dry markets and statistical arbitrage bounds for european derivatives

8/8/2019 Dry Markets and Statistical Arbitrage Bounds for European Derivatives

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Dry Markets and Statistical Arbitrage Bounds for

European Derivatives

J o~ao Amaro deMatosFaculdade de Economia

Universidade Nova de LisboaCampus de Campolide

1099-032 Lisboa, Portugal

Ana LacerdaDepartment of Statistics

Columbia University1255 Amsterdam Avenue

New York, NY 100027, USA

J anuary 2, 2006

A bstract

We derive statistical arbitr age bounds for the buying and selling

price of European derivativesunder incomplete markets. In this paper,

incompleteness is generated due to the fact that the market is dry, i.e.,

the underlying asset cannot betransacted at certain points in time. In

particular, we re¯ne the notion of statistical arbitrage in order to ex-

tend the procedure for the case where dryness is random, i.e., at each

point in time the asset can be transacted with a given probability. Weanalytically characterize several properties of the statistical arbitrage-

free interval, show that it is narrower than the super-replication in-

terval and dominates somehow alternative intervals provided in the

lit erature. Moreover, we show that, for su±ciently incomplete mar-

kets, the statistical arbitrage interval contains the reservation price of

the derivative.

1 I ntr oduction

In complete markets and under the absence of arbitrage opportunities, thevalue of a European derivative must be the sameas the cheapest portfoliothat replicates exactly its valueat any given point in time. However, in thepresence of some market imperfections, markets may become incomplete,and it is not possibleto exactly replicatethevalueof theEuropeanderivativeat all times anymore. Nevertheless, it is possible to derive an arbitrage-freerange of variation for the value of the derivative. This interval depends on

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two di®erent factors. First, on thenatureof market incompleteness; second,

on thenotion of arbitrage opportunities.I n what follows we consider that market incompleteness is generated

by the fact that agents cannot trade the underlying asset on which thederivative is written whenever they please. In fact, and as opposed to thetraditional asset pricing assumptions, markets are very rarely liquid andimmediacy is not always available. As Longsta®(1995, 2001, 2004) recalls,the relevance of this fact is pervasive through many ¯nancial markets. Themarkets for many assets such as human capital, business partnerships, pen-sion plans, saving bonds, annuities, trusts, inheritances and residential realestate, among others, are generally very illiquid and long periods of time(months, sometimes years) may be required to sell an asset. This point

becomes extremely relevant for the case of option pricing when we considerthat it is an increasingly common phenomenon even in well-established se-curities markets, as illustrated by the 1998 Russian default crisis, leadingmany traders to betrapped in risky positions they could not unwind.

To address the impact of this issue on derivatives' pricing, we considera discrete-time setting such that transactionsare not possible within a sub-set of points in time. Although clearly very stylized, the advantage of thissetting is that it incorporates in a very simple way the notion of marketilliquidity as the absence of immediacy. Under such illiquidity we say thatmarkets are dry. I n this framework, dryness changes what is otherwise acomplete market into a dynamically incomplete market. T his was also the

approach in Amaro de Matos and Ant~ao (2001) when characterizing thespecī c superreplication bounds for options in such markets and its impli-cations. We further extend this setting by assuming that transactions occurat each point in time with a given probability, re°ecting a more realisticex-ante uncertainty about themarket.

As stressed above, there is not a uniquearbitrage-free value for a deriva-

tive under market incompleteness. However, for any given derivative, port-folioscan befound that havethesamepayo®as thederivativein somestatesof nature, and higher payo®s in the other states. Such portfolios are said tobe superreplicating. Holdingsuch a portfolio should beworth morethan theoption itself and therefore, thevalueof thecheapest of such portfolios should

be seen as an upper bound on the selling value of the option. Similarly, alower bound for the buying price can be constructed. The nature of thesuperreplicating bounds is well characterized in the context of incompletemarkets in the papers by El Karoui and Quenez (1991,1995), Edirisinghe,Naik and Uppal (1993) and K aratzas and Kou (1996). T he superreplicat-ing bounds establish the limits of the interval for arbitrage-free value of the

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option. I f the price is outside this range, then a positive prō t is attain-

able with probability one. T herefore, the equilibrium prices at which thederivative is transacted should lie between those bounds.

Most of the times, however, these superreplicating boundsare trivial, inthe sense that they are too broad, not allowing a useful characterization of equilibrium prices' vicinity. As an alternative, Bernardo and Ledoit (2000)propose a utility-based approach, restricting the no-arbitrage condition torule out investment opportunities o®ering high gain-loss ratios, where gain(loss) is the expected positive (negative) part of excess payo®. In this way,narrower bounds are obtained. Analogously, Cochrane and Sa¶a-Requejo(2000) alsorestrict theno-arbitragecondition bynot allowing transactionsof \ good deals", i.e. assets with very high Sharpe ratio. Following Hansen and

J agannathan (1991), they show that thisrestrictionimposes an upper boundon the pricing kernel volatility and leads to narrower pricing implicationswhen markets are incomplete.

