4 are present values of existing in-vestment at each decision point respectively Rrecminus1 Rrecminus2 andRrecminus3 are the investment residual values that can be paid back ifthe development is abandoned In most cases one will notabandon completion after finishing drilling and the decision islimited to either immediate or deferred completion

After obtaining distributions of the value at each deci-sion point through backward calculation based on theprobability tree we can further compute the expectedeconomic value with respect to the value distribution Forinstance at the first decision point that is before decision-making upon regional exploration the incurred cost is theacquisition cost of the oil and gas asset I1 If the developmentis implemented the obtained revenue is C2 In the case ofdeferred development the investment changes into the assetof which the value is assumed to be equal to the acquisitioncost Under the circumstance that the development isabandoned part of previous investment will be lost Giventhe aforementioned information the expected value at theldquobefore regional explorationrdquo decision point is shown in thefollowing equation

E C1( 1113857 1113946P10

0Rrecminus1 minus I1 + 1113946

P10+P12

P10

0+ 11139461

P10+P12

C2

P10 P C2 lt 0( 1113857

P12 P C2 asymp 0( 1113857

P11 P C2 gt 0( 1113857

(9)

e expectation of the value of extraction at otherintermediate decision points can be calculated in asimilar approach and thus detailed description is dis-carded here

Regional exploration Trap exploration Drilling engineering Completion engineering Production

P0

P10

P11

P12

P20

P21

P30

P31

P32

P41

P42

P52

P51

P50

Yes

No

Wait

Yes Yes

No

Yes

No

Yes

No

Wait Wait Wait

Figure 5 Probability tree of overseas oil and gas extraction (there are three possibilities of whether to proceed to the next stage Yes meansimmediate development No means immediate abandonment and Wait means waiting for opportunity)

8 Mathematical Problems in Engineering

43 Application of the Modified Method In this paper theconventional decision-based binomial model is replaced bythe probability-based one For purposes of understandinghow this model can be applied this paper takes an overseasoil and gas extraction project as an example to brieflydemonstrate the application of the modified method

Sudan X is an oil and gas block located in Africa thatCNPC plans to invest inis block is seen with no elaborategeological data and only part of parameters required forasset appraisal can be determined on the basis of analogousblocks Acquisition of the exploration and developmentinterest of this block costs three million USD and thediscovery cost per barrel oil Iexpbbl is 12 $bbl e con-struction investment per barrel oil is 50 $bbl while theoperating cost per barrel oil is 17 $bbl e composite taxrate is about 20 e estimated period of exploration lastsfor two years and production capacity construction needsthree years followed by a sustained recovery of 17 yearsOther factors are all uncertain with dispersed probabilitydistribution and high uncertainty (Table 5)

It is easy to compute the probability distribution ofannual production of each year on the basis of the distri-butions of the resource reserves utilization rate productionrate and decline rate equation (1) and the assumption thatproduction capacity establishment can be completely fin-ished within three yearse calculation process is illustratedin Figure 6

With respect to oil price volatility discount rate anddistribution of sale price variation induced resource qualitywe can compute the distribution of the present value ofearnings V5 at each future year using equation (7) Resultsare presented in Figure 7

It is seen that V5gt 0 is highly likely and thus at thedecision point ldquoproductionrdquo strategies of deferring orabandoning are seldom chosen is is consistent with ourcommon observation Once all investments of explorationand development are down oil and gas operators barelydecide to abandon production

With equations (1)ndash(3) and distributions of relevantparameters the probability distributions of I4 and I5 can becomputed which subsequently leads to the distributions ofC4 and C5

Figures 7 and 8 indicate high odds of the case of C4gt 0and C5gt 0 is means that once oil and gas resources arefound during exploration probability of the proceedingdevelopment is very high Given this the values of the lattertwo decision points are overridden which is inconsistentwith the view held in the conventional real option law thathigher uncertainty results in the higher asset value

Similarly the probability distribution of C3 can be de-termined with the help of I3 and C4 and is shown in Figure 9

Unlike other decision points the future value at thedecision point ldquotrap explorationrdquo presents negative valueswhich is decisive in whether or not to execute trap

exploration If the future cash value is below zero it iscertain that investment stops A recovery ratio of 20 isset for the total previous investment residual value that isRrecminus2 20 I A range of [minus20 I2 +20 I2] for C3 is alsoset to define the case in which cash value is about zerois range is man-made and in fact will not be usedgiven equation (8) since no deferring strategy exists at thetrap exploration decision point At last the probabilitydistribution of C2 can be computed on the basis of I2 andC3 (Figure 10)

From Figure 10 it is seen that the probability of C2 belowzero is significantly reduced with decision-making beforetrap exploration However negative values still exist becauseloss can be induced by volatility of uncertainty factors eventhough rational decision-making has been executed edecision-making behavior is able to only reduce such lossesand yet it is incapable of entirely avoiding such losses Fi-nally the probability distribution of C1 can be computedusing equation (8) as is shown in Figure 11

Figure 11 demonstrates that before initiation of regionalexploration thanks to the availability of a waiting strategythe calculated value distribution presents high probabilityfor the value of about zeroe resultant expectation value ofC1 is 1148 million USD

In the case of the DCF method with no consideration ofdecision rights the probability distribution of C1 calculateddirectly using equation (6) is plotted in Figure 12 e ex-pectation value of C1-DCF calculated in this way is 1058million USD

Table 5 Distribution of uncertainty factors in Block Sudan X

Uncertainty factor Distribution of affected parametersResource reserves Ln (Rpre)simN(9 0422)

Global oil price3 dLn Ps-price 08(4-Ln Ps-price)dt + 002dztdztsim(0 dt)

Resource depth IdevbblsimN (16 322)Resource quality iquasimN (4 0782)Utilization rate rutisimTra (08 085 094 1)Production rate rprosimTri (001 002 003)Decline rate RdelsimTri (015 02 022)Discount rate4 isim001T(4) + 012ere are three possibilities of whether to proceed to the next stage Yesmeans immediate development No means immediate abandonment andWait means waiting for opportunity 3Ps-price is the sales price of crude oilLn Ps-price refers to the logarithm of crude oil price and 4 is themean valueof the logarithm of Brent oil price during 1999-2019 08 is the reversionspeed of the mean value the larger the value is the faster the logarithm ofprice approaches the long-term mean value 002 is the logarithm volatilityobtained by calculating the logarithms of Brent oil price in recent 20 yearsdzt refers to the standard Brownian motion dztsim (0 dt) 4e design ofdiscount rate distribution is based on the discount rate currently used byCNPC data from Evaluation parameters of investment projects of CNPC(2019)

Mathematical Problems in Engineering 9

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

parameters in the production process which are influencedby the geological conditions technical level of producersand the preference of decision makers and have certainsubjectivity External environmental factors mainly refer tothe political and economic environment of the target areawhich are objective macrofactors Other factors howeverdue to the relatively low level of uncertainty are not con-sidered in this research

32 Uncertainty Transmission Uncertainty of each factor isultimately transmitted onto the extraction value which isembodied as the uncertainty of the value of overseas oil andgas extraction e transmission mechanism of uncertaintyfactors is illustrated in Figure 2

e reserves utilization rate production rate anddecline rate impact the ultimately recovered saleable re-sources e larger the reserves the higher the utilizationratio the faster the production rate the slower the declinerate then the larger the saleable resources However theincrease of production rate will also generally lead to theincrease of decline rate e production rate and declinerate together affect the production curve of the whole lifecycle which further results in influence upon the time valueof capital Even if it has the same total production the netpresent value of different production curves is different Itis obvious that earlier recovery of hydrocarbons is favorableto amortizing the capital and repaying the loan e re-source quality and the global oil price have effects upon thefinal sales price and heavy oil and light oil are seen withdifferent global prices Generally quality compensation isused to represent the quality of resources e worse theresource quality is the higher the amount of compensationis needed and the lower the value of extraction is Fur-thermore the burial depth of resources is one of the de-cisive factors for drilling cost Higher capitalized costscompromise the value of extraction

e above work clarifies the route of each uncertaintyfactor to influence the value of oil and gas extraction and yethow much the influence is cannot be calculated through thepresented figure us we need to further investigate theefficiency of such a transmissionmechanisme value of oiland gas extraction is dependent on the cash flow generatedby future oil and gas production and the cash flow com-ponent is shown in Figure 3

e roles played by uncertainty factors in terms of cash flowgeneration are illustrated in Figure 3 For cash outflow theamount of the predicted resources is related to the explorationarea which will determine the exploration investment andreserves discovery cost is generally used to quantify the in-vestment e initial annual production will affect the facilities

construction investment and the total output will affect variousCAPEXs (capital expenditure) and OPEXs (operating expense)and then indirectly affect the expenditure of taxes of course allexpenditures are influenced by the investment environment taxlaws and local communities such as the level of local priceswhich will affect the workersrsquo wage For cash inflows global oilprices and the resource quality will affect sales prices whileproduction and sales prices determine the sales revenue Effortscould bemade to formulize these routes of cash flow generationwhich is in accordance with financial appraisal of oil and gasproduction

Many studies on oil and gas investment appraisal [19 23]have discussed those formulas and here a complete for-mulized description of cash inflow and outflow for oil andgas development is presented

In terms of exploration investment it is related topredicted recoverable reserves and also the discovery costper barrel oil erefore it can be expressed as

Iexp Iexpbbl times Rrec Rrec Rpre times rrec rrec 1113944

n

t1rpro 1 minus rdel( 1113857

tminus 1

(1)

where Iexp is the exploration investment Iexpbbl is the discoverycost per barrel oil Rrec are the predicted recoverable reservesRpre are the predicted development pending resources rrec is therecovery factor rpro stands for the production rate rdel repre-sents the decline rate n is the estimated recovery lifecycle

As for development investment it is primarily depen-dent on the predicted reserves and utilization and pro-duction rates and can be calculated as follows

Idev Idevbbl times Ptotal

Ptotal Rpre times ruti times rrec(2)

where Idev is the development investment including in-vestment on drilling fracturing and completion Idevbbl isthe development investment per barrel oil Ptotal is the totalproduction ruti is the utilization rate

en for construction investment it mainly involvesconstruction of surface oil and gas processing facilitieswhich is correlated to the maximum annual hydrocarbonoutput instead of total production Higher annual pro-duction results in higher construction investment which canbe computed using the following equation

Icon Iconbbl times Pini

Pini Rpre times ruti times rpro(3)

where Icon is the construction investment Iconbbl is theconstruction investment per barrel oil Pini stands for theinitial annual production

Table 1 Uncertainty factors of overseas oil and gas extraction

Categories Uncertainty factorsResource geology Oil and gas resource quantity quality depthProduction parameters Rate of utilization production declineExternal environment Politics economics sovereignty credit global oil priceOthers Geography topography hydrology operational proficiency

4 Mathematical Problems in Engineering

e sales revenue is determined by the global oil pricequality of produced oil and gas and produced amount of oiland gas Calculation can be done using the following equation

Rsal Psminusprice times Ptotal

Psminusprice Pgminusprice + iqua(4)

where Rsal is the sales revenue Ps-price is the sale price Pg-price isthe global oil price iqua is the price variation dependent onquality of hydrocarbon resources which may be negative

Calculating the operating cost and tax is relatively simpleand can be shown via the following equation

Ttax Ttaxbbl times Ptotal

Copex Copexbbl times Ptotal(5)

where Ttax stands for taxesCopex is the operating cost Ttaxbblis the average tax per barrel oil Copexbbl is the operating costper barrel oil

To sum up the composition of future cash flow of oil andgas assets development is shown in Table 2

33 Distribution Patterns of Uncertainty Factors Havingformulized the path of influence of each uncertainty factor

for future case flow of oil and gas extraction and clarified thetransmission mechanism of the uncertainty factor for thevalue of extraction we are still facing undefined degrees ofuncertainty for each factor itself In other words investi-gation of the distribution pattern of each factor should becarried out eg volatility of oil prices

In reference to volatility of each uncertainty factor manyfactors have available literatures for references Regardingreserves extensive research in the petroleum engineeringindustry [24 25] suggests that the oil reserve is undoubtedlyfound with the logarithmic normal distribution instead ofthe normal distribution assumed by some researchers in the

Recovery factorPredicted resources Predicted recoverable reserves

Utilization resources

Production rate

Utilization rate

Initial production Decline Rate Total production

Exploration investment

Facilities construction investment Investment environment amp society

Development investment Deep

Operating cost

Global oil prices Sale pricequality

Government Local tax

Sale revenue

Cash

flow

Figure 3 Components of the future cash flow of overseas oil and gas extraction

Table 2 Cash inflow and outflow of oil and gas assets

Categories Cash flows Notation Workflow stage

Outflow

Exploration investment Iexp ExplorationDevelopmentinvestment Idev Development

Construction investment Icon DevelopmentOperating cost Copex Sale

Taxes Ttax SaleInflow Sales revenue Rsal Sale

DepthRecoveryrate

Declinerate Reserves Utilization

rate Quality Global oilprices

Production Sales priceTime value

Overseas oil amp gas value

Sovereign credit

Investmentenvironment

Risk discount rate Investment

Resource conditions

Uncertainty factors

Local conditions Economic condition

Figure 2 Transmission of uncertainty with respect to the value of overseas oil and gas extraction

