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Quantitative Finance

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Hedging strategies for energy derivatives

P. Leoni , N. Vandaele & M. Vanmaele

To cite this article: P. Leoni , N. Vandaele & M. Vanmaele (2014) Hedging strategies for energyderivatives, Quantitative Finance, 14:10, 1725-1737, DOI: 10.1080/14697688.2013.836294

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Published online: 25 Oct 2013.

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Quantitative Finance , 2014Vol. 14, No. 10, 1725–1737, http://dx.doi.org/10.1080/14697688.2013.836294

Hedging strategies for energy derivativesP. LEONI ∗†, N. VANDAELE ‡ and M. VANMAELE ‡

†Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium‡Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281 S9,

B-9000 Ghent, Belgium

( Received 6 October 2011; accepted 2 August 2013 )

In this article, we dene a hedging strategy in a setting typical for the commodity market. Firstly, we

prove the existence of the locally risk-minimizing (LRM) hedging strategy for payment streams inthis setting. Next, a three-step procedure is described to determine the LRM hedging strategy. Thenthe procedure is illustrated for stochastic volatility models, as these models are a special case of thenon-traded situation which frequently occurs in the commodity markets. Finally, we introduce the(adjusted) LRMhedging strategy in the non-traded setting and for this specic setting we numericallyshow the outperformance of this strategy compared with current market practice.

Keywords : Energy derivatives; Hedging strategies; Non-traded assets; Local risk minimization;Stochastic volatilty models

JEL Classication : G13, G19

1. Introduction

Hedging under restrictions is a problem of great practicalimportance. It can become relevant in all nancial markets butit is extremely important in energy markets where liquidity canbe poor. Traditionally, a lot of academic attention has gone tosetting up risk-neutral spot models that immediately model theshort-to-delivery contracts, suchas a day-aheadforward, underthe assumption that hedging can be done perfectly. However,the market does not trade these short-to-delivery contacts veryfaraheadintime.Infact,mostofthetradingisfocusedonlowergranular delivery periods such as calendar years or seasons. Inthis paper, we will focus on the dynamics of such forward

contracts and establish how to hedge derivatives on smallerdelivery periods, such as months or days by using a coarse-grained trading approach. Of course, this approach introducesa basis risk asthe claims are written on a different asset than theone that is being used to hedge with. But in energy markets, theextra risk is worth takingbecause theliquidity constraint wouldeither not allow a direct hedging procedure to be executed, orthe associated cost would be too high.

We will dene an (adjusted) locally risk-minimizing (LRM)hedging strategy to a setting that is a good representation of everyday practice in an energy market. The obtained LRMhedging strategy is compared to a few common practices andshown to outperform signicantly.

Since energy such as electricity or gas are non-storablecommodities, the trading market has been organized around

∗Corresponding author. Email: [email protected]

futures and forwards. These contracts provide an agreementbetween the two transacting parties to deliver the commodityover a xed period of time rather than ensuring an instanta-neous delivery. This means that the variety of delivery periodsis enormous and although the correlation between them is notalways strong, usually only a few contracts are liquid enoughto execute trading strategies.

Since theunderlying asset is a commoditythat gets deliveredphysically in a certain volume, it is natural that the pricesare denominated in currency per unit of volume per unit of time. Over the years, energy markets have attained a specicstructure suitable for handling this ow nature. One of theunique features of the forward curve is its decomposition, or

bucketing, into different granularities. Far ahead in the future,the only forward contracts traded are forwards for deliveryof power over a complete calendar year. Once the calendaryear approaches, these contracts gradually break down intoquarterly contracts in a ‘cascading’process. Closer to delivery,these quarterly contracts will break up into monthly, weeklyand even daily forward contracts.

We discuss this setting in more detail for the electricitymarket and for the gas market separately.

(i) Electricity market: Besides the delivery period dur-ing which European power or electricity gets deliv-ered,one oftendistinguishesthree differentproducts:

base, peak and off-peak. Those are best explainedby means of an example. A CAL-14 peak pro-duct is a contract that will deliver electricity during

© 2013 Taylor & Francis


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1726 P. Leoni et al.

the entire calendar year 2014, but only duringthe peak hours of the day. That means thatduring the weekend no power will be deliveredand during the weekdays, the delivery only takesplace during the day (peak hours). A base contractensures delivery of power during every single hourof the delivery period, without exception. The speci-c denition of which hour isa peak hour or off-peak hour, depends on the market in question.As indicated, prices are denominated in currencyper unit of volume per unit of time. For example, aCAL-14 base product for a volume of 10MW(megawatt) could have a price of e 50/MWh. Thismeans that per delivered hour and per MW, the priceis e 50. There are 8760 h in the year 2014, so thetotal premium that is paid for the delivered power ise 50 × 8760 × 10 = e 4380000.A peak contract for the same delivery period 2014could have a price of e 80/MWh, which is higher

per active unit, but since the number of active hoursis much less, the total premium is still lower com-pared to the base contract. Roughly speaking thenumber of peak hours is one-third of the amountof base hours. This means that the premium ise 80 × 8760 / 3 × 10 = e 2 336 000.It is clear that there is a relationship between thepeak, off-peak and base price in the market. If onebuys electricity for delivery during peak hours andat the same time buys a contract that ensures deliv-ery during the off-peak hours, it is obvious that thepower is delivered without interruption and this is

equivalent to a base contract.If we denote the forward price of a peak contract byF ( p) and the forward price of an off-peak contractas F (o) , then the forward price of the base contractF (b) is given by

F (b) = w ( p) F ( p) + w (o) F (o)

where the weights w( p) and w (o) depend on thenumber of peak and off-peak hours of the market.Since all prices are normalized to one unit of powerandone hour,the typical weights are w ( p) = 1/ 3andw (o) = 2/ 3, where we should never forget that theactual cash ows will take into account the numberof hours (see examples above).In terms of liquidity, peak or off-peak contracts arenot as liquid as base, especially far ahead in thefuture. This has its implications for the writer of an option on a peak (or off-peak) forward contract.Although it is one of the very basic assumptions inderivatives theory, energy market traders often ndthemselves in a situation where they sell options al-though theunderlying contract is not liquidly traded.In the application of the theory, we will assume thata claim on peak power is transacted. It could be thatat the time of this transaction, the value of this peak

contract is known but that the spread between thebid price and the offer price is too big to efcientlyhedge this position. For this reason, it is commonpractice to hedge with a base product rather than a

peak product until closer to maturity of the optionthe strategy is swapped into a strategy in the peak product, when the liquidity has increased. We willcall the peak product non-tradable for reasons of illiquidity and in section 5 we will study the effectof different hedging strategies.

(ii) Gas market: The gas market is highly seasonal withsummer prices usually substantially cheaper thanwinterprices.Because of this, it is easy to understandthat the most liquid products are forward contractsfor the delivery of gas during the winter or during thesummer. Gas by itself is more storable than powerbecause the pressure differences in the network al-low for the variations in demand during the day.Because of this, there is no peak andoff-peakmarketfor gas.The interest in option contracts in the European gasmarket (e.g. UK or NBP market) is increasing, sincethese kind of contracts provide an interesting way of

hedging the portfolios of big gas players in the mar-ket. However, for historical reasons, the most liquidoption contract is a so-called seasonal option, thatactually consists of a strip of 6 monthly options. Sothere are six underlying levels F (1) , F (2) , . . . , F (6)

that are relevant for the pricing and hedging of sucha contract. However, at the time when the optionsare written, not all of these monthly forward pricesare known and the hedging has to be done by meansof the season forward, which is in fact given by

F =6

i = 1

w (i ) F (i ) . (1)

In general, we will denote the underlyings as a vectornamely (F (1) , . . . , F (d ) ) and the corresponding weights as(w (1) , . . . , w (d ) ) .