Given a set of pricing kernels compatible with the absenceof arbitrageopportunities, Cochrane and S¶aa-Requejo excludepricing kernels implyingvery high Sharpe ratios, whereas Bernardo and Ledoit exclude pricing ker-nels implying very high gain-lossratios for a benchmark utility. Noticethat,for a di®erent utility, BernardoandLedoit would excludeadi®erent subset of pricingkernels, for the samelevels of acceptablegain-lossratios. Alsonoticethat the interval designed by Cochrane and S¶aa-Requejo is not necessarilyarbitrage free, and therefore does not necessarily contain the equilibrium

price.I n order to avoid ad-hoc thresholds in either Sharpe or gain-loss ratios,

or to makesome parametric assumptions about a benchmark pricing kernel,asin Bernardo and Ledoit (2000), thework of Bondarenko (2003) introducesthe notion of statistical arbitrage opportunity, by imposing a weak assump-tion on a functional form of admissible pricing kernels, yielding narrower

pricing implicationsas compared to the superreplication bounds. A statisti-cal arbitrage opportunity is characterized as a zero-cost trading strategy forwhich (i) the expected payo®is positive, and (ii) the conditional expectedpayo®in each ¯nal state of the economy is nonnegative. Unlike a pure arbi-trageopportunity, a statistical arbitrageopportunity may allow for negative

payo®s, provided that theaverage payo®in each ¯nal state is nonnegative.In particular, ruling out statistical arbitrage opportunities imposes a novelmartingale-type restriction on thedynamics of securitiesprices. T hei mpor-tant propertiesof the restriction are that it is model-free, in the sense thatit requires no parametric assumptions about the true equilibrium model,and continues to hold when investors' beliefs are mistaken. Although Bon-

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darenko's interval can be shown to be in the arbitrage-free region, it does

not necessarily contain the equilibrium value of the derivative.I n this paper we extend thenotion of statistical arbitrageopportunity to

the case where the underlyingasset can be transacted at each point in timewith a given probability, and compare the statistical arbitrage-free boundswith thesuperreplication bounds. Weshow that thestatistical arbitrage-freeinterval is narrower than thepure arbitrage bounds, and show alsothat, forsu±ciently incompletemarkets(probability not toocloseto 1), thestatisticalarbitrage interval contains the reservation price of the derivative. We alsoprovide examples that allow comparison with the results of Cochrane andSa¶a-Requejo (2000) and discuss the comparison with Bernardo and Ledoit(2000).

T his paper is organized as follows. In section 2, the model is presentedand the pure arbitrage results are derived. In section 3 the notion of sta-tistical arbitrage in the spirit of Bondarenko (2003) is dē ned. In section4, the main results are presented. In Section 5 we ¯rst characterize thereservation prices and then show that, in a su±ciently dry market, they arecontained in the statistical arbitrageinterval. I n Section 6 we illustrate howthe statistical arbitrage-free interval somehow dominates alternative inter-vals provided in the literature. I n section 7 some numerical examples arepresented in order to illustrate some important properties of the bounds. Inthe last section several conclusions are presented.

2 T he M odel

Consider a discrete-time economy with T periods, with a risky asset anda riskless asset. At each point in time the price of the risky asset can bemultiplied either by U or by D to get the price of the next point in time.

Equili briumrequiresthat U > R > D, where R denotesoneplustherisk-freeinterest rate. At time t = 0 and t = T transactions are certainly possible.However, at t = 1;:::;T ¡ 1 there is uncertainty about the possibility of transaction of the risky asset. Transactions will occur with probability pat each of these points in time. A European Derivativewith maturity T isconsidered.

Consider the Binomial tree process followed by the price of the riskyasset. Let the set of nodes at date t be denoted by I t; and let each of thet +1 elements of I t be denoted by i t = 1;::: ;t + 1. For any t0< t, let I

itt0

denote the set of all the nodes at time t0

that are predecessors of a givennode it: A path on the event tree is a set of nodes w = [ t2f 0;1;:::;Tgit such

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that each element in the union satis̄ es it¡ 1 2 Iitt¡ 1: Let denote the set of

all paths on the event tree: T he payo®s of a European derivative, at each terminal node, will be de-

noted Gi T T : At each nodeit; thestock priceis given by Si tt = U

t+1¡ it Dit ¡ 1S0:Moreover, at each nodeit; thereis a number ¢

i tt representing the number of

shares bought (or sold, if negative), and a number Bi tt denoting theamountinvested (or borrowed, if negative) in the risk-free asset. Hence, at t thereare t + 1 values of ¢ i tt , composing a vector ¢ t ´ (¢

1t ;::: ;¢

t+1t ) 2 R

t+1:Similarly, weconstruct the vector B t ´ (B1t ;:: : ;B

t+1t ) 2 R

t+1:

Dē nition 1 A trading strategy is a portfolio process µt = (¢ t; Bt) ; com-posedof ¢ t units of the risky asset andan amount Bt invested in the riskless

asset, such that the portfolio's cost is ¢ tSt + Bt for t = 0;1;:::T ¡ 1:

I n order to ¯nd the upper (lower) bound of the arbitrage-free rangeof variation for the value of a European derivative we consider a ¯nancialinstitution that wishes to befully hedged when selling (buying) that deriva-

tive. T heobjective of theinstitution is to minimize (maximize) thecost of replicating the exercise value of thederivativeat maturity. T hevalue deter-mined under such optimization procedure avoids what is known asarbitrageopportunities, re°ecting the possibility of certain pro¯ts at zero cost.

T his section is organized as follows. We ¯rst characterize the upperbound, and then the lower bound for the interval of no-arbitrage admissi-

ble prices. For each bound, we ¯rst deal with the complete market case,and then with the fully incomplete market case, ¯nally introducing randomincompleteness.