Mathematical Problems in Engineering 5

real option field For the distribution of oil prices anagreement in understanding among extensive scholars hasnot been reached yet Some [15 26] believe that oil pricevariation should be a type of geometric Brownian motionwhile others [20 27ndash30] conduct oil price forecast using thesupport vector machine Bayesian model system simulationor a combination of multiple approaches with variouscorresponding results is paper tends to believe that oilprice complies with the Mean-Reversion with Jumps [26]which means that oil price is endowed with a mean-re-version nature and the mean oil price will gradually growwith time in case of no unexpected outburst events

In terms of quality and depth of resources both aredetermined according to the results of exploration experi-mental Wells e uncertainty of resource depth comes fromthe measurement error of experimental well depth and theuncertainty of resource quality comes from the measure-ment error of sulfur content and other indicators ereforethey can be considered as normal distribution and can beexpressed by quality compensation amount and drilling cost

Utilization rate production rate decline rate and otherfactors are to some extent subject to the subjective influence ofthe developer after consulting relevant experts we make thefollowing assumptions For the utilization rate it should bewithin [0 1] and we assume that it obeys the trapezoidaldistribution having probability within a certain subintervalmuch higher than the averagee production rate of resourcesis somewhat susceptible to subjectivity and meanwhile is alsoconstrained by geological conditions It should be within (0 1)with the existence of an optimal value and is therefore assumedto follow the triangular distribution e decline rate of hy-drocarbon recovery with a supposed range of (0 1) is related tothe production rate and also under the constraints of geologicalconditions Consequently it is also assumed to follow the tri-angular distributionWhen it comes to the discount rate of risksit is dependent on the local investment and financing envi-ronment sovereignty credit and politics Its distribution patternis still unclear For projects with low risks it may present theT-shaped distribution while for projects with higher risks itmay follow the normal distribution and we have not reached anagreement yet In addition in most cases the risk-free rate ofreturn is replaced with the long-term treasury bond rate (LTBR)of theUS and the risk-free rate plus the risk discount rate shouldbe the discount rate i used in calculating net present values

To sum up the distribution patterns of uncertaintyfactors are summarized in Table 3

4 AModified ApproachBased on the TrinomialTree Option Pricing Model

Upon accomplishment of identification of uncertainty fac-tors and investigation of transmission routes and distribu-tion patterns of probability we are able to calculate the netpresent value distribution on the basis of the establishedprobability density function and transmission route formulaof uncertainty factors e calculation is simple as is shownin equation (6) and the expectation value and variance of netpresent values can be obtained which is similar to theappraisal concept based on the DCF method Nonetheless

such practice still neglects the value of decision rights andthus we need to modify the trinomial tree option pricingmodel in a way inspired by the real options method

NPV 1113944n

t1Rsalminust minus Iexpminust minus Idevminust minus Iconminust minus Copexminust minus Ttaxminust1113872 1113873

times(1 + i)minus t

(6)

where NPV is the net present value of overseas oil and gasextraction i is the discount rate Rsal-t Iexp-t Idev-t Icon-t Copex-tand Ttax-t are the sales revenue exploration investment de-velopment investment construction investment operatingcost and taxes at the t-th year respectively (in case of noincome or expense under a specific term it should be zero)

41 Decision Points and Strategies during Appraisal (DeferredDevelopment Immediate Development and Sale of Assets)e real optionmethod can deal with the asset volatility ratiovia an approach combing the uncertainty factor and thetransmission route formula since the calculated net presentvalue presents itself as a distribution It should be noted thatthe resultant distribution does not necessarily follow thenormal or logarithmic normal distribution and thereforesome currently available option calculation models may beinapplicable e volatility ratio of the calculated distribu-tion does not solely depend on the oil price this singleuncertainty factor instead should be computed usingmultiple factors through the transmission route formulas

Another major disadvantage of the real option method isthat exercising rights cannot be done in a real-time manner Itis not like that one can immediately exercise the right at anymoment and there is no such thing as a simple switch forturning on and off lights to allow for immediate startupsuspension and termination of petroleum exploration anddevelopment For example the option to defer cannot beexercised in themiddle of drilling to instantaneously shut downthe development Oil and gas fields cannot be sold out duringexploration and development to exercise the option to aban-don Consequently the timing at which it is feasible to exercisethe option should be analyzed which is referred to as thedecision point in this paper At non-decision points optionscannot be exercised or partially exercised to defer or abandon

e general extraction workflow of oil and gas is illus-trated in Figure 4 At each decision point occurs a decision-making behavior which may have various strategy spacese corresponding decision space is concluded in Table 4

Here is a brief statement of the strategy space of eachdecision point listed in Table 4 For each major stage wehave three strategies namely starting investment deferringand waiting and abandoning investment right before ini-tiation of exploration development and sale In this regardthis paper is consistent with Tang et al [13] yet two ad-ditional intermediate decision points are considered in thispaper After accomplishment of regional exploration apreliminary appraisal is carried out before trap explorationwhich is consistent with the practice of oil companies Ifregional exploration presents favorable results exploration

6 Mathematical Problems in Engineering

goes on otherwise it will be abandoned Moreover afterfinishing the drilling engineering some oil and gas com-panies may decide not to perforate the payzone for the timebeing and wait for the right moment in accordance of theirown status and estimation of future oil price and supplytendencies It is based on this very fact that the decisionpoint is designed

In addition to the five main decision points mentionedabove in fact there are many possible accidental decisionpoints in the process of oil and gas extraction For examplelarge fluctuations of the oil price may delay or bring forwardthe exploration safety incidents may lead to the suspensionof someWells and oil and gas productionmay be suspendedfor political reasons However it is difficult to predictwhether these decision points will occur when they willoccur and how long a project may be suspended Since thispaper is only a method study there is no obvious differencein the application of the method whether it is 5 6 or moredecision points this model simplifies the actual situation andonly considers 5 main decision points that inevitably exist

42 Modification to the Trinomial Tree Option Pricing ModelOn the basis of the analysis on the decision-makingworkflow presented above it is found that investment onoverseas oil and gas extraction is characterized by limited

decision points and should be a type of Bermudan optionsto which the tree option model is applicable instead of theAmerican-style or European-style options

e conventional binomial modal develops the decision-making tree with respect to the probabilities of upward anddownward movements Magnitudes of upward and down-ward movements are dependent on the volatility ratio of thetotal asset erefore there are infinite decision points andthe resultant ultimate value of extraction follows theprobability distribution However decision points foroverseas oil and gas extraction are finite which means thatone is incapable of simulating the asset volatility throughinfinite decision points Moreover the NPV at each decisionpoint calculated using Table 3 and equations (1)ndash(6) presentsitself as a distribution Under such circumstances we are notable to plot and handle an N-ary tree with infinite upwardand downward points Given this some modification has tobe made upon the tree option pricing model e tree is notplotted in accordance with upward and downward move-ments instead it is developed in reference to the strategyspace Consequently we are able to calculate the probabilitydistribution of the pre-decision value of extraction in abackward manner as is shown in Figure 5

e probability tree-based method to estimate value stilladopts the concept of calculating the initial value of ex-traction in a backward manner although it is slightly dif-ferent with that based on the decision-making tree First thevalue at the last decision point ldquobefore productionrdquo iscomputed Upon accomplishment of exploration and de-velopment there are three available strategies namelyabandonment deferring and immediate production In thecase of the value at production higher than zero decisionmakers will choose immediate development in the case ofthe value at production lower than zero decisionmakers willchoose to abandon development and recover residual valuesor straightforward sale of assets with a production value ofabout zero decision makes will choose to defer developmentand wait for growing back of oil prices e specific boundlimit is dependent on preferences for utilities of investors

Table 3 Distribution patterns of uncertainty factors for overseas oil and gas extraction

Uncertainty factor Distribution pattern Source Range Additional remarksResource reserves Logarithmic normal distribution Reliable literature gt0 mdashSale price of oil Mean-reversion with jumps Available literature gt0 Mean value growing with timeResource depth Normal distribution mdash gt0 e measurement error is normally distributedResource quality Normal distribution mdash mdashUtilization rate Trapezoidal distribution Assumed [0 1] Consulting relevant expertsamp consider the realityRate of production Triangular distribution Assumed (0 1)Decline rate Triangular distribution Assumed (0 1)Discount rate T-shaped or normal distribution Assumed mdash Not in an agreement

Regional exploration Trap exploration Drilling engineering Completion engineering

Ground facility construction engineering

Production

Exploration Development Sale

Figure 4 Workflow of overseas oil and gas exploration and development

Table 4 Decision points and their strategy space for overseas oiland gas extraction

Stage Decision point Strategy space

ExplorationBefore regionalexploration Begin defer abandon

Before trap exploration Continue abandon

Development Before drilling Continue deferabandon

Before completion Continue defer

Sale Before production Continue deferabandon

Mathematical Problems in Engineering 7

e value at this decision point can be expressed using thefollowing equation

C5

V5 minus I5 V5 gt 0 V5 1113936n

tt5

Rsalminust minus Copexminust minus Ttaxminust1113872 1113873 times(1 + i)minus t

0 V5 asymp 0

Rrecminus5 minus I5 V5 lt 0 I5 1113936t5

tt4

Idev2minust + Icon2minust1113872 1113873 times(1 + i)minus t

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

where V5 is the present value of future earnings in case the oiland gas field is being developed I5 stands for the present valueof investment made in the previous stage which will becomethe sunk cost if the development is abandoned and the holdingasset if the development is deferred t5 is the time corre-sponding to the decision point ldquobefore productionrdquo C5 is thevalue at the decision point Rrecminus5 refers to the residual value ofprevious investment that can be recovered if the development isabandoned or the value that the asset can realize in case it issold out generally far less than I5 It should be noted that V5and I5 calculated in accordance with Table 3 are probabilitydistributions instead of specific numbers

Similarly value distributions at other decision points canbe computed as is shown in the following equation

C1

C2 C2 gt 0

0 C2 asymp 0

Rrecminus1 minus I1 C2 lt 0

⎧⎪⎨

⎪⎩

C2 C3 minus I2 C3 gt 0

Rrecminus2 minus I2 C3 le 01113896

C3

C4 minus I3 C4 gt 0

0 C4 asymp 0

Rrecminus3 minus I3 C4 lt 0

⎧⎪⎨

⎪⎩

C4 C5 minus I4 C5 gt 0

0 C5 le 01113896

I2 1113944

t1

t1Iexp1minust1113872 1113873 times(1 + i)

minus t

I3 1113944

t2

tt1

Iexp2minust1113872 1113873 times(1 + i)minus t

I4 1113944

t3

tt2

Idev1minust + Icon1minust1113872 1113873 times(1 + i)minus t

(8)

where I1 I2 I3 and I4 are present values of existing in-vestment at each decision point respectively Rrecminus1 Rrecminus2 andRrecminus3 are the investment residual values that can be paid back ifthe development is abandoned In most cases one will notabandon completion after finishing drilling and the decision islimited to either immediate or deferred completion

After obtaining distributions of the value at each deci-sion point through backward calculation based on theprobability tree we can further compute the expectedeconomic value with respect to the value distribution Forinstance at the first decision point that is before decision-making upon regional exploration the incurred cost is theacquisition cost of the oil and gas asset I1 If the developmentis implemented the obtained revenue is C2 In the case ofdeferred development the investment changes into the assetof which the value is assumed to be equal to the acquisitioncost Under the circumstance that the development isabandoned part of previous investment will be lost Giventhe aforementioned information the expected value at theldquobefore regional explorationrdquo decision point is shown in thefollowing equation

E C1( 1113857 1113946P10

0Rrecminus1 minus I1 + 1113946

P10+P12

P10

0+ 11139461

P10+P12

C2

P10 P C2 lt 0( 1113857

P12 P C2 asymp 0( 1113857

P11 P C2 gt 0( 1113857

(9)

e expectation of the value of extraction at otherintermediate decision points can be calculated in asimilar approach and thus detailed description is dis-carded here

Regional exploration Trap exploration Drilling engineering Completion engineering Production

P0

P10

P11

P12

P20

P21

P30

P31

P32

P41

P42

P52

P51

P50

Yes

No

Wait

Yes Yes

No

Yes

No

Yes

No

Wait Wait Wait

Figure 5 Probability tree of overseas oil and gas extraction (there are three possibilities of whether to proceed to the next stage Yes meansimmediate development No means immediate abandonment and Wait means waiting for opportunity)

8 Mathematical Problems in Engineering

43 Application of the Modified Method In this paper theconventional decision-based binomial model is replaced bythe probability-based one For purposes of understandinghow this model can be applied this paper takes an overseasoil and gas extraction project as an example to brieflydemonstrate the application of the modified method

Sudan X is an oil and gas block located in Africa thatCNPC plans to invest inis block is seen with no elaborategeological data and only part of parameters required forasset appraisal can be determined on the basis of analogousblocks Acquisition of the exploration and developmentinterest of this block costs three million USD and thediscovery cost per barrel oil Iexpbbl is 12 $bbl e con-struction investment per barrel oil is 50 $bbl while theoperating cost per barrel oil is 17 $bbl e composite taxrate is about 20 e estimated period of exploration lastsfor two years and production capacity construction needsthree years followed by a sustained recovery of 17 yearsOther factors are all uncertain with dispersed probabilitydistribution and high uncertainty (Table 5)