As we will show in this article, it is even not possible todetermine a delta hedge for these non-traded settings. Hencein practice adjusted delta hedges are used instead. We willpropose an (adjusted) LRM hedging strategy and compare itto the adjusted delta hedges used in practice.

The next section contains an overview of articles dealingwith non-traded assets. In section 3, we give a short introduc-tion to thetheory of (local) risk minimization. In section 3.1 , weshow how our setting ts into the very general setting found inliterature and discuss the existence of a LRM hedging strategyfor payment streams. Next, in section 3.2 we describe a three-step procedure to compute this strategy for the setting validfor the commodity market. We refer the readers interestedin the more technical details concerning the LRM hedgingstrategy to, e.g. Vandaele and Vanmaele (2008 ) and the refer-ences therein. We remark that stochastic volatility models are aspecial case of thenon-traded setting, we work with.Therefore,in section 3.3 we determine the LRM hedging strategy for thestochasticvolatilitymodels studied in Poulsen etal. (2009 ), butfollowing our three step approach. Another quadratic hedg-

ing strategy we could look at is the mean-variance hedging(MVH) strategy. This strategy has the advantage that it isself-nancing and that it minimizes the total cost. Althoughtheoretically speaking, the solution to this strategy is known,


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Hedging strategies for energy derivatives 1727

in practice it is still hard to nd the exact solution. For moredetails concerning the problems arising in the non-traded assetcase, we refer to Vandaele (2010 ). In the subclass of (afne)stochastic volatility models, explicit MVH strategies canbe determined, see e.g. Kallsen and Vierthauer (2009 ) andKallsen and Pauwels (2010 ).

Assumingspecic dynamics for the forward priceprocesses,the determination of the LRM and the adjusted LRM hedgingstrategies for the commodity setting described at the beginningof this section is given in section 4. In section 5, we comparethe different current market practices to the newly obtainedresults and show that the proposed adjusted LRM hedgingstrategies outperform, even in the simplest cases. Hence weshow with numerical results that the framework of LRM hedg-ing strategies is extremely relevant for managing the risksin contracts on non-traded assets. The ow of informationfrom mature nancial markets to energy markets has only justbegun. This paper is to our knowledge the rst to successfullyapply advanced mathematical results to the everyday practice

of hedging options in power and gas markets.

2. Literature

We found only few articles dealing with non-traded assets.The setting used in these articles differs from the one we work with, because they start from a different underlying problem.Furthermore, they all concentrate on the continuous setting,while we also allow discontinuous processes.

In literature, seee.g. Davis (2006 ), the termbasis riskis oftenused for the non-hedgeable risk which remains and cannotbe hedged away due to the fact that the asset on which theoption is written is not available for hedging. Hedging in thiscase can only be done by using some closely related asset.Sometimes, the underlying asset is available for hedging butis too expensive due to transactions costs.

We mention here some important papers, the interestedreader can also look at the references in those papers.

• Davis (2006 ) assumes that the underlying assetcannot be traded but is observable. Instead ‘a closelyrelated’ asset, with a continuous price process, istraded. This closely related asset is assumed to fol-low a Brownian motion which is correlated with theunderlying risky asset.The optimal hedging strategyis determined using exponential utility as a crite-rion. Numerical results for this case are derived inMonoyios (2004 ).

• Henderson (2002 ) and Henderson and Hobson(2002 ) work in the same setting. Furthermore, theyalso determine the power utility and give numericalresults.

• Hobson (2005 ) gives an upper bound for the utilityindifference price of a contingent claim on a non-tradedasset in thesame setting as describedby Davis(2006 ).

• Ankirchner et al. (2010 ) also calculate the exponen-tial utility-based indifference prices and correspond-ing hedges in a continuous setting. Their results areobtained in terms of solutions of forward–backward

stochastic differential equations. Hence, the opti-mal hedging strategies are described in terms of theindifference price gradient and the correlationcoefcients. Furthermore, the hedge can be seenas a generalization of the ‘delta-hedge’ in completemarkets.

• Horst et al. (2010 ) concentrate on transferring non-nancial risk, as for example depending on the tem-perature, to thecapital markets. They give numericalresults of equilibrium prices and optimal utilities ina continuous framework.

• Njoh (2007 ) determines quadratic hedging strate-gies for electricity spot prices with continuous priceprocesses when another related asset is used forhedging.

We also wish to point out the growing interest in literaturefor numerical comparisons between the delta hedge andthe quadratic hedging strategies, see Altmann et al. (2008 ),Denkl et al. (2009 ) and Brodén and Tankov (2011 ).

3. LRM hedging strategy

In this section, we concentrate on the existence and thedeterminationof theLRMhedgingstrategyforpayment streamsin terms of Radon–Nikodym derivatives by a three-step pro-cedure. Then, we illustrate our procedure by applying it tostochastic volatility models, as these models are a special caseof the non-traded situation which frequently occur in the

commodity market.In section 3.1, we rst state the results ensuring theexistence of the Föllmer–Schweizer (FS) decomposition basedon Choulli et al. (1998 ) and the existence of the LRM hedg-ing strategy in a multidimensional setting and allowing forpayment streams using Schweizer (2008 ). Assuming all theconditions for the existence of the LRM hedging strategy, wedetermine in section 3.2 the optimal number of risky assetsbased on Choulli et al. (2010 ), where an explicit form forthe FS decomposition is given in terms of the predictablecharacteristics. In fact the setting used in Choulli et al. (2010 )is much more general than is needed in the present paper. Thegoal in that article is thedetermination of theFS decompositionin a framework as general as possible, while the link withthe LRM hedging strategy was not described explicitly. TheLRM hedging strategy in continuous time only makes sense if the nite variation part is continuous and hence we make thisrestriction in our setting. Due to this assumption, we canobtainvery quickly a shorthand notation for the optimal number of risky assets, which will certainly help the reader to understandthe objective of this strategy and the FS decomposition. Weremark that the way we follow here is more linked with someof the intermediate results given in Černý and Kallsen (2007 )on their way to determining the MVH strategy.

LRM hedging strategies control less of the total risk than

mean-variance ones, but the advantage is that solutions forthe hedging strategies are much simpler. In the martingalesetting, there is no difference between the optimal number of risky assets given by the risk-minimizing (RM) and the MVH


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strategy. Both solutions are found through the Galtchouk–Kunita–Watanabe (GKW) decomposition.

For more details concerning quadratic strategies, we refer toe.g. Pham (2000 ), Schweizer (2001 ), Vandaele and Vanmaele(2008 ) and the references therein. We also remark that theresults of this section can beseenas an extension of the work byColwell and Elliott (1993 ). Colwell and Elliott looked at thecase when the contingent claim depends solely on the riskyassets in which one can invest, but at the time the articlewas written the theory of local risk minimization was lessdeveloped. We felt that it would be possible to describe aneasier, more straightforward approach to tackle not only thesetting described by Colwell and Elliott (1993 ), but an evenwider range of problems, including the problem of non-tradedassets and payment streams. Choulli et al. (2010 ) extendedColwell and Elliott (1993 ) in the sense that they describeda more direct way to describe the FS decomposition and thedynamics used to describe the process of the risky asset are farmore general.