2.1 T he upper bound in the case p = 0 and p= 1:

First, we present the well-known case where p= 1. The usual dē nition of

an arbitrage opportunity in our economy is as follows.

Dē nition 2 (P ure A rbitrage i n the case p= 1) I n this economy, anarbitrage opportunity is a zero cost trading strategy µt such that

1. thevalueof the portfolio is positive at any ¯nal node, i.e., ¢i T ¡ 1

T¡ 1Si T

T +

RBi T ¡ 1

T¡ 1 ¸ 0; for any i T¡ 1 2 Ii T

T¡ 1 and all i T 2 I T ; and

2. the portfolio is self-¯ nancing, i.e., ¢ i t¡ 1t¡ 1Si tt + RB

it¡ 1t¡ 1 ¸ ¢

i tt S

itt +B

i tt ;

for any it¡ 1 2 Ii tt¡ 1; all it 2 I t and all t 2 f 0;:::;T ¡ 1g:

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T he upper bound for the value of the European option is the maximum

value for which the derivative can be transacted, without allowing for ar-bitrage opportunities. This is the value of the cheapest portfolio that theseller of the derivative can buy in order to completely hedge his positionagainst the exercise at maturity, without theneed of additional ¯nancing atany rebalancing dates. Hence, for p = 1; the upper bound is C1u; given by

C1u = minf ¢ t;Btgt=0;::::;T ¡ 1

¢ 0S0+ B0

subject to ¢i T ¡ 1 T ¡ 1S

i T T + RB

i T ¡ 1 T¡ 1 ¸ G

i T T ; with i T ¡ 1 2 I

i T T¡ 1 and all i T 2 I T ;

and the self-¯nancing constraints ¢ i t¡ 1t¡ 1 Sitt + RB

i t¡ 1t¡ 1 ¸ ¢

itt S

itt + B

i tt ; for all

it¡ 1 2 Ii tt¡ 1; all it 2 I t and all t 2 f 0;:::;T ¡ 1g; where theconstraints re°ect

the absence of arbitrage opportunities. This problem leads to the familiarresult

C1u = 1

R T

TX

j=0

µ T

j

¶ µR ¡ D

U ¡ D

¶ j µ U ¡ RU ¡ D

¶ T¡ jG T+1¡ j T :

Consider now the case where p = 0. I n this case, the notion of a tradingstrate#i T gy satisfying the self-̄ nancing constraint is innocuous, since theportfolio µt cannot be rebalanced during the life of the option. Under theabsence of arbitrage opportunities, the upper bound for the value is C 0usatisfying

C0

u = minf ¢ 0;B0g¢ 0S0 +B0

subject to ¢ 0Si T

T + R T ¢ 0 ¸ G

i T T ; for all i T 2 I T : In this case, the bound

C0u can be shown to solve the maximization problem on a set of positiveconstants f #i T g; with

P T +1i T =1

#i T = 1;

C0u = max#i T

1

R T

X T+1i T =1

#i T Gi T T

subject to

S0 = 1

R T

X T+1i T =1

#i T Si T

T ;

For instance, if a call option with exerciseK is considered, we have1

C0u = 1

R T

· µR T ¡ D T

U T ¡ D T

¶¡

U T S0 ¡ K ¢+

+

µU T ¡ R T

U T ¡ D T

¶¡

D T S0 ¡ K ¢+

¸:

1See Amaro de Matos and Ant~ao (2001).

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2.2 T he lower bound in the case p = 0 and p= 1:

The lower bound for the value of an American derivative is the minimumvalue for which the derivative can be transacted without allowing for arbi-trageopportunities. T his is the value of themost expensive portfolio thatthe buyer of theoption can sell in order to be fully hedged, and without theneed of additional ¯nancing at rebalancing dates.

For p = 1, the lower bound for the value of the derivative under the

absence of arbitrageopportunities is thus C1l ; given by

C1l = maxf ¢ t;Btgt=0;::::;T ¡ 1

¢ 0S0+ B0

subject to ¢ i T ¡ 1 T ¡ 1Si T T + RBi T ¡ 1 T¡ 1 · Gi T T , with i T¡ 1 2 I i T T¡ 1 and all i T 2 I T ;and the self-¯nancing constraints ¢ i t¡ 1t¡ 1 S

itt + RB

i t¡ 1t¡ 1 · ¢

itt S

itt + B

i tt ; for all

it¡ 1 2 Ii tt¡ 1; all it 2 I t and all t 2 f 0;:::;T ¡ 1g; where theconstraints re°ect

the absence of arbitrage opportunities T his problem leads to the familiar

result

C1l = 1

R T

TX

j =0

µ T

j

¶ µR ¡ D

U ¡ D

¶ j µU ¡ R

U ¡ D

¶ T ¡ jG T+1¡ j T ; (1)

that coincides with the solution obtained for C1u:I n the case where p = 0; the lower bound for the value of the derivative

is C0l ; satisfying

C0l = maxf ¢ 0;B0g

¢ 0S0 +B0

subject to¢ 0S T +R T B0 · G

i T T : Asabove, it followsthat, for aset of positive

constants f #i T g; withP T +1

i T =1#i T = 1; this bound is given by

C0u = min#i T

1

R T

X T+1i T =1

#i T Gi T T

subject to

S0 = 1

R T

X T +1i T =1

#i T Si T

T :

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I n the case of a call option with exercise K ,we have2