It is easy to compute the probability distribution ofannual production of each year on the basis of the distri-butions of the resource reserves utilization rate productionrate and decline rate equation (1) and the assumption thatproduction capacity establishment can be completely fin-ished within three yearse calculation process is illustratedin Figure 6

With respect to oil price volatility discount rate anddistribution of sale price variation induced resource qualitywe can compute the distribution of the present value ofearnings V5 at each future year using equation (7) Resultsare presented in Figure 7

It is seen that V5gt 0 is highly likely and thus at thedecision point ldquoproductionrdquo strategies of deferring orabandoning are seldom chosen is is consistent with ourcommon observation Once all investments of explorationand development are down oil and gas operators barelydecide to abandon production

With equations (1)ndash(3) and distributions of relevantparameters the probability distributions of I4 and I5 can becomputed which subsequently leads to the distributions ofC4 and C5

Figures 7 and 8 indicate high odds of the case of C4gt 0and C5gt 0 is means that once oil and gas resources arefound during exploration probability of the proceedingdevelopment is very high Given this the values of the lattertwo decision points are overridden which is inconsistentwith the view held in the conventional real option law thathigher uncertainty results in the higher asset value

Similarly the probability distribution of C3 can be de-termined with the help of I3 and C4 and is shown in Figure 9

Unlike other decision points the future value at thedecision point ldquotrap explorationrdquo presents negative valueswhich is decisive in whether or not to execute trap

exploration If the future cash value is below zero it iscertain that investment stops A recovery ratio of 20 isset for the total previous investment residual value that isRrecminus2 20 I A range of [minus20 I2 +20 I2] for C3 is alsoset to define the case in which cash value is about zerois range is man-made and in fact will not be usedgiven equation (8) since no deferring strategy exists at thetrap exploration decision point At last the probabilitydistribution of C2 can be computed on the basis of I2 andC3 (Figure 10)

From Figure 10 it is seen that the probability of C2 belowzero is significantly reduced with decision-making beforetrap exploration However negative values still exist becauseloss can be induced by volatility of uncertainty factors eventhough rational decision-making has been executed edecision-making behavior is able to only reduce such lossesand yet it is incapable of entirely avoiding such losses Fi-nally the probability distribution of C1 can be computedusing equation (8) as is shown in Figure 11

Figure 11 demonstrates that before initiation of regionalexploration thanks to the availability of a waiting strategythe calculated value distribution presents high probabilityfor the value of about zeroe resultant expectation value ofC1 is 1148 million USD

In the case of the DCF method with no consideration ofdecision rights the probability distribution of C1 calculateddirectly using equation (6) is plotted in Figure 12 e ex-pectation value of C1-DCF calculated in this way is 1058million USD

Table 5 Distribution of uncertainty factors in Block Sudan X

Uncertainty factor Distribution of affected parametersResource reserves Ln (Rpre)simN(9 0422)

Global oil price3 dLn Ps-price 08(4-Ln Ps-price)dt + 002dztdztsim(0 dt)

Resource depth IdevbblsimN (16 322)Resource quality iquasimN (4 0782)Utilization rate rutisimTra (08 085 094 1)Production rate rprosimTri (001 002 003)Decline rate RdelsimTri (015 02 022)Discount rate4 isim001T(4) + 012ere are three possibilities of whether to proceed to the next stage Yesmeans immediate development No means immediate abandonment andWait means waiting for opportunity 3Ps-price is the sales price of crude oilLn Ps-price refers to the logarithm of crude oil price and 4 is themean valueof the logarithm of Brent oil price during 1999-2019 08 is the reversionspeed of the mean value the larger the value is the faster the logarithm ofprice approaches the long-term mean value 002 is the logarithm volatilityobtained by calculating the logarithms of Brent oil price in recent 20 yearsdzt refers to the standard Brownian motion dztsim (0 dt) 4e design ofdiscount rate distribution is based on the discount rate currently used byCNPC data from Evaluation parameters of investment projects of CNPC(2019)

Mathematical Problems in Engineering 9

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

e sales revenue is determined by the global oil pricequality of produced oil and gas and produced amount of oiland gas Calculation can be done using the following equation

Rsal Psminusprice times Ptotal

Psminusprice Pgminusprice + iqua(4)

where Rsal is the sales revenue Ps-price is the sale price Pg-price isthe global oil price iqua is the price variation dependent onquality of hydrocarbon resources which may be negative

Calculating the operating cost and tax is relatively simpleand can be shown via the following equation

Ttax Ttaxbbl times Ptotal

Copex Copexbbl times Ptotal(5)

where Ttax stands for taxesCopex is the operating cost Ttaxbblis the average tax per barrel oil Copexbbl is the operating costper barrel oil

To sum up the composition of future cash flow of oil andgas assets development is shown in Table 2

33 Distribution Patterns of Uncertainty Factors Havingformulized the path of influence of each uncertainty factor

for future case flow of oil and gas extraction and clarified thetransmission mechanism of the uncertainty factor for thevalue of extraction we are still facing undefined degrees ofuncertainty for each factor itself In other words investi-gation of the distribution pattern of each factor should becarried out eg volatility of oil prices

In reference to volatility of each uncertainty factor manyfactors have available literatures for references Regardingreserves extensive research in the petroleum engineeringindustry [24 25] suggests that the oil reserve is undoubtedlyfound with the logarithmic normal distribution instead ofthe normal distribution assumed by some researchers in the

Recovery factorPredicted resources Predicted recoverable reserves

Utilization resources

Production rate

Utilization rate

Initial production Decline Rate Total production

Exploration investment

Facilities construction investment Investment environment amp society

Development investment Deep

Operating cost

Global oil prices Sale pricequality

Government Local tax

Sale revenue

Cash

flow

Figure 3 Components of the future cash flow of overseas oil and gas extraction

Table 2 Cash inflow and outflow of oil and gas assets

Categories Cash flows Notation Workflow stage

Outflow

Exploration investment Iexp ExplorationDevelopmentinvestment Idev Development

Construction investment Icon DevelopmentOperating cost Copex Sale

Taxes Ttax SaleInflow Sales revenue Rsal Sale

DepthRecoveryrate

Declinerate Reserves Utilization

rate Quality Global oilprices

Production Sales priceTime value

Overseas oil amp gas value

Sovereign credit

Investmentenvironment

Risk discount rate Investment

Resource conditions

Uncertainty factors

Local conditions Economic condition

Figure 2 Transmission of uncertainty with respect to the value of overseas oil and gas extraction

Mathematical Problems in Engineering 5

real option field For the distribution of oil prices anagreement in understanding among extensive scholars hasnot been reached yet Some [15 26] believe that oil pricevariation should be a type of geometric Brownian motionwhile others [20 27ndash30] conduct oil price forecast using thesupport vector machine Bayesian model system simulationor a combination of multiple approaches with variouscorresponding results is paper tends to believe that oilprice complies with the Mean-Reversion with Jumps [26]which means that oil price is endowed with a mean-re-version nature and the mean oil price will gradually growwith time in case of no unexpected outburst events

In terms of quality and depth of resources both aredetermined according to the results of exploration experi-mental Wells e uncertainty of resource depth comes fromthe measurement error of experimental well depth and theuncertainty of resource quality comes from the measure-ment error of sulfur content and other indicators ereforethey can be considered as normal distribution and can beexpressed by quality compensation amount and drilling cost

Utilization rate production rate decline rate and otherfactors are to some extent subject to the subjective influence ofthe developer after consulting relevant experts we make thefollowing assumptions For the utilization rate it should bewithin [0 1] and we assume that it obeys the trapezoidaldistribution having probability within a certain subintervalmuch higher than the averagee production rate of resourcesis somewhat susceptible to subjectivity and meanwhile is alsoconstrained by geological conditions It should be within (0 1)with the existence of an optimal value and is therefore assumedto follow the triangular distribution e decline rate of hy-drocarbon recovery with a supposed range of (0 1) is related tothe production rate and also under the constraints of geologicalconditions Consequently it is also assumed to follow the tri-angular distributionWhen it comes to the discount rate of risksit is dependent on the local investment and financing envi-ronment sovereignty credit and politics Its distribution patternis still unclear For projects with low risks it may present theT-shaped distribution while for projects with higher risks itmay follow the normal distribution and we have not reached anagreement yet In addition in most cases the risk-free rate ofreturn is replaced with the long-term treasury bond rate (LTBR)of theUS and the risk-free rate plus the risk discount rate shouldbe the discount rate i used in calculating net present values

To sum up the distribution patterns of uncertaintyfactors are summarized in Table 3

4 AModified ApproachBased on the TrinomialTree Option Pricing Model

Upon accomplishment of identification of uncertainty fac-tors and investigation of transmission routes and distribu-tion patterns of probability we are able to calculate the netpresent value distribution on the basis of the establishedprobability density function and transmission route formulaof uncertainty factors e calculation is simple as is shownin equation (6) and the expectation value and variance of netpresent values can be obtained which is similar to theappraisal concept based on the DCF method Nonetheless

such practice still neglects the value of decision rights andthus we need to modify the trinomial tree option pricingmodel in a way inspired by the real options method

NPV 1113944n

t1Rsalminust minus Iexpminust minus Idevminust minus Iconminust minus Copexminust minus Ttaxminust1113872 1113873

times(1 + i)minus t

(6)

where NPV is the net present value of overseas oil and gasextraction i is the discount rate Rsal-t Iexp-t Idev-t Icon-t Copex-tand Ttax-t are the sales revenue exploration investment de-velopment investment construction investment operatingcost and taxes at the t-th year respectively (in case of noincome or expense under a specific term it should be zero)

41 Decision Points and Strategies during Appraisal (DeferredDevelopment Immediate Development and Sale of Assets)e real optionmethod can deal with the asset volatility ratiovia an approach combing the uncertainty factor and thetransmission route formula since the calculated net presentvalue presents itself as a distribution It should be noted thatthe resultant distribution does not necessarily follow thenormal or logarithmic normal distribution and thereforesome currently available option calculation models may beinapplicable e volatility ratio of the calculated distribu-tion does not solely depend on the oil price this singleuncertainty factor instead should be computed usingmultiple factors through the transmission route formulas

Another major disadvantage of the real option method isthat exercising rights cannot be done in a real-time manner Itis not like that one can immediately exercise the right at anymoment and there is no such thing as a simple switch forturning on and off lights to allow for immediate startupsuspension and termination of petroleum exploration anddevelopment For example the option to defer cannot beexercised in themiddle of drilling to instantaneously shut downthe development Oil and gas fields cannot be sold out duringexploration and development to exercise the option to aban-don Consequently the timing at which it is feasible to exercisethe option should be analyzed which is referred to as thedecision point in this paper At non-decision points optionscannot be exercised or partially exercised to defer or abandon

e general extraction workflow of oil and gas is illus-trated in Figure 4 At each decision point occurs a decision-making behavior which may have various strategy spacese corresponding decision space is concluded in Table 4

Here is a brief statement of the strategy space of eachdecision point listed in Table 4 For each major stage wehave three strategies namely starting investment deferringand waiting and abandoning investment right before ini-tiation of exploration development and sale In this regardthis paper is consistent with Tang et al [13] yet two ad-ditional intermediate decision points are considered in thispaper After accomplishment of regional exploration apreliminary appraisal is carried out before trap explorationwhich is consistent with the practice of oil companies Ifregional exploration presents favorable results exploration

6 Mathematical Problems in Engineering

goes on otherwise it will be abandoned Moreover afterfinishing the drilling engineering some oil and gas com-panies may decide not to perforate the payzone for the timebeing and wait for the right moment in accordance of theirown status and estimation of future oil price and supplytendencies It is based on this very fact that the decisionpoint is designed

In addition to the five main decision points mentionedabove in fact there are many possible accidental decisionpoints in the process of oil and gas extraction For examplelarge fluctuations of the oil price may delay or bring forwardthe exploration safety incidents may lead to the suspensionof someWells and oil and gas productionmay be suspendedfor political reasons However it is difficult to predictwhether these decision points will occur when they willoccur and how long a project may be suspended Since thispaper is only a method study there is no obvious differencein the application of the method whether it is 5 6 or moredecision points this model simplifies the actual situation andonly considers 5 main decision points that inevitably exist

42 Modification to the Trinomial Tree Option Pricing ModelOn the basis of the analysis on the decision-makingworkflow presented above it is found that investment onoverseas oil and gas extraction is characterized by limited

decision points and should be a type of Bermudan optionsto which the tree option model is applicable instead of theAmerican-style or European-style options

e conventional binomial modal develops the decision-making tree with respect to the probabilities of upward anddownward movements Magnitudes of upward and down-ward movements are dependent on the volatility ratio of thetotal asset erefore there are infinite decision points andthe resultant ultimate value of extraction follows theprobability distribution However decision points foroverseas oil and gas extraction are finite which means thatone is incapable of simulating the asset volatility throughinfinite decision points Moreover the NPV at each decisionpoint calculated using Table 3 and equations (1)ndash(6) presentsitself as a distribution Under such circumstances we are notable to plot and handle an N-ary tree with infinite upwardand downward points Given this some modification has tobe made upon the tree option pricing model e tree is notplotted in accordance with upward and downward move-ments instead it is developed in reference to the strategyspace Consequently we are able to calculate the probabilitydistribution of the pre-decision value of extraction in abackward manner as is shown in Figure 5

e probability tree-based method to estimate value stilladopts the concept of calculating the initial value of ex-traction in a backward manner although it is slightly dif-ferent with that based on the decision-making tree First thevalue at the last decision point ldquobefore productionrdquo iscomputed Upon accomplishment of exploration and de-velopment there are three available strategies namelyabandonment deferring and immediate production In thecase of the value at production higher than zero decisionmakers will choose immediate development in the case ofthe value at production lower than zero decisionmakers willchoose to abandon development and recover residual valuesor straightforward sale of assets with a production value ofabout zero decision makes will choose to defer developmentand wait for growing back of oil prices e specific boundlimit is dependent on preferences for utilities of investors