To further illustrate the usefulness of the method describedhere, we also refer to Poulsen et al. (2009 ). They determinethe LRM hedging strategy for a general class of stochasticvolatility models driven by continuous processes using thethree-step procedure introduced by El Karoui et al. (1997 ).Following that approach, the market is rst completed, thenthe hedging strategy is computed in the completed marketand nally this strategy is projected on the original market.In section 3.3, we obtain the same solution using our three stepapproach described here, which does not require the continuityof the process.

3.1. Existence of the LRM strategyWe work on the probability space ( , F , P ) . The ltration(F t )0≤ t ≤ T satises all the usual conditions and T ∈ [0, +∞ )is the xed time horizon.

When theunderlyings of thecontingent claim that onewantsto hedge are semimartingales we can no longer apply the RMhedging strategy, but we need instead to apply the LRM hedg-ing strategy described for the rst time by Schweizer in a seriesofarticles.The goal ofa RMhedgingstrategy is to minimize thevariance of the cost process C , while the portfolio should equalthe contingent claim at payment date. Hence, we minimize atany time t

E [(C T − C t )2| F t ],

with C t = V t − t 0 ξ u d S u where V denotes the value processof the portfolio and ·0 ξ u d S u is the gain process of tradingin the underlying S . The exact criterion, which is minimizedin the case of semimartingales, becomes a bit more involved.Therefore, we refer the interested reader to, e.g. Schweizer(1991 ). The RM hedging strategy is easily derived from theGKW decomposition:

Denition 3.1 Galtchouk–Kunita–Watanabe(GKW) decom-position: If S is a local martingale under the measure Q , then

H = H (0) +

T

0

ξ H u d S u + L H T Q-a.s.

is the GKW-decompositionof the square-integrable contingentclaim H if ξ H ∈ L ( S ) , if ξ · S and L H are local Q -martingalesand if L H is orthogonal to S , with L H 0 = 0.

Here we used theshorthand notation forstochastic integrals,namely, ξ · S stands for ·0 ξ u d S u . The optimal amount investedin the risky asset is given by ξ H in Denition 3.1 .

For the LRM hedging strategy, we dene rst the space S which we need in the denition of the FS decomposition.

Denition 3.2 The space S consists of all R d -valued pre-

dictable processes θ such that the stochastic integral processθ · S is well-dened and belongs to the space S 2 ( P ) of semimartingales. This means that

E T 0 θ s d M s θ s + T 0 θ s d As 2 < ∞ ,where denotes the transpose, · is the predictable quadraticvariation, M is the martingale part and A the nite variationpart of S .

The LRM hedging strategy is found through the FSdecomposition:

Denition 3.3 Föllmer–Schweizer (FS) decomposition:If S is a semimartingale under the measure P , then

H = H (0) + T 0 ξ H u d S u + L H T P -a.s.is the Föllmer–Schweizer decomposition of the contingentclaim H if the S -integrable process ξ H ∈ S and if L H isa square-integrable P -martingale orthogonal to the martingalepart M of S , with L H 0 = 0.

For continuous semimartingales the FS decomposition canbe deduced from the GKW-decomposition under the minimalmartingale measure (MMM), which is the unique measure

such that every local martingale L under P , which is alsoorthogonal to the martingale part M of S remains a martin-gale under the MMM. This approach is no longer valid fordiscontinuous semimartingales.

In Choulli et al. (2010 ), the relationship between the hedge-able part of the GKW decomposition under the MMM andthe one of the FS decomposition under the original measureis given and the FS decomposition is determined using thepredictable characteristics. In the next section, we will showhow we can determine the decomposition following a threestep procedure in terms of Radon–Nikodym derivatives whichis implicitly mentioned in Choulli et al. (2010 ). The aim hereis to show that our setting ts in the very general setting of Choulli et al. (2010 ) and to derive the existence conditions forthe LRM strategy from it. Since we concentrate here on thedetermination of the LRM strategy this less general setting,which is typical for thecommodity market, is not really restric-tive. Furthermore, we derive the hedging strategy for paymentstreams based on Schweizer (2008 ) where it is proved thateven in the multidimensional case and for payment streams theLRMhedging strategy is determined by the FS decomposition.In that paper also some conditions on the hedgeable risky assetS as given in Schweizer (1991 ) are relaxed.

Denition 3.4 A R d -valued semimartingale S = ( S t )0≤ t ≤ T satises the structure condition (SC) if S is special with canon-ical decomposition

S = S 0 + M + A = S 0 + M + d M λ,


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where M is a square integrable local martingale with M 0 = 0and λ is a R d -valued, predictable process that is locally squareintegrable w.r.t. M , so that the mean-variance tradeoff processK := λ d A = λ d M λ satises K T < ∞ P -a.s.We remark that the SC is related to the absence of arbitragecondition.

Denition 3.5 An L2-strategy is a pair φ = (ξ, η) , whereξ ∈ S and η is a real-valued adapted process such that thevalue process V (φ) := ξ S + η is right-continuous and square-integrable.Theorem 3.6 Schweizer (2008 ): Suppose the R d -valued semimartingale S satises theSC and themean-variancetrade-off process K = λ d M λ (or, equivalently, A) is continuous.Then a payment stream H admits a LRM L 2-strategy φ if and only if H T admits a FS decomposition. In that case φ = (ξ , η)is given by

ξ = θ H T , η = V H T − θ H T S

with

V H T t := H (0)T + t 0 θ H T s d S s + L H T t − H t , 0 ≤ t ≤ T ,

where the processes with superscript H T are coming from theFS decomposition of H T .

It is important to notice that this theorem guarantees thatif you nd a LRM hedging strategy then this strategy makessense and will really minimize the risk locally, but this doesnot guarantee the existence of this strategy.

Hereto we have to combine this with the existence of the FSdecomposition as proved in Choulli et al. (1998 ) under certainconditions for a very general class of so-called E -martingales.In our setting, the Girsanov density process Z describing thechange of measure from P to the MMM Q is given by E ( N )the Doléans-Dade exponential of N = − λ · M which is aspecial case of Choulli et al. (1998 ) where most of the proofsare given for an arbitrary N . From Proposition 3.7 in Choulliet al. (1998 ), we know that if N T ∈ L∞ then the conditionsfor the existence of a FS decomposition are satised. If weassume moreover that this process Z is a strictly positivesquare-integrable P -martingale then theclassof E -martingales

simplies to Q -martingales. Hence, we can conclude withthe following theorem which combines the main result fromSchweizer (1991 ) and Choulli et al. (1998 ):Theorem 3.7 Suppose S is an R d -valued special semi-martingale satisfying the SC and whose nite variation part A is continuous. If also λ · M T ∈ L ∞ then the LRM hedgingstrategy for a payment stream H and the FS decomposition of H T exist, and the L 2-strategy in which we should invest canbe found from the FS decomposition.