C0l = 1

R T

ÃR T ¡ U T¡ (i+1)D i+1

U T¡ i Di ¡ U T¡ (i+1)D i+1

! ¡U T¡ i Di S0 ¡ K

¢+(2)

+ 1

R T

µU T ¡ i Di ¡ R T

U T¡ iDi ¡ U T¡ (i+1)Di+1

¶ ¡U T¡ i¡ 1D i+1S0 ¡ K

¢+:

where i is dē ned as the unique integer satisfying Un¡ (i+1)D i+1 < Rn <

Un¡ iD i, and 0· i · n ¡ 1:

2.3 T he B ounds on P robabilistic M arkets

In the aforementioned cases we considered the cases where either p = 0 orp= 1. However, if p is not equal to neither 0 nor 1, theformulation has tobe adjusted. I f the risky asset can be transacted with a given probabilityp2 (0;1), then theusual dē nitionof arbitrageopportunity readsasfollows.

Dē nition 3 (P ure A rbitrage for p2 (0;1)) I n this economy, an arbi-trage opportunity is a zero cost trading strategy such that

1. the value of the portfolio is positive at any ¯nal node, i.e.,

¢ i tt Si T

T + R T¡ tB itt ¸ 0;

it 2 I

i T

t and all i T 2 I T ; and the self-̄ nancing constraints2. the portfolio is self-¯ nancing, i.e.,

¢i t¡ jt¡ j S

itt + R

j Bi t¡ jt¡ j ¸ ¢

itt S

itt + B

i tt ;

for all i t¡ j 2 Iitt¡ j ; all i t 2 I t and all t 2 f 0;:::;T ¡ 1g:

T he upper bound Cpu is thesolution of the following problem:

C pu = minf ¢ t ;Btgt=0;:::;T ¡ 1

¢ 0S0 +B0

where ¢ t;B t 2 R t+1;t = 0;:::; T ¡ 1; subject to the superreplicating condi-tions ¢ i tt S

i T T + R

T¡ tB i tt ¸ Gi T T ; with it 2 I

i Tt and all i T 2 I T ; and theself-

¯nancing constraints ¢it¡ jt¡ j S

itt +R

j Bi t¡ jt¡ j ¸ ¢

i tt S

itt +B

i tt for all it¡ j 2 I

i tt¡ j ;

all i t 2 I t and all t 2 f 0;:::; T ¡ 1g:On the other hand, the lower bound Cpl solves the following problem:

2See Amaro de Matos and Ant~ao (2001).

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C pl = maxf ¢ t ;Btgt=0;:::;T ¡ 1

¢ 0S0 +B0

where ¢ t; Bt 2 Rt+1;t = 0;:::;T ¡ 1; subject to the conditions ¢i tt S

i T T +

R T¡ tBitt · Gi T T ; with i t 2 I

i Tt and all i T 2 I T ; and the self-̄ nancing con-

straints ¢i t¡ jt¡ j S

i tt +R

j Bi t¡ jt¡ j · ¢

i tt S

itt +B

itt for all i t¡ j 2 I

itt¡ j ; all i t 2 I t and

all t 2 f 0;:::; T ¡ 1g:Noticethat the constraints in the above optimization problems are im-

plied by the absence of arbitrage opportunities and do not depend on theprobability p:3 Therefore, neither C pu nor C

pl will depend on p: We are now

in conditions to relatethese values to C0

u and C0

l as follows.

T heorem 4 For p 2 (0; 1) theupper and lower bound for the pri ces abovedonot depend on p: T he optimization problems above lead to the same solutionsas when p = 0:

P roof. Consider ¯rst the case of the upper bound. T he constraintscharacterizing Cpu includeall the constraints characterizing C0u: T hus, C

pu ¸

C0u. Now, let ¢00 and B

00 denote theoptimal valuesinvested, at time t = 0;

when p = 0. The trading strategy ¢ pt = ¢00 and B

pt = R

tB00, for allt = 1;:::;T ¡ 1, is an admissible strategy for any given p, hence Cpu = C0u:

The case of the lower bound is analogous.

T heintuition for this result is straightforward. Theupper (lower) boundof the European derivative remains the same as when p = 0; because withprobability 1 ¡ p it would not be possible to transact the stock at eachpoint in time. I n order to be fully hedged, as required by the absence of arbitrageopportunities, the worse scenario will be restrictive in spite of itspossibly low probability. T he fact that no intermediate transactions mayoccur dominates all other possibilities.

T he above result is strongly driven by the dē nition of arbitrage oppor-tunities. Nevertheless, if this notion is relaxed in an economic sensible way,a narrower arbitrage-free range of variation for the value of the European

derivativemay beobtained, possibly dependingnow on p: T hisis thesubjectof therest of thepaper.