Table 3 Distribution patterns of uncertainty factors for overseas oil and gas extraction

Uncertainty factor Distribution pattern Source Range Additional remarksResource reserves Logarithmic normal distribution Reliable literature gt0 mdashSale price of oil Mean-reversion with jumps Available literature gt0 Mean value growing with timeResource depth Normal distribution mdash gt0 e measurement error is normally distributedResource quality Normal distribution mdash mdashUtilization rate Trapezoidal distribution Assumed [0 1] Consulting relevant expertsamp consider the realityRate of production Triangular distribution Assumed (0 1)Decline rate Triangular distribution Assumed (0 1)Discount rate T-shaped or normal distribution Assumed mdash Not in an agreement

Regional exploration Trap exploration Drilling engineering Completion engineering

Ground facility construction engineering

Production

Exploration Development Sale

Figure 4 Workflow of overseas oil and gas exploration and development

Table 4 Decision points and their strategy space for overseas oiland gas extraction

Stage Decision point Strategy space

ExplorationBefore regionalexploration Begin defer abandon

Before trap exploration Continue abandon

Development Before drilling Continue deferabandon

Before completion Continue defer

Sale Before production Continue deferabandon

Mathematical Problems in Engineering 7

e value at this decision point can be expressed using thefollowing equation

C5

V5 minus I5 V5 gt 0 V5 1113936n

tt5

Rsalminust minus Copexminust minus Ttaxminust1113872 1113873 times(1 + i)minus t

0 V5 asymp 0

Rrecminus5 minus I5 V5 lt 0 I5 1113936t5

tt4

Idev2minust + Icon2minust1113872 1113873 times(1 + i)minus t

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

where V5 is the present value of future earnings in case the oiland gas field is being developed I5 stands for the present valueof investment made in the previous stage which will becomethe sunk cost if the development is abandoned and the holdingasset if the development is deferred t5 is the time corre-sponding to the decision point ldquobefore productionrdquo C5 is thevalue at the decision point Rrecminus5 refers to the residual value ofprevious investment that can be recovered if the development isabandoned or the value that the asset can realize in case it issold out generally far less than I5 It should be noted that V5and I5 calculated in accordance with Table 3 are probabilitydistributions instead of specific numbers

Similarly value distributions at other decision points canbe computed as is shown in the following equation

C1

C2 C2 gt 0

0 C2 asymp 0

Rrecminus1 minus I1 C2 lt 0

⎧⎪⎨

⎪⎩

C2 C3 minus I2 C3 gt 0

Rrecminus2 minus I2 C3 le 01113896

C3

C4 minus I3 C4 gt 0

0 C4 asymp 0

Rrecminus3 minus I3 C4 lt 0

⎧⎪⎨

⎪⎩

C4 C5 minus I4 C5 gt 0

0 C5 le 01113896

I2 1113944

t1

t1Iexp1minust1113872 1113873 times(1 + i)

minus t

I3 1113944

t2

tt1

Iexp2minust1113872 1113873 times(1 + i)minus t

I4 1113944

t3

tt2

Idev1minust + Icon1minust1113872 1113873 times(1 + i)minus t

(8)

where I1 I2 I3 and I4 are present values of existing in-vestment at each decision point respectively Rrecminus1 Rrecminus2 andRrecminus3 are the investment residual values that can be paid back ifthe development is abandoned In most cases one will notabandon completion after finishing drilling and the decision islimited to either immediate or deferred completion

After obtaining distributions of the value at each deci-sion point through backward calculation based on theprobability tree we can further compute the expectedeconomic value with respect to the value distribution Forinstance at the first decision point that is before decision-making upon regional exploration the incurred cost is theacquisition cost of the oil and gas asset I1 If the developmentis implemented the obtained revenue is C2 In the case ofdeferred development the investment changes into the assetof which the value is assumed to be equal to the acquisitioncost Under the circumstance that the development isabandoned part of previous investment will be lost Giventhe aforementioned information the expected value at theldquobefore regional explorationrdquo decision point is shown in thefollowing equation

E C1( 1113857 1113946P10

0Rrecminus1 minus I1 + 1113946

P10+P12

P10

0+ 11139461

P10+P12

C2

P10 P C2 lt 0( 1113857

P12 P C2 asymp 0( 1113857

P11 P C2 gt 0( 1113857

(9)

e expectation of the value of extraction at otherintermediate decision points can be calculated in asimilar approach and thus detailed description is dis-carded here

Regional exploration Trap exploration Drilling engineering Completion engineering Production

P0

P10

P11

P12

P20

P21

P30

P31

P32

P41

P42

P52

P51

P50

Yes

No

Wait

Yes Yes

No

Yes

No

Yes

No

Wait Wait Wait

Figure 5 Probability tree of overseas oil and gas extraction (there are three possibilities of whether to proceed to the next stage Yes meansimmediate development No means immediate abandonment and Wait means waiting for opportunity)

8 Mathematical Problems in Engineering

43 Application of the Modified Method In this paper theconventional decision-based binomial model is replaced bythe probability-based one For purposes of understandinghow this model can be applied this paper takes an overseasoil and gas extraction project as an example to brieflydemonstrate the application of the modified method

Sudan X is an oil and gas block located in Africa thatCNPC plans to invest inis block is seen with no elaborategeological data and only part of parameters required forasset appraisal can be determined on the basis of analogousblocks Acquisition of the exploration and developmentinterest of this block costs three million USD and thediscovery cost per barrel oil Iexpbbl is 12 $bbl e con-struction investment per barrel oil is 50 $bbl while theoperating cost per barrel oil is 17 $bbl e composite taxrate is about 20 e estimated period of exploration lastsfor two years and production capacity construction needsthree years followed by a sustained recovery of 17 yearsOther factors are all uncertain with dispersed probabilitydistribution and high uncertainty (Table 5)

It is easy to compute the probability distribution ofannual production of each year on the basis of the distri-butions of the resource reserves utilization rate productionrate and decline rate equation (1) and the assumption thatproduction capacity establishment can be completely fin-ished within three yearse calculation process is illustratedin Figure 6

With respect to oil price volatility discount rate anddistribution of sale price variation induced resource qualitywe can compute the distribution of the present value ofearnings V5 at each future year using equation (7) Resultsare presented in Figure 7

It is seen that V5gt 0 is highly likely and thus at thedecision point ldquoproductionrdquo strategies of deferring orabandoning are seldom chosen is is consistent with ourcommon observation Once all investments of explorationand development are down oil and gas operators barelydecide to abandon production

With equations (1)ndash(3) and distributions of relevantparameters the probability distributions of I4 and I5 can becomputed which subsequently leads to the distributions ofC4 and C5

Figures 7 and 8 indicate high odds of the case of C4gt 0and C5gt 0 is means that once oil and gas resources arefound during exploration probability of the proceedingdevelopment is very high Given this the values of the lattertwo decision points are overridden which is inconsistentwith the view held in the conventional real option law thathigher uncertainty results in the higher asset value

Similarly the probability distribution of C3 can be de-termined with the help of I3 and C4 and is shown in Figure 9

Unlike other decision points the future value at thedecision point ldquotrap explorationrdquo presents negative valueswhich is decisive in whether or not to execute trap

exploration If the future cash value is below zero it iscertain that investment stops A recovery ratio of 20 isset for the total previous investment residual value that isRrecminus2 20 I A range of [minus20 I2 +20 I2] for C3 is alsoset to define the case in which cash value is about zerois range is man-made and in fact will not be usedgiven equation (8) since no deferring strategy exists at thetrap exploration decision point At last the probabilitydistribution of C2 can be computed on the basis of I2 andC3 (Figure 10)

From Figure 10 it is seen that the probability of C2 belowzero is significantly reduced with decision-making beforetrap exploration However negative values still exist becauseloss can be induced by volatility of uncertainty factors eventhough rational decision-making has been executed edecision-making behavior is able to only reduce such lossesand yet it is incapable of entirely avoiding such losses Fi-nally the probability distribution of C1 can be computedusing equation (8) as is shown in Figure 11

Figure 11 demonstrates that before initiation of regionalexploration thanks to the availability of a waiting strategythe calculated value distribution presents high probabilityfor the value of about zeroe resultant expectation value ofC1 is 1148 million USD

In the case of the DCF method with no consideration ofdecision rights the probability distribution of C1 calculateddirectly using equation (6) is plotted in Figure 12 e ex-pectation value of C1-DCF calculated in this way is 1058million USD

Table 5 Distribution of uncertainty factors in Block Sudan X

Uncertainty factor Distribution of affected parametersResource reserves Ln (Rpre)simN(9 0422)

Global oil price3 dLn Ps-price 08(4-Ln Ps-price)dt + 002dztdztsim(0 dt)

Resource depth IdevbblsimN (16 322)Resource quality iquasimN (4 0782)Utilization rate rutisimTra (08 085 094 1)Production rate rprosimTri (001 002 003)Decline rate RdelsimTri (015 02 022)Discount rate4 isim001T(4) + 012ere are three possibilities of whether to proceed to the next stage Yesmeans immediate development No means immediate abandonment andWait means waiting for opportunity 3Ps-price is the sales price of crude oilLn Ps-price refers to the logarithm of crude oil price and 4 is themean valueof the logarithm of Brent oil price during 1999-2019 08 is the reversionspeed of the mean value the larger the value is the faster the logarithm ofprice approaches the long-term mean value 002 is the logarithm volatilityobtained by calculating the logarithms of Brent oil price in recent 20 yearsdzt refers to the standard Brownian motion dztsim (0 dt) 4e design ofdiscount rate distribution is based on the discount rate currently used byCNPC data from Evaluation parameters of investment projects of CNPC(2019)

Mathematical Problems in Engineering 9

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

real option field For the distribution of oil prices anagreement in understanding among extensive scholars hasnot been reached yet Some [15 26] believe that oil pricevariation should be a type of geometric Brownian motionwhile others [20 27ndash30] conduct oil price forecast using thesupport vector machine Bayesian model system simulationor a combination of multiple approaches with variouscorresponding results is paper tends to believe that oilprice complies with the Mean-Reversion with Jumps [26]which means that oil price is endowed with a mean-re-version nature and the mean oil price will gradually growwith time in case of no unexpected outburst events

In terms of quality and depth of resources both aredetermined according to the results of exploration experi-mental Wells e uncertainty of resource depth comes fromthe measurement error of experimental well depth and theuncertainty of resource quality comes from the measure-ment error of sulfur content and other indicators ereforethey can be considered as normal distribution and can beexpressed by quality compensation amount and drilling cost

Utilization rate production rate decline rate and otherfactors are to some extent subject to the subjective influence ofthe developer after consulting relevant experts we make thefollowing assumptions For the utilization rate it should bewithin [0 1] and we assume that it obeys the trapezoidaldistribution having probability within a certain subintervalmuch higher than the averagee production rate of resourcesis somewhat susceptible to subjectivity and meanwhile is alsoconstrained by geological conditions It should be within (0 1)with the existence of an optimal value and is therefore assumedto follow the triangular distribution e decline rate of hy-drocarbon recovery with a supposed range of (0 1) is related tothe production rate and also under the constraints of geologicalconditions Consequently it is also assumed to follow the tri-angular distributionWhen it comes to the discount rate of risksit is dependent on the local investment and financing envi-ronment sovereignty credit and politics Its distribution patternis still unclear For projects with low risks it may present theT-shaped distribution while for projects with higher risks itmay follow the normal distribution and we have not reached anagreement yet In addition in most cases the risk-free rate ofreturn is replaced with the long-term treasury bond rate (LTBR)of theUS and the risk-free rate plus the risk discount rate shouldbe the discount rate i used in calculating net present values

To sum up the distribution patterns of uncertaintyfactors are summarized in Table 3

4 AModified ApproachBased on the TrinomialTree Option Pricing Model

Upon accomplishment of identification of uncertainty fac-tors and investigation of transmission routes and distribu-tion patterns of probability we are able to calculate the netpresent value distribution on the basis of the establishedprobability density function and transmission route formulaof uncertainty factors e calculation is simple as is shownin equation (6) and the expectation value and variance of netpresent values can be obtained which is similar to theappraisal concept based on the DCF method Nonetheless

such practice still neglects the value of decision rights andthus we need to modify the trinomial tree option pricingmodel in a way inspired by the real options method

NPV 1113944n

t1Rsalminust minus Iexpminust minus Idevminust minus Iconminust minus Copexminust minus Ttaxminust1113872 1113873

times(1 + i)minus t

(6)

where NPV is the net present value of overseas oil and gasextraction i is the discount rate Rsal-t Iexp-t Idev-t Icon-t Copex-tand Ttax-t are the sales revenue exploration investment de-velopment investment construction investment operatingcost and taxes at the t-th year respectively (in case of noincome or expense under a specific term it should be zero)