An important underlying condition, which is now hiddenbeneath other conditions, is the closedness of the space

G T ( S ) := { (θ · S ) T : θ ∈ S }. Monat and Stricker (1995 )proved that if λ · M T ∈ L ∞ then the space G T ( S ) is closedin L2 . Choulli et al. (1998 ) generalized this result to theclass of E -martingales.

3.2. Determination of the LRM strategy

Based on the previous section the following conditions areassumed to be satised:

• S is a special semimartingale satisfying the SC;• the nite variation part A is continuous;• λ · M T ∈ L ∞ , with M the P -martingale part of S ;• the density process Z = E (− λ · M ) is a strictly

positive square-integrable P -martingale.

The procedure to determine the LRM hedging strategy isbased on the proof of Theorem 5 .5 of Choulli et al. (1998 )but is much simpler because of the assumptions that we made.We denote by H the payment process for which we want todetermine the hedging strategy.

Step 1 Dene the Q -martingale Y with Y T = H T byY t =

E [ H T Z T |F t ] Z t

= E Q [ H T |F t ] on {t ∈ [0, T [}. (2)

The value of the hedging portfolio V H at time t is then thedifference of Y and the payment process H at t , i.e. V H t =Y t − H t .Comment: The existence and the form of this Y is given byProposition 3 .12 (iii) of Choulli et al. (1998 ) but in a simpliedform thanksto theassumptionthat Z neverbecomes zero. In thecase of a contingent claim with payoff H at T the payments attimes t ∈ [0, T ) are zero and the value of the hedging portfolioV H at time t equals Y t .

Step 2 Extract the P -martingale part I of the

Q -martingale

Y which equals

I = Y − Y (0) + I , N P

where ·, · P stands for the quadratic covariation of two P -martingales.Comment: Note that we made here the dependence of thequadratic (co)variation process on the measure explicit by us-ing a superscript. Furthermore we know that − I , N P is thenite variation part of the Q -martingale Y under P , see alsoProposition 4.2 in Choulli et al. (2010 ).From the FS decomposition of H T we may infer Y = Y (0) +ξ · S + L and hence I = ξ · M + L which is the GKWdecomposition of the P -martingale I .In the setting described here the nite variation part of Y isthus also continuous because − I , N P = ξ · M + L , N P =λξ · M , M P = ξ · A, see Denition 3.4.The problem of nding the FS decomposition of H T is nowreduced to the determination of the GKW decomposition of the P -martingale I .

Step3 Computethe optimal number ξ as theRadon–Nikodymderivative:

ξ = (d M , M P ) inv d I , M P . (3)

The strategy at time t is then given by (ξ t , ηt ) with ηt = Y t −ξ t S t − H t .


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Comment: The superscript inv denotes the inverse of the ma-trix. Here we mean the pseudoinverse of Moore–Penrose, seeAlbert (1972 ), whose existence is guaranteed.

The expression ( 3) follows by taking the angle bracket pro-cess with respect to M under P of both sides of I = ξ · M + Land solving for ξ :

I , M P = ξ · M , M P .To understand the difference between the LRM and the RMhedging strategy under the MMM Q , we also calculate theRM strategy:

ξ̃ t = (d S , S Qt )

inv d Y , S Qt (4)

and ηt = Y t − ξ̃ t S t − H t . Note thatwhen S isnot a Q -martingale,S , S Q has to be understood as the quadratic variation of theQ -martingale part of S . Similarly, Y , S Q is the quadraticcovariation of the Q -martingale part of S and the Q -martingaleY .Hence, one can easily determine the relationship betweenthe GKW and the FS decomposition in the following way:starting from Denition 3.1 applied to Y , we nd

Y , S Q = ξ̃ · S , S Q + LGKW , S Q = ξ̃ · S , S Q . (5)

On the other hand, starting from Denition 3.3 we have that

Y , S Q = ξ · S , S Q + LFS , S Q .

Solving for ξ and using ( 5), we obtain

ξ = (d S , S Q ) inv d Y , S Q − (d S , S Q ) inv d LFS , S Q

= ξ̃ − .

In this form, we obviously see that there is no equivalencebetween the GKW decomposition and the FS decompositionwhen the orthogonality between LFS and M under P , is notpreserved under Q . For an explicit counterexample where = (d S , S Q ) inv d LFS , S Q = 0, we refer to Choulli et al.(2010 ). Furthermore, = 0 in the continuous case, becausefor continuousprocesses theorthogonality is always preserved.

3.3. Stochastic volatility models

If the liquidity of the market increases, the liquidity in thederivatives follows as well. In an established option’s market,options with various strikes all become readily traded andone can observe smile and/or skew behaviour in theimplied volatilities. It is well known by now that this smilebehaviour of long-dated options can be well explained bythe introduction of another stochastic factor in the volatil-ity or variance process. In particular, options on oil-relatedmarkets such as Brent or WTI exhibit this kind of strike-dependent volatility and therefore in this context it is naturalto introduce a stochastic volatility model. The MVH strategyof claims on risky assets with a stochastic volatility is of-ten investigated in literature. As references we mention Bi-agini et al. (2000 ), Černý and Kallsen (2008 ), Chan et al.

(2009 ), Kallsen and Vierthauer (2009), Kallsen et al. (2010)and Kallsen and Pauwels (2010 , 2011 ). The references to theLRMhedging strategyfor stochastic volatilitymodels aremorerare: Frey and Runggaldier (1999 ) and Poulsen et al. (2009 ).

In Bertsimas et al. (2001 ), the -arbitrage strategy is deter-mined for stochastic volatility models.

We illustrate here the technique for the determination of the LRM strategy as described in section 3.2 on the classof stochastic volatility models used in Poulsen et al. (2009 ).They apply a three-step procedure introduced by El Karoui et al. (1997 ): the market is completed, the hedging strategy iscalculated in this completed market and then projected on theoriginal market. In fact our procedure is similar, in that thedetermination of the MMM is part of completing the market.However, our approach is more general because the approachof El Karoui et al. (1997 ) is limited to Brownian motions.Furthermore, we also know that in the continuous setting theFS decomposition can easily be deduced from the GKW de-composition. This will be shown explicitly on the exampleapplied in Poulsen et al. (2009 ).

The hedge is determined for European claims H ( S ∗T ) whenthe underlyingundiscounted riskyasset S ∗follows a stochasticvolatility model of the following form:

d S ∗( t )S ∗( t )

= µ dt + S ∗( t )γ f (V ( t ))

× 1 − ρ 2dW 1( t ) + ρ dW 2( t )dV ( t )V ( t )

= β ( V ( t )) dt + g(V ( t )) dW 2( t ),

with independent standard Brownian motions W 1 and W 2 . Werefer to Poulsen et al. (2009 ) for an overview of the modelscontained in this class and for more details concerning thefunctions/parameters β , γ , µ , ρ , g and f .With S we denote the discounted dynamics:

d S ( t )S ( t )

= (µ − r )dt + S ∗( t )γ f (V ( t ))

× 1 − ρ 2dW 1( t ) + ρ dW 2 ( t ) ,where r denotes the risk-free interest rate. The notation M isused for the martingale part of the risky asset S . We are ina Markovian market model and we can easily determine theMMM Q : Z t = E [

d Qd P |F t ] = E (− λ · M ) t withd M t = S ( t ) S ∗( t )γ f (V ( t )) 1 − ρ 2dW 1( t ) + ρ dW 2 ( t )d M , M Pt = S ( t )

2 S ∗( t )2γ f 2(V ( t )) dt

λ t = µ − r S ( t ) S ∗( t )2γ f 2 (V ( t ))

.