3 T his happens since, in order to have an arbitrage opportunity, we must ensure that,whether market exists or not at each time t 2 f 1;:::;T ¡ 1g; the agent will never losewealth. T herefore, the optimizati on problem cannot depend on p:

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3 Statistical A rbitrage O pportunity

Consider theeconomydescribedin theprevioussection. Let Tp = f 1;:::;T ¡ 1gdenote the set of points in time. At each of thesepoints there is market withprobability p; and there is no market with probability 1¡ p: The existence(or not) of the market at time t corresponds to the realization of a randomvariable yt that assumes the value 0 (when there is no market) and 1 (whenthere is market). T his random variable is dē ned for all t 2 Tp and it is

assumed to be independent of the ordinary sourceof uncertainty that gen-erates the price process. We can therefore talk about a market existenceprocess. I n order to construct one such process, let us start with the statespace. Let #( Tp) denote thenumber of points in Tp: At each of thesepoints,

market may either exist or not exist, leading to 2#( Tp) possible states of nature. We then have the collection of possible statesof nature denoted bŷ = f vi gi=1;:::;2# ( Tp) ; each vi corresponding to a distinct state. Moreover,

let F̂ = F̂1;::: ; F̂ T¡ 1; where F̂ t is the ¾-algebra generated by the randomvariable yt. Let py be the probability associated with the random variableyt: For all t 2 Tp; we have py (yt = 1) = p and py (yt = 0) = 1¡ p:

3.1 T he expected value of a portfolio

We now construct a random variable that allows to construct the expectedfuture value of a portfolio in this setting. For t < t0; let xt;t0 be a random

variable identifying thelast time that transactions take place beforedate t0;given that we are at time t; and transactions are currently possible. Let ̂ tbe the subset of ̂ such that ̂ t =

nvi 2 ̂ : yt (vi) = 1

o: Then,

xt;t0 : ̂ t !©

t;: :: ;t0¡ 1ª

Let pxt; t0

be the probability associated with xt;t0: Then,

pxt; t0¡

xt;t0 = t¢

= (1 ¡ p)t0¡ t¡ 1 :

Moreover, for a given s 2 (t;t0) ;

pxt;t0¡xt;t0 = s

¢= p(1¡ p)t0¡ s¡ 1 :

Also note that

X t0¡ 1s=t

pxt;t0¡

xt;t0 = s¢

= (1¡ p)t0¡ t¡ 1+

X t0¡ 1s=t

p(1¡ p)t0¡ s¡ 1 = 1;

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as it should.

Consider a given trading strategy (¢ t; Bt)t=0;::: ;T ; where (¢ t; Bt) ´(¢ itt ; B

i tt )i t2I t is a (t + 1)¡ dimensional vector. Consider a given path w

and (is)s2f t;:::;Tg ½ w. Suppose that the agent is at a given node it; whererebalancing is possible. As there is uncertainty about the existence of mar-ket at thefuturepoints in time, there is also uncertainty about theportfoliothat the agent will be holding at a future node it0: In fact, the portfolio at

it0 may beany of ³

¢ itt ; Bitt

´;

³¢

it+1t+1;B

i t+1t+1

´;::: ; or

³¢

i t0¡ 1t0¡ 1 ; B

i t0¡ 1t0¡ 1

´; where

(is)s2f t;:::;t0¡ 1g ½w:Clearly, the expected value of a given tradingstrategy at nodeit0, given

that we are at node it; is

E px t;t0

ith¢ ixx S

it0 + R

t0¡ xBixx i = X s=t;:::;t0¡ 1pxt; t0 ¡xt;t0 = s¢h¢

iss S

it0 +R

t0¡ sB iss i ;

where we use x to short notation for xt;t0:

3.2 Statistical versus pure arbitrage

A pure arbitrage opportunity is a zero-cost portfolio at time t, such that thevalue of each possible portfolio at node i T is positive, i.e.,

¢it+ jt+ j S

i T T +R

T ¡ t¡ j Bit+ jt+ j ¸ 0

for all i t+ j such that i t is a predecessor, j = 0;1;::: ;T ¡ t ¡ 1 and

Epxt;Ti t

h¢ ixx S

i T T +R

T¡ x Bixx

i> 0;

together with the self-̄ nancing constraints

¢ itt Sit+ jt+ j +R

j ¡ t B itt ¸ ¢it+ jt+ j S

i t+ jt+ j + B

it+ jt+ j ;

for all i t+ j such that i t is a predecessor, and j = 1;:: : ;T ¡ t ¡ 1:I f statistical arbitrage is considered, however, an arbitrage opportunity

requires only that, at node i T ; the expected value of the portfolio at T is

positive,

Epxt;Ti t

h¢ ixx S

i T T +R

T¡ x Bixx

i¸ 0;

together with weaker self-¯nancing conditions. Let us regard these latterconditions in some detail.

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Suppose that we are at a given node it: If there is market at the next

point in time we then have, for sure, the portfolio³

¢ itt ; B i tt

´at time t + 1.

Hence, if node it+1 is reached, the self-¯nancing condition is

¢ i tt Sit+1t+1 +RB

itt ¡

³¢

i t+1t+1S

i t+1t+1 + B

i t+1t+1

´¸ 0

Consider now that t +2 is reached. At time t there is uncertainty aboutthe existence of the market at time t + 1: Hence, at time t + 2 we can ei-

ther havethe portfolio³

¢ itt ; Bi tt

´or theportfolio

³¢

it+1t+1 ;B

it+1t+1

´: Under the

concept of statistical arbitrage, wewant to ensure that, in expected value,we are not going to lose at node it+2: Hence, the self-̄ nancing conditionbecomes

Xs=t;t+1

pxt; t+2 (xt;t+2 = s)³

¢ iss Si t+2t+2 +R

t+2¡ sB iss´

¸³

¢i t+2t+2S

t+2t+2+ B

i t+2t+2

´

More generally, for any t at which transaction occurs and t < t0< T; thestatistical self-̄ nancing condition becomes