41 Decision Points and Strategies during Appraisal (DeferredDevelopment Immediate Development and Sale of Assets)e real optionmethod can deal with the asset volatility ratiovia an approach combing the uncertainty factor and thetransmission route formula since the calculated net presentvalue presents itself as a distribution It should be noted thatthe resultant distribution does not necessarily follow thenormal or logarithmic normal distribution and thereforesome currently available option calculation models may beinapplicable e volatility ratio of the calculated distribu-tion does not solely depend on the oil price this singleuncertainty factor instead should be computed usingmultiple factors through the transmission route formulas

Another major disadvantage of the real option method isthat exercising rights cannot be done in a real-time manner Itis not like that one can immediately exercise the right at anymoment and there is no such thing as a simple switch forturning on and off lights to allow for immediate startupsuspension and termination of petroleum exploration anddevelopment For example the option to defer cannot beexercised in themiddle of drilling to instantaneously shut downthe development Oil and gas fields cannot be sold out duringexploration and development to exercise the option to aban-don Consequently the timing at which it is feasible to exercisethe option should be analyzed which is referred to as thedecision point in this paper At non-decision points optionscannot be exercised or partially exercised to defer or abandon

e general extraction workflow of oil and gas is illus-trated in Figure 4 At each decision point occurs a decision-making behavior which may have various strategy spacese corresponding decision space is concluded in Table 4

Here is a brief statement of the strategy space of eachdecision point listed in Table 4 For each major stage wehave three strategies namely starting investment deferringand waiting and abandoning investment right before ini-tiation of exploration development and sale In this regardthis paper is consistent with Tang et al [13] yet two ad-ditional intermediate decision points are considered in thispaper After accomplishment of regional exploration apreliminary appraisal is carried out before trap explorationwhich is consistent with the practice of oil companies Ifregional exploration presents favorable results exploration

6 Mathematical Problems in Engineering

goes on otherwise it will be abandoned Moreover afterfinishing the drilling engineering some oil and gas com-panies may decide not to perforate the payzone for the timebeing and wait for the right moment in accordance of theirown status and estimation of future oil price and supplytendencies It is based on this very fact that the decisionpoint is designed

In addition to the five main decision points mentionedabove in fact there are many possible accidental decisionpoints in the process of oil and gas extraction For examplelarge fluctuations of the oil price may delay or bring forwardthe exploration safety incidents may lead to the suspensionof someWells and oil and gas productionmay be suspendedfor political reasons However it is difficult to predictwhether these decision points will occur when they willoccur and how long a project may be suspended Since thispaper is only a method study there is no obvious differencein the application of the method whether it is 5 6 or moredecision points this model simplifies the actual situation andonly considers 5 main decision points that inevitably exist

42 Modification to the Trinomial Tree Option Pricing ModelOn the basis of the analysis on the decision-makingworkflow presented above it is found that investment onoverseas oil and gas extraction is characterized by limited

decision points and should be a type of Bermudan optionsto which the tree option model is applicable instead of theAmerican-style or European-style options

e conventional binomial modal develops the decision-making tree with respect to the probabilities of upward anddownward movements Magnitudes of upward and down-ward movements are dependent on the volatility ratio of thetotal asset erefore there are infinite decision points andthe resultant ultimate value of extraction follows theprobability distribution However decision points foroverseas oil and gas extraction are finite which means thatone is incapable of simulating the asset volatility throughinfinite decision points Moreover the NPV at each decisionpoint calculated using Table 3 and equations (1)ndash(6) presentsitself as a distribution Under such circumstances we are notable to plot and handle an N-ary tree with infinite upwardand downward points Given this some modification has tobe made upon the tree option pricing model e tree is notplotted in accordance with upward and downward move-ments instead it is developed in reference to the strategyspace Consequently we are able to calculate the probabilitydistribution of the pre-decision value of extraction in abackward manner as is shown in Figure 5

e probability tree-based method to estimate value stilladopts the concept of calculating the initial value of ex-traction in a backward manner although it is slightly dif-ferent with that based on the decision-making tree First thevalue at the last decision point ldquobefore productionrdquo iscomputed Upon accomplishment of exploration and de-velopment there are three available strategies namelyabandonment deferring and immediate production In thecase of the value at production higher than zero decisionmakers will choose immediate development in the case ofthe value at production lower than zero decisionmakers willchoose to abandon development and recover residual valuesor straightforward sale of assets with a production value ofabout zero decision makes will choose to defer developmentand wait for growing back of oil prices e specific boundlimit is dependent on preferences for utilities of investors

Table 3 Distribution patterns of uncertainty factors for overseas oil and gas extraction

Uncertainty factor Distribution pattern Source Range Additional remarksResource reserves Logarithmic normal distribution Reliable literature gt0 mdashSale price of oil Mean-reversion with jumps Available literature gt0 Mean value growing with timeResource depth Normal distribution mdash gt0 e measurement error is normally distributedResource quality Normal distribution mdash mdashUtilization rate Trapezoidal distribution Assumed [0 1] Consulting relevant expertsamp consider the realityRate of production Triangular distribution Assumed (0 1)Decline rate Triangular distribution Assumed (0 1)Discount rate T-shaped or normal distribution Assumed mdash Not in an agreement

Regional exploration Trap exploration Drilling engineering Completion engineering

Ground facility construction engineering

Production

Exploration Development Sale

Figure 4 Workflow of overseas oil and gas exploration and development

Table 4 Decision points and their strategy space for overseas oiland gas extraction

Stage Decision point Strategy space

ExplorationBefore regionalexploration Begin defer abandon

Before trap exploration Continue abandon

Development Before drilling Continue deferabandon

Before completion Continue defer

Sale Before production Continue deferabandon

Mathematical Problems in Engineering 7

e value at this decision point can be expressed using thefollowing equation

C5

V5 minus I5 V5 gt 0 V5 1113936n

tt5

Rsalminust minus Copexminust minus Ttaxminust1113872 1113873 times(1 + i)minus t

0 V5 asymp 0

Rrecminus5 minus I5 V5 lt 0 I5 1113936t5

tt4

Idev2minust + Icon2minust1113872 1113873 times(1 + i)minus t

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

where V5 is the present value of future earnings in case the oiland gas field is being developed I5 stands for the present valueof investment made in the previous stage which will becomethe sunk cost if the development is abandoned and the holdingasset if the development is deferred t5 is the time corre-sponding to the decision point ldquobefore productionrdquo C5 is thevalue at the decision point Rrecminus5 refers to the residual value ofprevious investment that can be recovered if the development isabandoned or the value that the asset can realize in case it issold out generally far less than I5 It should be noted that V5and I5 calculated in accordance with Table 3 are probabilitydistributions instead of specific numbers

Similarly value distributions at other decision points canbe computed as is shown in the following equation

C1

C2 C2 gt 0

0 C2 asymp 0

Rrecminus1 minus I1 C2 lt 0

⎧⎪⎨

⎪⎩

C2 C3 minus I2 C3 gt 0

Rrecminus2 minus I2 C3 le 01113896

C3

C4 minus I3 C4 gt 0

0 C4 asymp 0

Rrecminus3 minus I3 C4 lt 0

⎧⎪⎨

⎪⎩

C4 C5 minus I4 C5 gt 0

0 C5 le 01113896

I2 1113944

t1

t1Iexp1minust1113872 1113873 times(1 + i)

minus t

I3 1113944

t2

tt1

Iexp2minust1113872 1113873 times(1 + i)minus t

I4 1113944

t3

tt2

Idev1minust + Icon1minust1113872 1113873 times(1 + i)minus t

(8)

where I1 I2 I3 and I4 are present values of existing in-vestment at each decision point respectively Rrecminus1 Rrecminus2 andRrecminus3 are the investment residual values that can be paid back ifthe development is abandoned In most cases one will notabandon completion after finishing drilling and the decision islimited to either immediate or deferred completion

After obtaining distributions of the value at each deci-sion point through backward calculation based on theprobability tree we can further compute the expectedeconomic value with respect to the value distribution Forinstance at the first decision point that is before decision-making upon regional exploration the incurred cost is theacquisition cost of the oil and gas asset I1 If the developmentis implemented the obtained revenue is C2 In the case ofdeferred development the investment changes into the assetof which the value is assumed to be equal to the acquisitioncost Under the circumstance that the development isabandoned part of previous investment will be lost Giventhe aforementioned information the expected value at theldquobefore regional explorationrdquo decision point is shown in thefollowing equation

E C1( 1113857 1113946P10

0Rrecminus1 minus I1 + 1113946

P10+P12

P10

0+ 11139461

P10+P12

C2

P10 P C2 lt 0( 1113857

P12 P C2 asymp 0( 1113857

P11 P C2 gt 0( 1113857

(9)

e expectation of the value of extraction at otherintermediate decision points can be calculated in asimilar approach and thus detailed description is dis-carded here

Regional exploration Trap exploration Drilling engineering Completion engineering Production

P0

P10

P11

P12

P20

P21

P30

P31

P32

P41

P42

P52

P51

P50

Yes

No

Wait

Yes Yes

No

Yes

No

Yes

No

Wait Wait Wait

Figure 5 Probability tree of overseas oil and gas extraction (there are three possibilities of whether to proceed to the next stage Yes meansimmediate development No means immediate abandonment and Wait means waiting for opportunity)

8 Mathematical Problems in Engineering

43 Application of the Modified Method In this paper theconventional decision-based binomial model is replaced bythe probability-based one For purposes of understandinghow this model can be applied this paper takes an overseasoil and gas extraction project as an example to brieflydemonstrate the application of the modified method

Sudan X is an oil and gas block located in Africa thatCNPC plans to invest inis block is seen with no elaborategeological data and only part of parameters required forasset appraisal can be determined on the basis of analogousblocks Acquisition of the exploration and developmentinterest of this block costs three million USD and thediscovery cost per barrel oil Iexpbbl is 12 $bbl e con-struction investment per barrel oil is 50 $bbl while theoperating cost per barrel oil is 17 $bbl e composite taxrate is about 20 e estimated period of exploration lastsfor two years and production capacity construction needsthree years followed by a sustained recovery of 17 yearsOther factors are all uncertain with dispersed probabilitydistribution and high uncertainty (Table 5)

It is easy to compute the probability distribution ofannual production of each year on the basis of the distri-butions of the resource reserves utilization rate productionrate and decline rate equation (1) and the assumption thatproduction capacity establishment can be completely fin-ished within three yearse calculation process is illustratedin Figure 6

With respect to oil price volatility discount rate anddistribution of sale price variation induced resource qualitywe can compute the distribution of the present value ofearnings V5 at each future year using equation (7) Resultsare presented in Figure 7

It is seen that V5gt 0 is highly likely and thus at thedecision point ldquoproductionrdquo strategies of deferring orabandoning are seldom chosen is is consistent with ourcommon observation Once all investments of explorationand development are down oil and gas operators barelydecide to abandon production

With equations (1)ndash(3) and distributions of relevantparameters the probability distributions of I4 and I5 can becomputed which subsequently leads to the distributions ofC4 and C5

Figures 7 and 8 indicate high odds of the case of C4gt 0and C5gt 0 is means that once oil and gas resources arefound during exploration probability of the proceedingdevelopment is very high Given this the values of the lattertwo decision points are overridden which is inconsistentwith the view held in the conventional real option law thathigher uncertainty results in the higher asset value

Similarly the probability distribution of C3 can be de-termined with the help of I3 and C4 and is shown in Figure 9

Unlike other decision points the future value at thedecision point ldquotrap explorationrdquo presents negative valueswhich is decisive in whether or not to execute trap

exploration If the future cash value is below zero it iscertain that investment stops A recovery ratio of 20 isset for the total previous investment residual value that isRrecminus2 20 I A range of [minus20 I2 +20 I2] for C3 is alsoset to define the case in which cash value is about zerois range is man-made and in fact will not be usedgiven equation (8) since no deferring strategy exists at thetrap exploration decision point At last the probabilitydistribution of C2 can be computed on the basis of I2 andC3 (Figure 10)

From Figure 10 it is seen that the probability of C2 belowzero is significantly reduced with decision-making beforetrap exploration However negative values still exist becauseloss can be induced by volatility of uncertainty factors eventhough rational decision-making has been executed edecision-making behavior is able to only reduce such lossesand yet it is incapable of entirely avoiding such losses Fi-nally the probability distribution of C1 can be computedusing equation (8) as is shown in Figure 11

Figure 11 demonstrates that before initiation of regionalexploration thanks to the availability of a waiting strategythe calculated value distribution presents high probabilityfor the value of about zeroe resultant expectation value ofC1 is 1148 million USD

In the case of the DCF method with no consideration ofdecision rights the probability distribution of C1 calculateddirectly using equation (6) is plotted in Figure 12 e ex-pectation value of C1-DCF calculated in this way is 1058million USD

Table 5 Distribution of uncertainty factors in Block Sudan X

Uncertainty factor Distribution of affected parametersResource reserves Ln (Rpre)simN(9 0422)

Global oil price3 dLn Ps-price 08(4-Ln Ps-price)dt + 002dztdztsim(0 dt)