Therefore Z t = E [− µ − r

S ∗(t )γ f (V (t )) ( 1 − ρ 2dW 1( t )+ ρ dW 2( t ))].We start by deriving the hedging strategy for the claim H ( S ∗T )on the basis of the GKW decompostion.Step 1 We dene the Q -martingale according to (2). Bythe Markovian property, we know that at time t only dependson S ∗( t ) and V ( t ):

( t , S ∗( t ), V ( t ))

= e− r (T − t ) E Q [ H ( S ∗(T )) | S ∗( t ), V ( t )]. (6)

Applying Itô’s formula to this Q -martingale givesd ( t , S ∗( t ), V ( t )) = S ∗d ( S ∗)m, Q ( t )+ V V ( t )g( V ( t )) dW 2, Q ( t ),


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with W 2, Q the Q -Brownian motion originating from W 2 and

with ( S ∗)m , Q the Q -martingale part of S ∗with the following

dynamics

d ( S ∗)m, Q (t ) = S ∗(t )γ + 1 f (V (t ))

×

1 − ρ 2dW 1, Q ( t ) + ρ dW 2, Q (t ) ,

with W 1, Q the Q -Brownian motion originating from W 1 .

Step 2 I = since we work under the MMM Q .Step 3 The angle bracket process of S under Q is given byd S , S Q = S ( t )2 S ∗( t )2γ f (V ( t )) 2dt .The number of risky assets invested in S originating from theGKW decomposition is in view of 5 given by

ξ GKWt =d , S Qt

d S , S Qt = S

∗d ( S ∗)m, Q , S Qt + V S ∗( t )γ S ( t )g (V ( t )) f (V ( t )) V ( t )ρ dt

d S , S Qt

= S ∗ + ρV ( t )g (V ( t ))

S ∗( t )γ + 1 f (V ( t ))V

S ∗( t )S ( t )

. (7)

Hence, the amount we have to invest in S ∗ exactly equalsthe one given by Poulsen et al. (2009 ):

S ∗ + ρV ( t )g( V ( t ))

S ∗( t )γ + 1 f (V ( t ))V ,

where therst term is theamount wehave to investaccording tothe delta hedge and the second term can be seen as a correctionterm. Thus the hedging strategy in this case is (ξ GKW , η) withη = t − ξ GKWt S t .

Next, we determine the LRM strategy for the claim underthe original measure P .Step 1 This is the same as Step 1 above.

Step 2 The dynamics of the P -martingale part I of (6) are

d I ( t ) = S ∗d ( S ∗)m , P ( t ) + V V ( t )g( V ( t )) dW 2 ( t ),

where d ( S ∗)m , P ( t ) = S ∗(t )

S (t ) d M ( t ) .

Step 3 Taking the P -angle bracket of M with I gives

d I , M Pt = S ∗d ( S ∗)m , P , M Pt

+ V V ( t )g (V ( t )) d W 2 , M Pt ,

with d ( S ∗)m, P , M Pt =S ∗( t )S ( t )

d M , M Pt and

d W 2 , M Pt = S ∗( t )γ S ( t ) f (V ( t ))ρ dt .

Therefore the units of stock for the LRM hedging strategyequal

ξ FSt =d I , M Pt

d M , M Pt

= S ∗ + ρV ( t )g (V ( t ))

S ∗( t )γ + 1 f (V ( t ))V

S ∗( t )S ( t )

,

which is exactly the number found from the determination of the GKW decomposition given in ( 7). The LRM strategy is

(ξ FS

, η) with η = t − ξ FSt S t .

This demonstrates again the equality between the GKWdecomposition under the MMM and the FS decompositionunder the original measure in the continuous case.

We remark that we can easily extend this model in severalways. We can add for example jumps, a stochastic interestrate or use a vector to describe the stochastic volatility. Wewill not do this explicitly for this setting, because all thesestochastic volatility models can be seen as a special case of non-traded assets. Namely choose for F (1) the risky asset andfor ( F (2) , . . . , F (d ) ) the (vector of) stochastic volatilities, fur-thermore as weights we take the vector (1, 0, . . . , 0) . For anexample in which the LRM hedging strategy for the Batesmodel is calculated, we refer to Hubalek and Sgarra (2007 ).

4. Application to non-traded assets

The xed combination in which we can invest will be denotedby F := d i = 1 w

(i ) F (i ) , with F (i ) the forward prices. We

remark that the ltration F contains the information of thenon-traded assets with prices F (i ) , i = 1, . . . , d , as well astheinformationof thetraded assetwith price F ,inthesensethatthe ltration is generated by the underlying driving processes.By , we denote the price of the claim written on the non-traded assets under a martingale measure which we do notspecify here yet.

4.1. Strategy derived from the delta hedge

In practice, non-traded assets are often hedged using a strategybased on the delta hedge. We will use the intuitively obtainedhedging strategies to compare them with the LRM hedgingstrategy.

In this section, we restrict to the two-dimensional case forillustrations but the discussion is easily extended to higherdimensions.

In thestandard two-dimensionalcase, we trade ξ (i ) forwards

with price F (i), i = 1, 2, such that the risk originating from the

rate of change of the claim price with respect to these forwardprices equals

∂∂ F (1)

− ξ (1) d F (1) + ∂∂ F (2)

− ξ (2) d F (2) . (8)

This risk can be completely eliminated by choosing ξ (i ) equalto ∂∂ F (i ) , i = 1, 2.

For non-traded assets, we can only invest in ξ assets withprice F . It is impossible to eliminate the risk exposure com-pletely because the following equations should be satisedby ξ

ξ (1) = w (1) ξ and ξ (2) = w (2) ξ

and so we must search for the most optimal ξ . We give someintuitively-based solutions for ξ :


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• Total volume-neutral strategy:The number ξ to invest in F equals

ξ = max ∂

∂ F (1),

∂

∂ F (2).

• Volume-neutral strategy:The power market has two natural units of volumesince the commodity is delivered in a certain mag-nitude over a period of time. The magnitude is ex-pressed in MW and the time in hours. So instead of focusing on the MW position, one could also focuson the total volume, taking into account the lengthof the delivery period:

ξ = w (1) ∂∂ F (1)

+ w (2) ∂∂ F (2)

.

• Price-adjusted strategy:

ξ =

w (1) ∂∂ F (1) F (1) + w (2) ∂∂ F (2) F

(2)

w (1) F (1) + w (2) F (2) .

• Delta hedging with minimal risk exposure:Restricting ( 8) to the setting of non-traded assets andcalculatingthe differential of the portfolio consistingof the claim and ξ forwards with price F , we ndthe following:

∂∂ F (1)

− ξ w (1) d F (1) + ∂∂ F (2)

− ξ w (2) d F (2) .

Here we have left out the d t -part because this partis not risky.The variance of this remaining risk is in vectornotation:

var (ξ ) =∂∂ F

− ξ w d F , F P∂∂ F

− ξ w ,

where ∂∂ F is the gradient of , w is the vector con-taining the weights and F := ( F (1) , F (2) ) . We min-imize this variance to obtain the optimal ξ :

d var (ξ )d ξ

= − w d F , F P∂∂ F

− ξ w

−∂∂ F

− ξ w d F , F P w = 0.