Epx

t; t0

i t

h¢ ixx S

i t0t0 +R

t0¡ xB ixx

i¸

³¢

it0t0 S

i t0t0 +B

i t0t0

´

Dē nition 5 4 A statistical arbitrage opportunity is a zero-cost tradingstrategy for which

1. At any node it, the expected value of the portfolio at any ¯nal nodeis positive, i.e.,

Epxt;¶ Tit

h¢ ixxt; T S

i T + R

T¡ xB ixxt; T

i¸ 0

4 T his notion of Arbitrage Opportunity is in the spirit of Bondarenko (2003). In hisdē nition 2, a Statistical Arbitrage Opportunity (SAO) is dē ned as a zero-cost trading

strategy with a payo®Z T = Z (F T ), such that

(i) E [Z T jF0] > 0; and(ii) E [Z T jF0;» T ] ¸ 0; for all » T ;

where »t denotes the state of the Nature at time t; and F t = (»1; : : : ; »t ) is the marketinformation set, with F0 = Á. A lso, t he second expectation is taken at time t = 0 andis conditional to the terminal state » T . However, notice that eliminating SAO 's at timet = 0 does not imply the absence of SAO's at futuretimes t 2 [1; T ¡ 1]. Hence, in orderto incorporate a dynamically consistent absence of SAO's, we re¯ne the dē nition of a

SAO as a zero-cost trading strategy with a payo®Z T = Z (F T ), such that

(i) E [Z T jF0] > 0; and(ii) E [Z T jF t;» T ] ¸ 0; for all » T and all t 2 [0;T ¡ 1] :

12


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for any it 2 I t and t 2 f 0;1;:: : ;T ¡ 1g; and

2. The portfolio is statistical ly self-̄ nancing, i.e:,

Epx

t;¶t0

it

h¢ ixxt; t0S

it0 + R

t0¡ xB ixxt; t0

i¡

³¢

i t0t0 S

i t0t0 +B

i t0t0

´¸ 0

for any it 2 I t; t0> t, t 2 f 0;1;:: : ;T ¡ 2g and t02 f 1;::: ;T ¡ 1g:

T he two dē nitions of arbitrage are related in the following.

T heorem 6 If there are no statistical arbitrage opportunities, then thereare no pure arbitrage opportunities.

P roof. If there is a pure arbitrage opportunity then the inequalitiespresent in thedē nition of arbitrage opportunity, dē nition 3, arerespected.

Hence, as theseexpressions are theterms under expectation in thedē nitionof Statistical Arbitrageopportunity, presented in dē nition 5, there is alsoa statistical arbitrage opportunity.

T he set of portfolios that represent a pure arbitrage opportunity is asubset of theportfoliosthat represent astatistical arbitrageopportunity, i.e.,there are portfolios that, in spite of not being a pure arbitrage opportunity,represent a statistical arbitrage opportunity.

I n order to have a statistical arbitrage opportunity it is not necessary(although it is su±cient) that thevalue of theportfolio at the ¯nal date ispositive. It is onlynecessary that, for all t, theexpectedvalue of theportfolioat the ¯nal date is positive.

Consider now the self-̄ nancing conditions under statistical arbitrage.When rebalancing the portfolio it is not necessary (although it is su±cient)that the value of the new portfolio is smaller than the value of the old one.

Thishappens becausefuturerebalancing isuncertain, leading to uncertainty

about theportfolio that the agent will beholdingin any future moment. Inorder to avoid a statistical arbitrage opportunity it is only necessary thatthe expected valueof theportfolio at a given point in time is larger than thevalue of the rebalancing portfolio.

Finally, notice that the concept of statistical arbitrage opportunity re-duces to the usual concept of arbitrage opportunity in the limiting casep= 0:

3.3 A ugmented measures

For technical reasons, we now dē ne an augmented probability spaceQ on . In order to do that, we dē nea semipath m from it to i t0, which is a set

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of nodes m = [ k2f t;:::;t0gik such that i k 2 Iik+1k : Let

+i t ;it0

denote the set of

semipaths from it to i t0:

Dē nition 7 An augmented probability space in is a set of probabilitiesnq(i T ;T );m(i t ;t)

osuch that i t 2 I t; m 2

+it ;i T

, t = 0;::: ;T and

X

i T

T¡ 1X

t=0

X

i t

X

m

q(i T ;T);m(it ;t)

= 1;

Dē nition 8 A modī ed martingale probability measure is an augmentedprobability measure Q 2 Q which satis¯es

(i)

S0 = 1R T

Xf i T 2I T g

qi T Si T T

where

qi T = T¡ 1X

t=0

X

nit :i t2 I

i Tt

o

X

nm2- +it ;i T

oq(i T ;T);m(i t ;t)

Si T T ;

(ii)

Si T ¡ 1 T¡ 1 =

1

R

X

ni T :i T ¡ 12I

i T

T ¡ 1

o¼(i T ;T );m(it;t)

Si T T

withX

ni T :i T ¡ 12I

i T T ¡ 1

o¼(i T ;T);m(i t;t)

= 1

and

¼(i T ;T);m(it ;t)

= 1

¥

T ¡ 1X

t=0

pxt; T (xt;T = T ¡ 1)X

ni t :i t2I

i T ¡ 1t

o

X

nm2- +

i t ;i T:i T ¡ 12m

oq(i T ;T);m(i t;t)

where

¥ = T¡ 1X

t=0

pxt; T (xt;T = T ¡ 1)X

ni T :i T ¡ 12I

i T T ¡ 1

o

X

nit:i t2I

i T ¡ 1t

o

X

nm2- +

it ;i T:i T ¡ 12m

oq(i T ;T );m(i t ;t)