Resource depth IdevbblsimN (16 322)Resource quality iquasimN (4 0782)Utilization rate rutisimTra (08 085 094 1)Production rate rprosimTri (001 002 003)Decline rate RdelsimTri (015 02 022)Discount rate4 isim001T(4) + 012ere are three possibilities of whether to proceed to the next stage Yesmeans immediate development No means immediate abandonment andWait means waiting for opportunity 3Ps-price is the sales price of crude oilLn Ps-price refers to the logarithm of crude oil price and 4 is themean valueof the logarithm of Brent oil price during 1999-2019 08 is the reversionspeed of the mean value the larger the value is the faster the logarithm ofprice approaches the long-term mean value 002 is the logarithm volatilityobtained by calculating the logarithms of Brent oil price in recent 20 yearsdzt refers to the standard Brownian motion dztsim (0 dt) 4e design ofdiscount rate distribution is based on the discount rate currently used byCNPC data from Evaluation parameters of investment projects of CNPC(2019)

Mathematical Problems in Engineering 9

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

goes on otherwise it will be abandoned Moreover afterfinishing the drilling engineering some oil and gas com-panies may decide not to perforate the payzone for the timebeing and wait for the right moment in accordance of theirown status and estimation of future oil price and supplytendencies It is based on this very fact that the decisionpoint is designed

In addition to the five main decision points mentionedabove in fact there are many possible accidental decisionpoints in the process of oil and gas extraction For examplelarge fluctuations of the oil price may delay or bring forwardthe exploration safety incidents may lead to the suspensionof someWells and oil and gas productionmay be suspendedfor political reasons However it is difficult to predictwhether these decision points will occur when they willoccur and how long a project may be suspended Since thispaper is only a method study there is no obvious differencein the application of the method whether it is 5 6 or moredecision points this model simplifies the actual situation andonly considers 5 main decision points that inevitably exist

42 Modification to the Trinomial Tree Option Pricing ModelOn the basis of the analysis on the decision-makingworkflow presented above it is found that investment onoverseas oil and gas extraction is characterized by limited

decision points and should be a type of Bermudan optionsto which the tree option model is applicable instead of theAmerican-style or European-style options

e conventional binomial modal develops the decision-making tree with respect to the probabilities of upward anddownward movements Magnitudes of upward and down-ward movements are dependent on the volatility ratio of thetotal asset erefore there are infinite decision points andthe resultant ultimate value of extraction follows theprobability distribution However decision points foroverseas oil and gas extraction are finite which means thatone is incapable of simulating the asset volatility throughinfinite decision points Moreover the NPV at each decisionpoint calculated using Table 3 and equations (1)ndash(6) presentsitself as a distribution Under such circumstances we are notable to plot and handle an N-ary tree with infinite upwardand downward points Given this some modification has tobe made upon the tree option pricing model e tree is notplotted in accordance with upward and downward move-ments instead it is developed in reference to the strategyspace Consequently we are able to calculate the probabilitydistribution of the pre-decision value of extraction in abackward manner as is shown in Figure 5

e probability tree-based method to estimate value stilladopts the concept of calculating the initial value of ex-traction in a backward manner although it is slightly dif-ferent with that based on the decision-making tree First thevalue at the last decision point ldquobefore productionrdquo iscomputed Upon accomplishment of exploration and de-velopment there are three available strategies namelyabandonment deferring and immediate production In thecase of the value at production higher than zero decisionmakers will choose immediate development in the case ofthe value at production lower than zero decisionmakers willchoose to abandon development and recover residual valuesor straightforward sale of assets with a production value ofabout zero decision makes will choose to defer developmentand wait for growing back of oil prices e specific boundlimit is dependent on preferences for utilities of investors

Table 3 Distribution patterns of uncertainty factors for overseas oil and gas extraction

Uncertainty factor Distribution pattern Source Range Additional remarksResource reserves Logarithmic normal distribution Reliable literature gt0 mdashSale price of oil Mean-reversion with jumps Available literature gt0 Mean value growing with timeResource depth Normal distribution mdash gt0 e measurement error is normally distributedResource quality Normal distribution mdash mdashUtilization rate Trapezoidal distribution Assumed [0 1] Consulting relevant expertsamp consider the realityRate of production Triangular distribution Assumed (0 1)Decline rate Triangular distribution Assumed (0 1)Discount rate T-shaped or normal distribution Assumed mdash Not in an agreement

Regional exploration Trap exploration Drilling engineering Completion engineering

Ground facility construction engineering

Production

Exploration Development Sale

Figure 4 Workflow of overseas oil and gas exploration and development

Table 4 Decision points and their strategy space for overseas oiland gas extraction

Stage Decision point Strategy space

ExplorationBefore regionalexploration Begin defer abandon

Before trap exploration Continue abandon

Development Before drilling Continue deferabandon

Before completion Continue defer

Sale Before production Continue deferabandon

Mathematical Problems in Engineering 7

e value at this decision point can be expressed using thefollowing equation

C5

V5 minus I5 V5 gt 0 V5 1113936n

tt5

Rsalminust minus Copexminust minus Ttaxminust1113872 1113873 times(1 + i)minus t

0 V5 asymp 0

Rrecminus5 minus I5 V5 lt 0 I5 1113936t5

tt4

Idev2minust + Icon2minust1113872 1113873 times(1 + i)minus t

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

where V5 is the present value of future earnings in case the oiland gas field is being developed I5 stands for the present valueof investment made in the previous stage which will becomethe sunk cost if the development is abandoned and the holdingasset if the development is deferred t5 is the time corre-sponding to the decision point ldquobefore productionrdquo C5 is thevalue at the decision point Rrecminus5 refers to the residual value ofprevious investment that can be recovered if the development isabandoned or the value that the asset can realize in case it issold out generally far less than I5 It should be noted that V5and I5 calculated in accordance with Table 3 are probabilitydistributions instead of specific numbers

Similarly value distributions at other decision points canbe computed as is shown in the following equation

C1

C2 C2 gt 0

0 C2 asymp 0

Rrecminus1 minus I1 C2 lt 0

⎧⎪⎨

⎪⎩

C2 C3 minus I2 C3 gt 0

Rrecminus2 minus I2 C3 le 01113896

C3

C4 minus I3 C4 gt 0

0 C4 asymp 0

Rrecminus3 minus I3 C4 lt 0

⎧⎪⎨

⎪⎩

C4 C5 minus I4 C5 gt 0

0 C5 le 01113896

I2 1113944

t1

t1Iexp1minust1113872 1113873 times(1 + i)

minus t

I3 1113944

t2

tt1

Iexp2minust1113872 1113873 times(1 + i)minus t

I4 1113944

t3

tt2

Idev1minust + Icon1minust1113872 1113873 times(1 + i)minus t

(8)

where I1 I2 I3 and I4 are present values of existing in-vestment at each decision point respectively Rrecminus1 Rrecminus2 andRrecminus3 are the investment residual values that can be paid back ifthe development is abandoned In most cases one will notabandon completion after finishing drilling and the decision islimited to either immediate or deferred completion

After obtaining distributions of the value at each deci-sion point through backward calculation based on theprobability tree we can further compute the expectedeconomic value with respect to the value distribution Forinstance at the first decision point that is before decision-making upon regional exploration the incurred cost is theacquisition cost of the oil and gas asset I1 If the developmentis implemented the obtained revenue is C2 In the case ofdeferred development the investment changes into the assetof which the value is assumed to be equal to the acquisitioncost Under the circumstance that the development isabandoned part of previous investment will be lost Giventhe aforementioned information the expected value at theldquobefore regional explorationrdquo decision point is shown in thefollowing equation

E C1( 1113857 1113946P10

0Rrecminus1 minus I1 + 1113946

P10+P12

P10

0+ 11139461

P10+P12

C2

P10 P C2 lt 0( 1113857

P12 P C2 asymp 0( 1113857

P11 P C2 gt 0( 1113857

(9)

e expectation of the value of extraction at otherintermediate decision points can be calculated in asimilar approach and thus detailed description is dis-carded here

Regional exploration Trap exploration Drilling engineering Completion engineering Production

P0

P10

P11

P12

P20

P21

P30

P31

P32

P41

P42

P52

P51

P50

Yes

No

Wait

Yes Yes

No

Yes

No

Yes

No

Wait Wait Wait

Figure 5 Probability tree of overseas oil and gas extraction (there are three possibilities of whether to proceed to the next stage Yes meansimmediate development No means immediate abandonment and Wait means waiting for opportunity)

8 Mathematical Problems in Engineering

43 Application of the Modified Method In this paper theconventional decision-based binomial model is replaced bythe probability-based one For purposes of understandinghow this model can be applied this paper takes an overseasoil and gas extraction project as an example to brieflydemonstrate the application of the modified method

Sudan X is an oil and gas block located in Africa thatCNPC plans to invest inis block is seen with no elaborategeological data and only part of parameters required forasset appraisal can be determined on the basis of analogousblocks Acquisition of the exploration and developmentinterest of this block costs three million USD and thediscovery cost per barrel oil Iexpbbl is 12 $bbl e con-struction investment per barrel oil is 50 $bbl while theoperating cost per barrel oil is 17 $bbl e composite taxrate is about 20 e estimated period of exploration lastsfor two years and production capacity construction needsthree years followed by a sustained recovery of 17 yearsOther factors are all uncertain with dispersed probabilitydistribution and high uncertainty (Table 5)

It is easy to compute the probability distribution ofannual production of each year on the basis of the distri-butions of the resource reserves utilization rate productionrate and decline rate equation (1) and the assumption thatproduction capacity establishment can be completely fin-ished within three yearse calculation process is illustratedin Figure 6

With respect to oil price volatility discount rate anddistribution of sale price variation induced resource qualitywe can compute the distribution of the present value ofearnings V5 at each future year using equation (7) Resultsare presented in Figure 7

It is seen that V5gt 0 is highly likely and thus at thedecision point ldquoproductionrdquo strategies of deferring orabandoning are seldom chosen is is consistent with ourcommon observation Once all investments of explorationand development are down oil and gas operators barelydecide to abandon production

With equations (1)ndash(3) and distributions of relevantparameters the probability distributions of I4 and I5 can becomputed which subsequently leads to the distributions ofC4 and C5

Figures 7 and 8 indicate high odds of the case of C4gt 0and C5gt 0 is means that once oil and gas resources arefound during exploration probability of the proceedingdevelopment is very high Given this the values of the lattertwo decision points are overridden which is inconsistentwith the view held in the conventional real option law thathigher uncertainty results in the higher asset value

Similarly the probability distribution of C3 can be de-termined with the help of I3 and C4 and is shown in Figure 9

Unlike other decision points the future value at thedecision point ldquotrap explorationrdquo presents negative valueswhich is decisive in whether or not to execute trap

exploration If the future cash value is below zero it iscertain that investment stops A recovery ratio of 20 isset for the total previous investment residual value that isRrecminus2 20 I A range of [minus20 I2 +20 I2] for C3 is alsoset to define the case in which cash value is about zerois range is man-made and in fact will not be usedgiven equation (8) since no deferring strategy exists at thetrap exploration decision point At last the probabilitydistribution of C2 can be computed on the basis of I2 andC3 (Figure 10)

From Figure 10 it is seen that the probability of C2 belowzero is significantly reduced with decision-making beforetrap exploration However negative values still exist becauseloss can be induced by volatility of uncertainty factors eventhough rational decision-making has been executed edecision-making behavior is able to only reduce such lossesand yet it is incapable of entirely avoiding such losses Fi-nally the probability distribution of C1 can be computedusing equation (8) as is shown in Figure 11

Figure 11 demonstrates that before initiation of regionalexploration thanks to the availability of a waiting strategythe calculated value distribution presents high probabilityfor the value of about zeroe resultant expectation value ofC1 is 1148 million USD

In the case of the DCF method with no consideration ofdecision rights the probability distribution of C1 calculateddirectly using equation (6) is plotted in Figure 12 e ex-pectation value of C1-DCF calculated in this way is 1058million USD

Table 5 Distribution of uncertainty factors in Block Sudan X

Uncertainty factor Distribution of affected parametersResource reserves Ln (Rpre)simN(9 0422)

Global oil price3 dLn Ps-price 08(4-Ln Ps-price)dt + 002dztdztsim(0 dt)

Resource depth IdevbblsimN (16 322)Resource quality iquasimN (4 0782)Utilization rate rutisimTra (08 085 094 1)Production rate rprosimTri (001 002 003)Decline rate RdelsimTri (015 02 022)Discount rate4 isim001T(4) + 012ere are three possibilities of whether to proceed to the next stage Yesmeans immediate development No means immediate abandonment andWait means waiting for opportunity 3Ps-price is the sales price of crude oilLn Ps-price refers to the logarithm of crude oil price and 4 is themean valueof the logarithm of Brent oil price during 1999-2019 08 is the reversionspeed of the mean value the larger the value is the faster the logarithm ofprice approaches the long-term mean value 002 is the logarithm volatilityobtained by calculating the logarithms of Brent oil price in recent 20 yearsdzt refers to the standard Brownian motion dztsim (0 dt) 4e design ofdiscount rate distribution is based on the discount rate currently used byCNPC data from Evaluation parameters of investment projects of CNPC(2019)