Solving this equation for this ξ , gives

ξ =∂∂ F d F , F

P w

w d F , F P w.

This is exactly the result we will obtain when weapply the LRM hedging theory when is the priceof the claim under the MMM linked with F . So weachieved here an intuitive explanation for the rathercomplicated theory of local risk minimization. Weremark that we cannot follow blindly this intuitiveapproach, because in this way it is for example notclear why we have to use the MMM to price theclaim.We note that Poulsen et al. (2009 ) independentlymade an analogous conclusion.

These intuitively-based solutions will be used for compa-rison with the solutions to the local risk minimization.

4.2. (Adjusted) LRM hedging strategy

The theoretical determination of the LRM hedging strategyin this context is rather straightforward once one understandsproperly the existence conditions. A major concept for LRMhedging strategies is the MMM. Concentrating on our setting,it is important to understand that one must consider martingalemeasures with respect to the underlying asset which is usedfor hedging and NOT with respect to the assets on which theclaim depends. More concretely, we need to determinethe MMM of F and not of F (1) , . . . , F (d ) . This illustrates theincompleteness of the market we work in, even in the case withtwo driving Brownian motions (there is only one martingalemeasure to be determined for the vector (F (1) , F (2) ) frominnitely many for F ). Due to the results described in section

3.2 , the determination of the LRM hedging strategy for non-traded assets is reduced to a straightforward application of theformulas given there.

We distinguish two different cases: the continuous case andthe discontinuous case.

• In the continuous case, the LRM hedging strategyequals that of the RM hedging strategy under theMMM Q , see (4). Hence, the optimal position in thehedging asset F is given by

ξ =d , F Q

d F , F Q=

∂∂ F d F , F

Q w

w d F , F Q w.

To see this, apply Itô’s formula to relying on itscontinuity and martingale property.

• In the discontinuous case, we apply formula ( 3) andhence the optimal position in the hedging asset F isgiven by

ξ =d I , M P

d M , M P =

d I , F m P ww d F m , F m P w

, (9)

with I the P -martingale part of and where M and F m stand for the P -martingale part of F , respec-tively F .

To nd the explicit position in risky assets, we need to ll in thedynamics of the processes in the brackets. Due to the typicalsetting it is not possible to express the optimal amount in termsof the cumulant function by using Fourier transformation asHubalek et al. (2006 ) did,see Vandaele (2010 ) formore details.Therefore, we proceed in the following way:

If the forwards have the following dynamics under theoriginal measure P

d F (i )t = F (i )t α( i )dt + σ

(i ) dW (i )t , i = 1, 2

then by applying Itô’s formula to the expected price of theclaim, taken under the MMM of F = w (1) F (1) + w (2) F (2) , weobtain the dynamics of the P -martingale part I of :

d I t = f (1) σ (1)t F

(1)t dW

(1)t + f (2) σ

(2)t F

(2)t dW

(2)t ,


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where f (i ) stands for ∂∂ F (i ) , i = 1, 2. Inserting these

dynamics in ( 3) leads to the following value for ξ :

ξ =w (1) σ (1) F (1) f (1) σ

(1) F (1) + f (2) ρσ (2) F (2) + w (2) σ (2) F (2) f (1) ρσ

(1) F (1) + f (2) σ (2) F (2)

w (1) σ (1) F (1) 2 + 2w (1) w (2) ρσ (1) F (1) σ (2) F (2) + w (2) σ (2) F (2) 2 ,

where ρ represents the correlation between the two Brownianmotions W (1) and W (2) .

We apply this exact formula when F (1) and F (2) , and hencealso F , aremartingales under theoriginal measure. If F is not amartingale under theoriginal measure then claim pricesshouldbe calculated under the MMM linked with F . To this end, apartial differential equation (PDE) has to be solved. However,the determination of this claim price by solving a PDE at everytrading date is really time consuming. Therefore, we proposean adjustment of the amount ξ , which can be calculated fastwithout solving a PDE. Namely, we use here as martingale

measure not the MMM, but the unique martingale measurewhich ensures that both F (1) and F (2) become martingales.Hence, the optimal value for ξ in the adjusted LRM hedgingstrategy is given by

ξ =w (1) σ (1) F (1) (1) σ (1) F (1) + (2) ρσ (2) F (2) + w (2) σ (2) F (2) (1) ρσ (1) F (1) + (2) σ (2) F (2)

w (1) σ (1) F (1)2 + 2w (1) w (2) ρσ (1) F (1) σ (2) F (2) + w (2) σ (2) F (2) 2

, (10)

where the amounts (i ) := ∂∂ F (i ) are calculated using themartingale measure for F (1) and F (2) separately. In fact theyshould be calculated under the MMM related to F in the semi-

martingale case. Still we nd that this makes sense in oursetting, because after some time F (1) and F (2) will becomeboth liquidon themarketand then themartingale measure usedhere is the correct one. Furthermore as we will discuss later on,nding the correct drift is almost impossible, hence nding thecorrect minimal martingale measure is equally difcult.

In the presence of jumps in the dynamics of F (i ) , i = 1, 2,the claim price solves a partial integro differential equation(PIDE). So it becomes even more involved.Hence, we will alsouseformula ( 10) in thediscontinuous case, butthen ρ nolongerstands for the correlation between the two Brownian motions.Instead, ρ is the parameter used to express the correlation be-tween thetwo forwardswith prices F (1) and F (2) . Furthermore,the martingale measure for F (1) and F (2) is no longer uniquebecause of the incompleteness and we choose to work underthe mean-correcting measure (see e.g. Schoutens (2003 )) forF (1) and F (2) separately to calculate the (i ) := ∂∂ F (i ) .

5. Numerical results

The theory can successfully be applied to a wide variety of problems, such as the hedging of a strip of seasonly optionsin the gas market where there are 6 underlyings as mentionedin section 1 or the hedging of an option on peak power for

which one would execute the hedging with the base product.In the rst example, one has to consider 6 different options,each with its own expiry, its own underlying price and henceeach with its own hedging ratio. The underlying quantity one

would hedge with, would be the average of the 6 underlyingcontracts ( 1), i.e. the season contract.

The critical timepoints in this framework are the points intime where the underlying monthly forwards become trade-able, the moment where the liquidly traded average forwardexpires as well as the expiries of the individual options con-tracts in the strip. In practice, right after the rst option ex-pires, the average contract with price F , as well as the rstmonthly forward with price F (1) will expire. The market thencascades into having a liquid universe of 2 monthly forwardsF (2) and F (3) , as well as a quarterly forward that consists of the (weighted) average of the last 3 monthly contracts in the

strip.Although perfectly sensible, the numerical example will

require the determination of more variables, in particular the

6 volatilities and 15 pair-wise correlations. Moreover, the cor-relation structure for such gas contracts typically changes aseach contract is about to expire because the weather forecasts

have a 2-week horizon and the impact on price for near-to-delivery forwards is crucial. One can then observe a signicantdecorrelationof thiscontract.The LRMframeworkstillappliesand is in fact a very powerful technique.