;

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(iii) there existsn

®(i t0;t

0);m(i t;t)

o;for all it0 2 I t0; it 2 I t; m 2

+i t ;i T

and

t0> t for all t = 0;::: ;T ¡ 1 such that, for all 0 < k < T ;

Sikk = 1

R T ¡ k

X

ni T :ik2 I

i Tk

oµ(i T ;T);m(i t ;t) S

i T T +

X

t0>k

1

Rt0¡ k"(it0;t

0);m(it;t)

Si t0t0 ;

whereX

ni T :ik2 I

i Tk

oµ(i T ;T);m(i t;t)

+X

t0>k

"(i t0;t

0);m(i t;t)

= 1

and

µ(i T

;T);m(i t ;t) =

1

£

kX

t=0pxt; T (xt;T = k)

Xni t:it2I

i kt

o

X

nm2- +

i t ;i T:ik2m

o q(i T

;T);m(i t;t)

"(it0;t

0);m

(it ;t) =

1

£

X

tk

X

t


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where

¢ t;B t 2 Rt+1;t = 0; :::;T ¡ 1

subject to the conditions of a positiveexpected payo®

Epxt;¶ Tit

£¢ ixx S

i T + R

T¡ xB ixx¤

¸ Gi T T ;

for any it 2 I t and t 2 f 0;1;::: ;T ¡ 1g5; and self-̄ nancing conditions

Epx

t;¶t0

i t

h¢ ixx S

it0 +R

t0¡ xB ixx

i¡

³¢

it0t0 S

it0t0 + B

i t0t0

´¸ 0

for any it 2 I t; t0> t, t 2 f 0; 1;::: ;T ¡ 2g and t02 f 1;::: ;T ¡ 1g6:

Example 9 I llustration of the optimization problem with T = 3. The evo-lution of the price underlying asset can be represented by the tree in ¯gure1.

t=3t=2t=1t=0 t=3t=2t=1t=0

Figure1: Evolution of the undelying asset' price.

I n what concerns the evolution of thepriceprocessthere are eight di®er-ent states, i.e., = f wi gi=1;:::;8 : T he problem that must be solved in orderto ¯nd the upper bound is the following.

Cu = minf ¢ t ;Btgt=0;:::;2

¢ 0S0+ B0

where

f ¢ 0; B0g = f (¢0;B0)g

f ¢ 1; B1g =

©¡

¢

1

1;B

1

1

¢

;

¡

¢

2

1; B

2

1

¢ª

f ¢ 2; B2g =©¡¢ 12;B

12¢;¡¢ 22; B

22

¢;¡¢ 32; B32

¢ª

5For each it there are 2( T ¡ t) paths, and asa result, 2( T ¡ t) (t+ 1) restrictions at ti me t. T he total number of restrictions is

P T ¡ 1t=0 2

( T ¡ t)(t+ 1):

6For each it there areP T ¡ 1

t0= t+12t

0¡ t: Hence, for each t there are (t+ 1)P T ¡ 1

t0=t+12t

0¡ t :

Hence, there areP

T ¡ 2t=0

(t+ 1)P

T ¡ 1t0=t+1

2t0¡ t restrictions.

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subject to the conditions of a positive expected payo®

h¢ i22 S

i33 + RB

i22

i¸ Gi33 ;

for all i3 2 I 3 and i2 2 I 2 such that i2 2 Ii32 : (these are 6 constraints);

ph

¢ i22 Si33 +RB

i22

i+ (1¡ p)

h¢ i11 S

i33 + R

2B i11

i¸ Gi33

for all i3 2 I 3; i2 2 I 2 and i1 2 I 1 such that i2 2 Ii32 : and i1 2 I

i21 ; i.e., i1; i2

and i3 belong to the same path (these are 8 constraints) and

£p2+ p(1¡ p)

¤h¢ i2

2

Si3

3

+ RB i2

2

i+p(1¡ p)

h¢ i1

1

Si3

3

+ R2B i1

1

i+

+(1¡ p)2h¢ 0Si33 + R

3B i22i ¸ Gi33

for all i3 2 I 3; i2 2 I 2 and i1 2 I1 such that i2 2 Ii32 : and i1 2 I

i21 ; i.e., i1;

i2 and i3 belong to the same path (these are 8 constraints). Moreover, theself-̄ nancing constraints must also be considered

¢ 0Si11 +RB0 ¸ ¢

i11 S

i11 + B

i11

for any i1 2 I 1 (2 constraints),

(1¡ p)

h

¢ 0Si2

2 + R2

Bi2

2

i

+ p

h

¢i1

1 Si2

2 +RBi1

1

i

¸ ¢i2

2 Si2

2 + Bi2

2

for any i2 2 I2 and i1 2 I1 such that i1 2 Ii21 ; i.e., i1;and i2 belong to the

same path (these are 4 constraints) and, ¯nally,

¢ i11 Si22 + RB

i11 ¸ ¢

i22 S

i22 + B

i22

for any i2 2 I2 and i1 2 I1 such that i1 2 Ii21 ; i.e., i1;and i2 belong to the

same path (these are 4 constraints).