Mathematical Problems in Engineering 9

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

e value at this decision point can be expressed using thefollowing equation

C5

V5 minus I5 V5 gt 0 V5 1113936n

tt5

Rsalminust minus Copexminust minus Ttaxminust1113872 1113873 times(1 + i)minus t

0 V5 asymp 0

Rrecminus5 minus I5 V5 lt 0 I5 1113936t5

tt4

Idev2minust + Icon2minust1113872 1113873 times(1 + i)minus t

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(7)

where V5 is the present value of future earnings in case the oiland gas field is being developed I5 stands for the present valueof investment made in the previous stage which will becomethe sunk cost if the development is abandoned and the holdingasset if the development is deferred t5 is the time corre-sponding to the decision point ldquobefore productionrdquo C5 is thevalue at the decision point Rrecminus5 refers to the residual value ofprevious investment that can be recovered if the development isabandoned or the value that the asset can realize in case it issold out generally far less than I5 It should be noted that V5and I5 calculated in accordance with Table 3 are probabilitydistributions instead of specific numbers

Similarly value distributions at other decision points canbe computed as is shown in the following equation

C1

C2 C2 gt 0

0 C2 asymp 0

Rrecminus1 minus I1 C2 lt 0

⎧⎪⎨

⎪⎩

C2 C3 minus I2 C3 gt 0

Rrecminus2 minus I2 C3 le 01113896

C3

C4 minus I3 C4 gt 0

0 C4 asymp 0

Rrecminus3 minus I3 C4 lt 0

⎧⎪⎨

⎪⎩

C4 C5 minus I4 C5 gt 0

0 C5 le 01113896

I2 1113944

t1

t1Iexp1minust1113872 1113873 times(1 + i)

minus t

I3 1113944

t2

tt1

Iexp2minust1113872 1113873 times(1 + i)minus t

I4 1113944

t3

tt2

Idev1minust + Icon1minust1113872 1113873 times(1 + i)minus t

(8)

where I1 I2 I3 and I4 are present values of existing in-vestment at each decision point respectively Rrecminus1 Rrecminus2 andRrecminus3 are the investment residual values that can be paid back ifthe development is abandoned In most cases one will notabandon completion after finishing drilling and the decision islimited to either immediate or deferred completion

After obtaining distributions of the value at each deci-sion point through backward calculation based on theprobability tree we can further compute the expectedeconomic value with respect to the value distribution Forinstance at the first decision point that is before decision-making upon regional exploration the incurred cost is theacquisition cost of the oil and gas asset I1 If the developmentis implemented the obtained revenue is C2 In the case ofdeferred development the investment changes into the assetof which the value is assumed to be equal to the acquisitioncost Under the circumstance that the development isabandoned part of previous investment will be lost Giventhe aforementioned information the expected value at theldquobefore regional explorationrdquo decision point is shown in thefollowing equation

E C1( 1113857 1113946P10

0Rrecminus1 minus I1 + 1113946

P10+P12

P10

0+ 11139461

P10+P12

C2

P10 P C2 lt 0( 1113857

P12 P C2 asymp 0( 1113857

P11 P C2 gt 0( 1113857

(9)

e expectation of the value of extraction at otherintermediate decision points can be calculated in asimilar approach and thus detailed description is dis-carded here

Regional exploration Trap exploration Drilling engineering Completion engineering Production

P0

P10

P11

P12

P20

P21

P30

P31

P32

P41

P42

P52

P51

P50

Yes

No

Wait

Yes Yes

No

Yes

No

Yes

No

Wait Wait Wait

Figure 5 Probability tree of overseas oil and gas extraction (there are three possibilities of whether to proceed to the next stage Yes meansimmediate development No means immediate abandonment and Wait means waiting for opportunity)

8 Mathematical Problems in Engineering

43 Application of the Modified Method In this paper theconventional decision-based binomial model is replaced bythe probability-based one For purposes of understandinghow this model can be applied this paper takes an overseasoil and gas extraction project as an example to brieflydemonstrate the application of the modified method

Sudan X is an oil and gas block located in Africa thatCNPC plans to invest inis block is seen with no elaborategeological data and only part of parameters required forasset appraisal can be determined on the basis of analogousblocks Acquisition of the exploration and developmentinterest of this block costs three million USD and thediscovery cost per barrel oil Iexpbbl is 12 $bbl e con-struction investment per barrel oil is 50 $bbl while theoperating cost per barrel oil is 17 $bbl e composite taxrate is about 20 e estimated period of exploration lastsfor two years and production capacity construction needsthree years followed by a sustained recovery of 17 yearsOther factors are all uncertain with dispersed probabilitydistribution and high uncertainty (Table 5)

It is easy to compute the probability distribution ofannual production of each year on the basis of the distri-butions of the resource reserves utilization rate productionrate and decline rate equation (1) and the assumption thatproduction capacity establishment can be completely fin-ished within three yearse calculation process is illustratedin Figure 6

With respect to oil price volatility discount rate anddistribution of sale price variation induced resource qualitywe can compute the distribution of the present value ofearnings V5 at each future year using equation (7) Resultsare presented in Figure 7

It is seen that V5gt 0 is highly likely and thus at thedecision point ldquoproductionrdquo strategies of deferring orabandoning are seldom chosen is is consistent with ourcommon observation Once all investments of explorationand development are down oil and gas operators barelydecide to abandon production

With equations (1)ndash(3) and distributions of relevantparameters the probability distributions of I4 and I5 can becomputed which subsequently leads to the distributions ofC4 and C5

Figures 7 and 8 indicate high odds of the case of C4gt 0and C5gt 0 is means that once oil and gas resources arefound during exploration probability of the proceedingdevelopment is very high Given this the values of the lattertwo decision points are overridden which is inconsistentwith the view held in the conventional real option law thathigher uncertainty results in the higher asset value

Similarly the probability distribution of C3 can be de-termined with the help of I3 and C4 and is shown in Figure 9

Unlike other decision points the future value at thedecision point ldquotrap explorationrdquo presents negative valueswhich is decisive in whether or not to execute trap

exploration If the future cash value is below zero it iscertain that investment stops A recovery ratio of 20 isset for the total previous investment residual value that isRrecminus2 20 I A range of [minus20 I2 +20 I2] for C3 is alsoset to define the case in which cash value is about zerois range is man-made and in fact will not be usedgiven equation (8) since no deferring strategy exists at thetrap exploration decision point At last the probabilitydistribution of C2 can be computed on the basis of I2 andC3 (Figure 10)

From Figure 10 it is seen that the probability of C2 belowzero is significantly reduced with decision-making beforetrap exploration However negative values still exist becauseloss can be induced by volatility of uncertainty factors eventhough rational decision-making has been executed edecision-making behavior is able to only reduce such lossesand yet it is incapable of entirely avoiding such losses Fi-nally the probability distribution of C1 can be computedusing equation (8) as is shown in Figure 11

Figure 11 demonstrates that before initiation of regionalexploration thanks to the availability of a waiting strategythe calculated value distribution presents high probabilityfor the value of about zeroe resultant expectation value ofC1 is 1148 million USD

In the case of the DCF method with no consideration ofdecision rights the probability distribution of C1 calculateddirectly using equation (6) is plotted in Figure 12 e ex-pectation value of C1-DCF calculated in this way is 1058million USD

Table 5 Distribution of uncertainty factors in Block Sudan X

Uncertainty factor Distribution of affected parametersResource reserves Ln (Rpre)simN(9 0422)

Global oil price3 dLn Ps-price 08(4-Ln Ps-price)dt + 002dztdztsim(0 dt)

Resource depth IdevbblsimN (16 322)Resource quality iquasimN (4 0782)Utilization rate rutisimTra (08 085 094 1)Production rate rprosimTri (001 002 003)Decline rate RdelsimTri (015 02 022)Discount rate4 isim001T(4) + 012ere are three possibilities of whether to proceed to the next stage Yesmeans immediate development No means immediate abandonment andWait means waiting for opportunity 3Ps-price is the sales price of crude oilLn Ps-price refers to the logarithm of crude oil price and 4 is themean valueof the logarithm of Brent oil price during 1999-2019 08 is the reversionspeed of the mean value the larger the value is the faster the logarithm ofprice approaches the long-term mean value 002 is the logarithm volatilityobtained by calculating the logarithms of Brent oil price in recent 20 yearsdzt refers to the standard Brownian motion dztsim (0 dt) 4e design ofdiscount rate distribution is based on the discount rate currently used byCNPC data from Evaluation parameters of investment projects of CNPC(2019)

Mathematical Problems in Engineering 9

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

43 Application of the Modified Method In this paper theconventional decision-based binomial model is replaced bythe probability-based one For purposes of understandinghow this model can be applied this paper takes an overseasoil and gas extraction project as an example to brieflydemonstrate the application of the modified method

Sudan X is an oil and gas block located in Africa thatCNPC plans to invest inis block is seen with no elaborategeological data and only part of parameters required forasset appraisal can be determined on the basis of analogousblocks Acquisition of the exploration and developmentinterest of this block costs three million USD and thediscovery cost per barrel oil Iexpbbl is 12 $bbl e con-struction investment per barrel oil is 50 $bbl while theoperating cost per barrel oil is 17 $bbl e composite taxrate is about 20 e estimated period of exploration lastsfor two years and production capacity construction needsthree years followed by a sustained recovery of 17 yearsOther factors are all uncertain with dispersed probabilitydistribution and high uncertainty (Table 5)

It is easy to compute the probability distribution ofannual production of each year on the basis of the distri-butions of the resource reserves utilization rate productionrate and decline rate equation (1) and the assumption thatproduction capacity establishment can be completely fin-ished within three yearse calculation process is illustratedin Figure 6

With respect to oil price volatility discount rate anddistribution of sale price variation induced resource qualitywe can compute the distribution of the present value ofearnings V5 at each future year using equation (7) Resultsare presented in Figure 7

It is seen that V5gt 0 is highly likely and thus at thedecision point ldquoproductionrdquo strategies of deferring orabandoning are seldom chosen is is consistent with ourcommon observation Once all investments of explorationand development are down oil and gas operators barelydecide to abandon production

With equations (1)ndash(3) and distributions of relevantparameters the probability distributions of I4 and I5 can becomputed which subsequently leads to the distributions ofC4 and C5

Figures 7 and 8 indicate high odds of the case of C4gt 0and C5gt 0 is means that once oil and gas resources arefound during exploration probability of the proceedingdevelopment is very high Given this the values of the lattertwo decision points are overridden which is inconsistentwith the view held in the conventional real option law thathigher uncertainty results in the higher asset value

Similarly the probability distribution of C3 can be de-termined with the help of I3 and C4 and is shown in Figure 9

Unlike other decision points the future value at thedecision point ldquotrap explorationrdquo presents negative valueswhich is decisive in whether or not to execute trap

exploration If the future cash value is below zero it iscertain that investment stops A recovery ratio of 20 isset for the total previous investment residual value that isRrecminus2 20 I A range of [minus20 I2 +20 I2] for C3 is alsoset to define the case in which cash value is about zerois range is man-made and in fact will not be usedgiven equation (8) since no deferring strategy exists at thetrap exploration decision point At last the probabilitydistribution of C2 can be computed on the basis of I2 andC3 (Figure 10)

From Figure 10 it is seen that the probability of C2 belowzero is significantly reduced with decision-making beforetrap exploration However negative values still exist becauseloss can be induced by volatility of uncertainty factors eventhough rational decision-making has been executed edecision-making behavior is able to only reduce such lossesand yet it is incapable of entirely avoiding such losses Fi-nally the probability distribution of C1 can be computedusing equation (8) as is shown in Figure 11

Figure 11 demonstrates that before initiation of regionalexploration thanks to the availability of a waiting strategythe calculated value distribution presents high probabilityfor the value of about zeroe resultant expectation value ofC1 is 1148 million USD

In the case of the DCF method with no consideration ofdecision rights the probability distribution of C1 calculateddirectly using equation (6) is plotted in Figure 12 e ex-pectation value of C1-DCF calculated in this way is 1058million USD

Table 5 Distribution of uncertainty factors in Block Sudan X

Uncertainty factor Distribution of affected parametersResource reserves Ln (Rpre)simN(9 0422)

Global oil price3 dLn Ps-price 08(4-Ln Ps-price)dt + 002dztdztsim(0 dt)

Resource depth IdevbblsimN (16 322)Resource quality iquasimN (4 0782)Utilization rate rutisimTra (08 085 094 1)Production rate rprosimTri (001 002 003)Decline rate RdelsimTri (015 02 022)Discount rate4 isim001T(4) + 012ere are three possibilities of whether to proceed to the next stage Yesmeans immediate development No means immediate abandonment andWait means waiting for opportunity 3Ps-price is the sales price of crude oilLn Ps-price refers to the logarithm of crude oil price and 4 is themean valueof the logarithm of Brent oil price during 1999-2019 08 is the reversionspeed of the mean value the larger the value is the faster the logarithm ofprice approaches the long-term mean value 002 is the logarithm volatilityobtained by calculating the logarithms of Brent oil price in recent 20 yearsdzt refers to the standard Brownian motion dztsim (0 dt) 4e design ofdiscount rate distribution is based on the discount rate currently used byCNPC data from Evaluation parameters of investment projects of CNPC(2019)