However, the numerical illustration is a lot simpler andclearer when we focus on the electricity forwards where baseand peak contracts are for delivery in the same period anddo not have different expiries. We restrict ourselves to a set-ting in which the claim is depending on F (1) but the hedgingwill be done with F = w(1) F (1) + w (2) F (2) . We investi-gate the hedging for an at-the-money call on F (1) as this isthe most challenging example. It is well known that hedgingof far in-the-money or out-of-the-money options is relativelyeasy because the rebalancing frequency is low. The depen-dency of such options on the underlying forward is either com-plete (in-the-money) or weak (out-of-the-money). The changeper time-step of the required hedging ratios is hence verysmall.

Note that both F (1) and F (2) are contracts for delivery overdifferent periods of time, while the delivery period of F coversboth periods. The total premiums should always be adjustedto the delivery period. We take the example of the base/peak problem, see section 1. This means that theweights are roughlyspeaking w (1) = 1/ 3 for peak and w (2) = 2/ 3 for off-peak power. The cashow corresponding to a purchase or sale of

such a contract is w(i )

F (i )

to adjust correctly for the deliveryperiod.Hence, it is clear that buying the base contract F delivers

power during the peak and off-peak hours, corresponding to a


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position in both assets F (1) and F (2) . This means that there aretwo intuitively choices for hedging the claim on peak powerF (1) by using base. One could try and focus on the volume risk or on the price risk, see section 4.1.

5.1. Set-upWe introduce the following notations: ( F (1) ) stands for theprice of the claim while (1) = ∂∂ F (1) and

(2) = ∂∂ F (2)

represent its partial derivatives with respect to the peak andoff-peak contract prices. Note that in our example the optiononly depends on F (1) and thus (2) = 0. The amount of riskyassets, that are used as a hedge for the claim, is denoted by ξ .For convenience, we will assume that the interest rate r = 0.We will observe the expected value and the standard deviationof the total cost, which is dened as the sum of the initial costand the nal deviation of the hedging portfolio from the optionin case of a self-nancing portfolio. Hence, the L2-hedging

error, which is often calculated in literature, equals the sum of the variance of the total cost with the square of the differencebetween the mean of total cost and the initial cost. For a non-self-nancing portfolio the total cost is dened as the sum of all the costs on every rebalance date. Given the specic natureof this problem, we assume a lifetime of the claim of T = 3years, where for the rst T 1 = 0.5 year, a strategy in the baseasset is followed. This is inspired by the fact that at some point,liquidity grows in the peak contract. We call the time T 1 theroll-over point. We assume that after this time, the claim canbe hedged further with a classical delta hedge or any otherhedging strategy on the asset F (1) itself. In a Brownian settingfrom this point onwards we will hedge perfectly and there isno need in an analysis beyond this point.

The price of the claim at time zero is such that the expectedtotal cost of the strategy is zero, where the price of the option atroll-over time is determined under the unique martingale mea-sure in the Brownian motion case, while in the discontinuouscase the mean-correcting martingale measure is used. Due tothe zero interest rate, we can restrict ourselves to observingthe total cost over the lifetime at the roll-over time. This costof hedging will be neutralized by the initial premium of theclaim. We will show that the uncertainty over the outcomeof the different strategies is quite large and therefore we willalso study the standard deviation of the hedging cost in those

different strategies. The one with the lowest variance is clearlyto be preferred in practice.

Both the peak and the off-peak contracts are assumed tofollow a geometric Brownian motion:

F (i )t = F (i )0 exp µ

(i ) t + σ (i ) W (i )t , i = 1, 2

where the correlation between the Brownian motions is givenby d W (1) , W (2) t = ρ dt .

As parameters we choose σ (1) = 40%, σ (2) = 30% andρ = 75%. For the drift we look at two different situations. Therst and most easy one is where we assume both assets to bemartingales. Hence µ (i ) = r − 0.5(σ (i ) )2 and we will call this

the martingale case. In a second example, the semimartingalecase, we introduce a drift by setting µ (1) = 0.07 and µ (2) =0.05. For both cases, we will look at the performance of thedifferent strategies.

As starting levels for the prices, we assume that F (1) =e 90/MWh and F (2) = e 60/MWh, and hence the base assetis worth F = e 70/MWh. If we normalize the time of thebase contract to one, the cash ows would be given by e 70for baseload of which e 30 is coming from the peak contractand e 40 from the off-peak. Note that although the price foroff-peak is lower, the total cashow of the off-peak power ishigher compared to the one of the peak contract because theamount of delivered hours during off-peak is higher.

5.2. Different strategies

In this section, we repeat the strategies described in section4.1 and section 4.2 , but adjusted to the setting described here,namely where the claim only depends on F (1) . The controlstrategy, which we cannot follow in practice, is added.

5.2.1. Control strategy. In order to verify our results, wecalculate the classical strategy. This means that we are hedgingthe claim on F (1) by effectively taking positions in this asset.

5.2.2. Totalvolume-neutralstrategy. Thenumber ξ is hereequal to

ξ = max ( (1) , (2) ) = (1) .

We basically focus on the total volume of the peak contract.If the derivative of the claim with respect to F (1) requiresa certain amount in F (1) , this same amount is taken in F ,

ensuring that the volume during the peak hours is correct.However, the residual risk that comes into the picture, is thevolume taken in the off-peak asset.

In fact, the volatility of the off-peak asset is lower, andtherefore ignoring this asset is safe. Clearly, the risk in thisstrategy is coming mostly from the second risky asset F (2) .

5.2.3. Volume-neutral strategy. The optimal amount ξ isgiven by

ξ = w (1) (1) + w (2) (2) = w (1) (1) .

In this strategy, it is assumed that if we need 3MW of peak power, one can replace this by 1MW of base power, becausethe total amount of power over the delivery period is thenroughly the same. Or in other words, it is assumed that we canreplace volume in the peak hours by volume in the off-peak hours.

It will become clear that this is the worst strategy.

5.2.4. Price-adjusted strategy. If we take

ξ =w (1) (1) F (1) + w (2) (2) F (2)

w (1) F (1) + w (2) F (2) =

w (1) (1) F (1)

F ,

the value of the hedge in F and the value of the (theoretical)hedge in F (1) areequal.This ensures that thecash-owsduringthe hedging strategy are the ones one would have from thehedging strategy in F (1) .


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5.2.5. Risky strategy. If one wants to fully understand theconcept of hedging, one should always be prepared to take onestep back and ask oneself if the riskyness really decreased bysetting up a strategy. Therefore, we compare the strategies tothe strategy of doing nothing and waiting until the roll-overtime before starting to hedge the claim. In this case, the fullrisk is taken and ξ = 0.

5.2.6. Adjusted LRM hedging strategy. In none of theabove strategies, the volatility or correlation between F (1) andF (2) played a role. It is however very natural that this shouldhave an effect on thestrategy oneshould follow. The(adjusted)LRM strategy captures this completely, see section 4.2 andin fact outperforms all of the above strategies. In this settingwhere (2) = 0, the optimal amount equals

ξ =σ (1) F (1) (1) (w (1) σ (1) F (1) + w (2) σ (2) F (2) ρ)

w (1) σ (1) F (1)2 + 2w (1) w (2) ρσ (1) F (1) σ (2) F (2) + w (2) σ (2) F (2) 2

.

5.3. Results

To obtain the results, we simulated 25 000 paths and rebalancetwice a week.