Mathematical Problems in Engineering 9

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

We have calculated the oil and gas asset value at eachdecision point without options and made a comparisonbetween the oil and gas asset value at the decision point withand without options e comparison results are shown inTable 6

erefore it is safe to say that for Block Sudan X de-cision rights at each decision point do possess values whichare about 09 million USD in total For Block Sudan Xdecision rights at decision points C1 C2 and C3 do possess

practical values and it does not seem that the value ofdecision rights at the latter two decision points would havegreat increase due to continuous development Furthermorethe disperse degree of the value probability distribution canrepresent the degrees of the aggregate risk of the investmentFigures 11 and 12 present varied disperse degrees in termsof the distribution Figure 11 is observed to be more con-centrated in the portion above zero which suggests rea-sonable future decisions can reduce risks

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Prob

abili

ty

Iognormal distribution----predicted reserves Trapezoidal distribution--utilization rate

Trangular distribution-initial production rate Triangular distribution-lapse rate

300

000

600

000

900

000

120

000

0

150

000

0

180

000

0

210

000

0

240

000

0

270

000

0

080

082

084

086

088

090

092

094

096

098

100

001

002

003

015

016

017

018

019

020

021

022

amp Decline rate ()Initial production rate ()

amp Utilization rate ()Predict reserves (104 bbl)

Prob

abili

ty

035

03

025

02

015

01

005

0100 90 80 70 60 50 40 30 20 10 0

Production (10 4 bbl)2022 2024 2026 2028 2030 2032 2034 2036 2038 2040 2042

Year

Figure 6 Probability distributions of predicted reserves utilization rate production rate decline rate and annual production of each year

10 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

100

000

0

120

000

0

140

000

0

160

000

0

180

000

0

200

000

0

PV (104$)

Prob

abili

ty

ndash64

0

ndash60

0

ndash56

0

ndash52

0

ndash48

0

ndash44

0

ndash40

0

ndash36

0

ndash32

0

ndash28

0

ndash24

0

ndash20

0

ndash16

0

Normal distribution-quality discount

Quality compensation ($bbl)

Prob

abili

ty0

04

005

006

007

008

009

010

011

012

013

014

015

016

T distribution-discount rate

Discount rate ()

Global oil price ($bbl)

004

0035

003

0025

002

0015

001

0005

0

Prob

abili

ty

20152020

20252030

20352040

Year

20 30 40 50 60 70 80 90 100 110 120

Price ($bbl)

Figure 7 Probability distributions of the global oil price quality compensation discount rate and the present value V5 of future productionearnings at the decision point ldquoproductionrdquo

Mathematical Problems in Engineering 11

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

000

001

002

003

004

Prob

abili

ty

000

001

002

003

004

005

Prob

abili

ty

800000600000400000200000PV (104$)

12000001000000800000600000PV (104$)

C5 C4

Figure 8 Probability distributions of C4 and C5

000

001

002

003

004

005

Prob

abili

ty

ndash220000 ndash110000 440000 550000330000220000110000000PV (104$)

C3

Abandon Continue

Figure 9 Probability distribution of the future value of extraction at the decision point ldquotrap explorationrdquo

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 400000300000200000100000000PV (104$)

C2

ContinueAbandon

Defer

Figure 10 Probability distribution of C2

12 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

5 Conclusions

e value of overseas oil and gas assets includes the cash flowvalue of future oil and gas development and the value ofdecision rights at each decision point e distribution andtransmission efficiency of uncertainty factors of assets will

affect the value of decision rights is paper holds that thereare five major decision points and three major decisionstrategies in the oil and gas asset development Combinedwith the discounted cash flow technique and the TrinomialTree Option Pricing Model a probability tree was con-structed to replace the traditional decision tree thereby

000

001

002

003

004

005

006

007Pr

obab

ility

ndash100000 400000300000200000100000000PV (104$)

C1

Figure 11 Probability distribution of C1

000

001

002

003

004

005

Prob

abili

ty

ndash200000 ndash100000 000 100000 200000 300000 400000ndash300000PV (104$)

C1-DCF

Figure 12 Probability distribution of C1 without considering the decision rights

Table 6 A comparison between oil and gas asset value at each decision point with and without the value of decision rights (million USdollars)

Decision point C5 C4 C3 C2 C1

With options 8293 4687 1364 1252 1148Without options 8293 4687 1317 1174 1058Cumulative value of decision rights 0 0 046 078 09

Mathematical Problems in Engineering 13

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

obtaining an evaluation method that can be used to calculatethe probability distribution of the value of decision rights ateach decision point in inverse order Taking into account thevalue of decision rights and avoiding the determination ofvolatility of oil and gas assets this method can maximallyutilize the original information about uncertainty factorsand reflect the risks of oil and gas asset development to acertain extent e details are as follows

(1) Future decision rights of overseas oil and gas ex-traction are valuable and the asset valuation shouldincorporate composite values of future cash flow anddecision rights How much the decision right shouldbe valued is dependent on the probability distribu-tion of assets affected by uncertainty factors It isembodied as that deferring or abandoning strategiescan be implemented in the case of high odds of futurelosses However it is not the case that higher un-certainty leads to higher values of decision rightsbecause the value of extraction probability distri-bution is under joint effects of the strategy space andfuture cash flow

(2) e value of extraction presents varied volatilityratios at different time points e value volatility isnot only affected by oil prices but also dependent onsuperposition of situations of each factor Accordingto the quantification of the impact of uncertaintieson development value the available information canbe to the greatest extent exploited by using theprobability distribution e resultant calculation ofvalue distribution can efficiently deal with the issuearoused by difficulties in determining volatility ratiosof real option methods

(3) e reality decides that oil and gas extraction similarto the Bermuda option is seen with infinite decisionpoints and thus the tree option pricing model isapplicable is paper replaces the decision-basedtree model with the probability-based tree model tosolve the problem that we are not able to plot andhandle an N-ary tree with infinite upward anddownward points and precisely computes theprobability distribution of the value of extraction ateach decision point in a backward mannere valueprobability distribution can not only represent howmuch the asset should be valued but also representhow high risks are with respect to its disperse degreeBy comparison incorporation of decision-makingalters the probability distribution of the value ofextraction which makes it more practical and isconducive to asset valuation and investment deci-sion-making

Finally an in-depth study on the distribution types ofuncertainty factors of oil and gas assets needs to be carriedout If the types of uncertainty factors can be further dividedand the distribution types of uncertainty factors can beproved more accurately then the calculated value of oil andgas assets will be more accurate In future research the teamwill analyze and discuss each kind of uncertainty factors and

propose a processing method to deal with possible non-inevitable decision points in order to build a more accurateand practical model

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was funded by a National Science and Technologymajor project (2016ZX05042-002-004)

Supplementary Materials

e Excel sheet explains the calculation process of the casepresented in Section 43 By using the Crystal Ball plugin ofExcel the uncertainty factor is simulated to calculate thevalue of each decision point (Supplementary Materials)

References

[1] Q Wang S Y Li and R R Li ldquoChinarsquos dependency onforeign oil will exceed 80 by 2030 developing a novelNMGM-ARIMA to forecast Chinarsquos foreign oil dependencefrom two dimensionsrdquo Energy vol 163 pp 151ndash167 2018

[2] L X Mu J Pan Z Tian Z Ji G Hu and S Yuan ldquoeoverseas hydrocaron resources strategy of Chinese oil-gascompaniesrdquo Acta Petrolei Sinica vol 34 no 5 pp 1023ndash10302013 in chinese

[3] X P Hu ldquoCarrying out the globe resource strategygoingabroad to develope overseas resourcesrdquo Geology and Pro-specting vol 39 pp 74ndash76 2003 in chinese

[4] L Mu Z Fan and A Xu ldquoDevelopment characteristicsmodels and strategies for overseas oil and gas fieldsrdquo Pe-troleum Exploration And Development vol 45 no 4pp 735ndash744 2018

[5] G F Fa R E Yuan J Lan Q Zou and Z Y Li ldquoNet reservesevaluation and sensitivity analysis of shale gas project underroyalty amp tax system in British Columbia Canadardquo 6irdInternational Conference On Energy Engineering And Envi-ronmental Protection vol 227 2019

[6] Y Song X H Qiu H J Li and L X Sui ldquoFlexibility value inMampA decision making for overseas oil and gas assetsrdquo inProceedings of the International Conference On Energy AndEnvironment Engineering (ICEEE 2015) IEEE NanjingChina pp 565ndash571 April 2015

[7] W Li D Luo and J Yuan ldquoA new approach for the com-prehensive grading of petroleum reserves in China twonatural gas examplesrdquo Energy vol 118 pp 914ndash926 2017

[8] R Guo D Luo X Zhao and J Wang ldquoIntegrated evaluationmethod-based technical and economic factors for interna-tional oil exploration projectsrdquo Sustainability vol 8 no 22016

[9] R Weijermars ldquoEconomic appraisal of shale gas plays inContinental Europerdquo Applied Energy vol 106 pp 100ndash1152013

14 Mathematical Problems in Engineering

[10] H Liu ldquoStudy on flaws and improvement of discounted cashflow theory in mergers and acquisitionsrdquo in Proceedings of the2008 4th IEEE International Conference on Management ofInnovation and Technology pp 1337ndash1341 Bangkok ai-land September 2008

[11] D G Carmichael ldquoA cash flow view of real optionsrdquo 6eEngineering Economist vol 61 no 4 pp 265ndash288 2016

[12] L M Abadie and J M Chamorro ldquoValuation of real optionsin crude oil productionrdquo Energies vol 10 no 8 2017

[13] B-J Tang H-L Zhou H Chen K Wang and H CaoldquoInvestment opportunity in Chinarsquos overseas oil project anempirical analysis based on real option approachrdquo EnergyPolicy vol 105 pp 17ndash26 2017

[14] J-Y Huang Y-F Cao H-L Zhou H Cao B-J Tang andN Wang ldquoOptimal investment timing and scale choice ofoverseas oil projects a real option approachrdquo Energies vol 11no 11 2018

[15] M N Fonseca E d O Pamplona V E d M ValerioG Aquila L C S Rocha and P Rotela Junior ldquoOil pricevolatility a real option valuation approach in an African oilfieldrdquo Journal of Petroleum Science and Engineering vol 150pp 297ndash304 2017

[16] D Lund and R Nymoen ldquoComparative statics for real optionson oil what stylized factsrdquo 6e Engineering Economistvol 63 no 1 pp 54ndash65 2017

[17] L Zhan C M Yang and S Hu Risk Assessment and Pre-vention in Oil-Gas Exploration Industry 6e Tarim Basin asthe Case Universe Academic Press Toronto Toronto Canada2008

[18] A Ghandi and C Lawell ldquoOn the rate of return and riskfactors to international oil companies in Iranrsquos buy-backservice contractsrdquo Energy Policy vol 103 pp 16ndash29 2017

[19] A Z Yin ldquoStudy on economic evaluation index system of oil-gas exploration projectrdquo in Advanced Research on Informa-tion Science Automation and Material System H ZhangG Shen and D Jin Eds pp 1693ndash1696 Trans Tech Publi-cations Ltd Stafa-Zurich Switzerland 2011

[20] H Xie Q Guo F Li et al ldquoPrediction of petroleum ex-ploration risk and subterranean spatial distribution of hy-drocarbon accumulationsrdquo Petroleum Science vol 8 no 1pp 17ndash23 2011

[21] C Wegener T Basse F Kunze and H-J von MettenheimldquoOil prices and sovereign credit risk of oil producing coun-tries an empirical investigationrdquo Quantitative Financevol 16 no 12 pp 1961ndash1968 2016

[22] Y Yang J Li X Sun and J Chen ldquoMeasuring external oilsupply risk a modified diversification index with country riskand potential oil exportsrdquo Energy vol 68 pp 930ndash938 2014

[23] G H Pei and X L Huang Application of NPV Method inOilfield Well Pattern Design Orient Acad Forum Marrick-ville Australia 2008

[24] D G Quirk and R Ruthrauff ldquoAnalysis of reserves discoveredin petroleum explorationrdquo Journal of Petroleum Geologyvol 29 no 2 pp 125ndash146 2006

[25] E D Attanasi and R R Charpentier ldquoComparison of twoprobability distributions used to model sizes of undiscoveredoil and gas accumulations does the tail wag the assessmentrdquoMathematical Geology vol 34 no 6 pp 767ndash777 2002

[26] Y Q Zhou and L Yan ldquoComparing two models for evalu-ating an oilfield development project mean-reversion withJumps geometric brownian motionrdquo Advanced MaterialsResearch vol 616-618 pp 1568ndash1572 2013

[27] X Zhu and Z Guo ldquoSimulation study on forecasting methodof oil price forecastingrdquo Computer Simulation vol 28 no 6pp 361ndash364 2011

[28] Y Zhang J He and T Yin ldquoResearch on petroleum priceprediction based on SVMrdquo Computer Simulation vol 29no 3 p 375 2012

[29] C Y Lee ldquoLong-term crude oil price forecast using thebayesian modelrdquo POSRI Business and Economic Reviewvol 11 no 2 pp 58ndash86 2011

[30] C Baumeister and L Kilian ldquoForecasting the real price of oilin a changing world a forecast combination approachrdquoJournal of Business amp Economic Statistics vol 33 no 3pp 338ndash351 2015

Mathematical Problems in Engineering 15