5.3.1. Hedging cost. For the various strategies, we deter-mine the hedging cost up to the roll-over time. We look atthis cost both for the martingale case as well as for the semi-

martingale case. Table 1 contains for each case the expectedcost and the standard deviation between brackets. The largerthis standard deviation, the more uncertainty and hence themore risk remains in the hedging procedure. Let us focus rston the martingale case. It is obvious that the LRM strategyoutperforms the current market practices. Compared to doingnothing, hence this is the full risk case, the improvement isvery good. The reason that the total volume-neutral strategyworks well is because a big part of the risk is concentrated inthe peak price since this contract has the highest volatility.

In practice, the hedging cost is considered as the fair valueprice of the option. From table 1, we deduce that the averagecost of hedging is almost identical across all the strategies,hence each strategy indicates the same fair value price for theoption at the start.

In the semimartingale case, we can observe that for thecontrol strategy, there is no effect. This is natural as we al-ready know that pricing is always done under a risk-neutralmartingale measure, which is unique in the continuous case.For all the other strategies, we see that the cost of the strategyis changing. At the same time, the uncertainty grows as well.However, once again the LRM behaves better than any of theothers.

Remark that in practice, the estimation of the drift term isvirtually impossible. Knowing the drift would mean, knowing

where the prices would go and often it might be possibleto distinguish trends in the short or extremely long run, butthe deviations from these kind of trends make it very hard toeven estimate the drift term correctly. Since the energy market

is a forward market, we can assume that the market priceseverything correctly, and hence that the quantities are indeedmartingales. If later, it turns out that there was a systematicdrift, we then hope that the margin taken at inception in theoption premium is sufcient to cover this.

We conclude that the LRM hedging strategy outperformsthe more intuitive approaches and even in case the assets areonly semimartingales, the method still works well.

In fact, we could even go one step further. We want tocalculate the cost of hedging in case the underlyings followa discontinuous price process, but in a very fast way and henceby avoiding again the use of PIDE’s. Therefore, we use theamount ξ described in ( 10), where the price of the claim andthe ’s are calculated under the mean-correcting martingalemeasure linked with the two processes F (1) and F (2) . For

this purpose, we assume that both peak and off-peak can bewritten as exponential variance gamma processes, where thecharacteristic function of a variance gamma function equals

φ( z) = 1 − i uθν +12

σ 2ν u2− 1/ν

,

see Schoutens (2003 ). As in Leoni and Schoutens (2008 ), weassume that the Gamma clock is equal for both assets andhence they jump at the same time. We will take the followingparameters:

µ (1)mart = − 0.0179 µ (2)mart = − 0.005µ (1)semi = 0.07 µ

(2)semi = 0.05

σ (1) = 40 .50% σ (2) = 30%θ (1) = − 0.10 θ (2) = − 0.05

ν = 0.25ρ = 74 .80%

These numbers ensure us that the option price, calculatedunder the mean-correcting measure for peak, leads to the sameprice as we had in the Brownian case. The correlation betweenthe Brownian components has been adjusted downwards suchthat the linear correlation coefcient between the logreturns of the assets remained around 75% as earlier. Furthermore, µ (i )in themartingale case is determined by the following equation:

µ (i )mart = r − log(1 − 0.5ν(σ (i ) )2 − θ (i ) ν)/ν, i = 1, 2.

Within this set-up, we obtain the results reported in table 2.In the martingale case, all the strategies have a lower cost of hedging, but with a greater uncertainty than in the Brownianmotion case. This can be explained by the fact that within a VGmodel, there are only small changes in the prices of the assetsuntil a signicant jump is noticeable.Thefat-tailed distribution(compared to the normal distribution) favours smaller movesmost of the time and some extreme jumps once in a while.

The interesting aspect of this analysis is that the controlstrategy becomes less good in the sense that the uncertaintybecomes bigger in the semi-martingale case. This can beexplainedeasily bynotingthat theintroduction of jumps induce


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Table 1. Hedging cost and standard deviation in the case of Brownian motions.

Strategy Martingale case Semimartingale case

Control 24.39 (0.39) 24.39 (0.40)Total volume-neutral 24.37 (8.24) 26.06 (9.30)Volume-neutral 24.34 (13.94) 27.84 (15.77)Price-adjusted 24.35 (12.95) 27.57 (14.59)(Adjusted) LRM 24.38 (5.95) 24.67 (6.33)Full risk 24.33 (17.15) 28.73 (19.38)

Table 2. Hedging cost and standard deviation in the case of a multivariate variance gamma process.

Strategy Martingale case Semimartingale case

Control 24.39 (2.41) 24.39 (2.43)Total volume-neutral 23.88 (9.08) 24.81 (9.49)Volume-neutral 23.63 (14.74) 25.54 (15.40)Price-adjusted 23.67 (13.74) 25.44 (14.33)(Adjusted) LRM 24.06 (6.65) 24.25 (6.97)

Full risk 23.51 (17.94) 25.91 (18.75)

a higher variance in the hedging process if the rebalancingfrequency remains equal.

We canalso observe that theimpact, at least relatively speak-ing on the other strategies of the introduction of such jumps isless important and the uncertainty remains completely drivenby the basis risk while hedging. Turning this around, one canalso understand that increasing the hedging frequency will noteliminate the variance in the RM strategy as one cannot expecttheerror to becomeinnitely small by just hedging more often.The variance or risk is introduced by the liquidity constraintand the fact that one chooses to hedge with a correlated, butdifferent asset.

When we turn to the semimartingale case, we can deducesimilar results as before. The cost of hedging depends on theactual drift of the process and in general thisis not a nicefeatureof a strategy because this drift is extremely hard to measure orestimate. However, it becomes clear in this case as well, thatthe LRM strategy is rather robust, making it the most suitablecandidate for real hedging of claims on non-tradable assets.

In fact, one could also argue that we do not know what thereal LRM hedging strategy would do and if this would not

behave even better. We think the real strategy will surely notperform better than in the continuous martingale case, and ouradjusted LRM hedging strategy only performs slightly worsethan the exact LRM hedging strategy used in the continuousmartingale case. Hence,we believe that thepossible increase inaccuracy of the results will not outweigh for the loss in com-putational speed. Furthermore, the reason why this adjustedstrategy works well, even in the discontinuous case, is becauseit still accurately captures the real correlation between the twoassets, although it does not use the exact angle bracket process.The correct LRM however, would require a transformed driftprocess, which would shift the expected cost of hedging (andhence the price of the claim), but the effect on the variancewould be smaller. For small drift terms, the approximationworks well enough for practical purposes.

It should be clear that the LRM theory can be applied to anycomplex process andnot just a variance gamma drivenprocess.Typically, more complex processes can provide better modelsbut the number of parameters will increase. Moreover, thevariance gamma process has the convenient property that thecovariance terms can be explictly written down, see Leoni andSchoutens (2008 ), eliminating a numericalcalibration.

6. Conclusion

Thefocus in this paper wasto nda numerically applicable andfast method to hedge options on non-traded assets. Thereto, werst derived theoretically the formulas for the LRM hedgingstrategy for this non-traded asset case. This theoretical resultmotivated the denition of the adjusted LRM hedging strategyas an approximation to the exact one. In the numerical part,we showed that this adjusted strategy outperforms the currentmarket practices in the Brownian motion case as well as in thevariance gamma case.

Acknowledgements

Nele Vandaele gratefully acknowledges the nancial supportby the Research Foundation-Flanders. The authors thank twoanonymousreferees and theeditor for the valuable suggestionsthat improved the paper considerably.

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