4.1.2 Solution

T heorem 10 There existsa modī ed martingale probability measure, qi T 2QS; such that the upper bound for arbitrage-free value of a European optioncan be written as

Cu = maxqi T 2QS

1

R T

X

f i T 2I T gqi T Gi T T : (3)

17


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P roof. See proof in appendix A.1.

Remark 11 I f a Call Option i s considered, the values for qi T ; in a modelwith two periods are explicitly calculated in appendix A.3. I n that case itcan be shown that for a strictly positive p; the q1; q2 and q3 are also strictlypositive.

I n what follows we characterize some relevant properties of Cu:

1. Cu · C0u

P roof. Let ¢ 00 and B00denote the optimal values invested, at time

t = 0; in the stock and in the risk-free asset respectively, when p = 0.

T hetrading strategy ¹¢ t = ¹¢ P =00 and ¹Bt = Rt ¹B p=00 , for t = 1;:::; T ¡ 1,is an admissible strategy for any given p. As a result, the solution of the problem for any p cannot belarger that thevalue of this portfolioat t = 0 (which is C0u).

2. Cu ¸ C1u:

P roof. Consider the trading strategy ¡

¹¢ ¤t ; ¹B¤t

¢t=0;::: ;T¡ 1

that solvesthe maximization problem that characterizes the upper bound for ap 2 (0;1) : This is an admissiblestrategy for the case p = 1, because

it is self-̄ nancing, i.e.,

¢i t¡ 1t¡ 1S

i tt + RB

it¡ 1t¡ 1 ¸ ¢

itt S

itt +B

i tt ;

and superreplicates thepayo®of theEuropean derivative at maturity,i.e.,

¢i T ¡ 1 T¡ 1S

i T T + RB

i T ¡ 1 T¡ 1 ¸ G

i T T :

Hence, the solution of the problem for p = 1 cannot be higher thanthevalue of this portfolio at t = 0 (which is Cu).

3. limp! 0Cu = C0u and limp! 1Cu = C

1u:

P roof. See Appendix A.4

An example for a Call Option and T=2 is also show in appendix A.4.

4. Cu is a decreasing function of p.

P roof. See Appendix A.4

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5. For a Call Option and T = 2, we can provethat

Cu · pC1u + (1¡ p)C

0u

meaning that the probabilistic upper bound is a convex linear combi-nation of the perfectly liquid upper bound and the perfectly illiquidupper bound.

P roof. See appendix A.4.

4.2 T he L ower B ound

The organization of this section is analogous to the section for the upper

bound.

4.2.1 T he P roblem

Theproblemof determining thelower bound of the statistical arbitrage-freerange of variation for thevalue of an European derivative, can bestated as

Cl = maxf ¢ t ;Btgt=0;:::;T ¡ 1

¢ 0S0 + B0

where

¢ t;B t 2 Rt+1;t = 0; :::;T ¡ 1

subject to the conditions of a positiveexpected payo®

Epxt;¶ Tit

£¢ ixx S

i T + R

T¡ xB ixx¤

· Gi T T ;

for any it 2 I t and t 2 f 0;1;::: ;T ¡ 1g7;and self-̄ nancing conditions

Epx

t;¶t0

i t

h¢ ixx S

it0 +R

t0¡ xB ixx

i¡

³¢

it0t0 S

it0t0 + B

i t0t0

´· 0

for any it 2 I t; t0> t, t 2 f 0; 1;::: ;T ¡ 2g and t02 f 1;::: ;T ¡ 1g8:

7As in the upper bound case, for each i t there are 2( T ¡ t) paths, and as a result ,2( T ¡ t)

(t+ 1) restrictions at time t. T he total number of restricti ons isP T ¡ 1t=0 2( T ¡ t)

(t+ 1):8As i n the upper bound case, for each i t there are

P T ¡ 1t0=t+1

2t0¡ t : Hence, for each t

there are (t + 1)P

T ¡ 1t0= t+1

2t0¡ t: Hence, there are

P T ¡ 2t=0

(t+ 1)P

T ¡ 1t0=t+1

2t0¡ t restrictions.

19


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4.2.2 Solution

T heorem 12 Thereexistsan modī ed martingale probabili tymeasure, qi T 2QS; such that the upper bound for arbitrage-free value of an European optioncan be written as

Cl = minqi T 2QS

1

R T

X

f i T 2 I Tgqi T Gi T T :

P roof. T heproof is analogous to the upper bound.

Remark 13 I f a call option is considered, the values for qi T ; in a modelwith two periods are explicitly calculated in appendix B.2. I n that case it

can be shown that for a strictly positive p; the q1

; q2

and q3

are also strictlypositive.

I n what follows we characterize some relevant properties of Cl :

1. Cl ¸ C0l :

2. Cl · C1l :

3. limp! 0Cl = C0l and limp! 0Cl = C1l :

An example for a call option and T = 2 is shown in appendix ??

4. Cl is a increasing function of p. T heproofs of theseproperties areanalogous to those presented for theupper bound.

5 U tility Functions and R eservation P rices

In this section we show that theprice for which any agent is indi®erent be-tween transacting or not transacting the derivative, to be called the reser-vation pri ce of the derivative, is contained within the statistical arbitragebounds derived above.

Let ut (:) denote a utility function representing the preferences of anagent at time t. T he argument of the utility function is assumed to be theconsumption at timet. Let y bethe initial endowment of the agent, and Zt

denote the vector of consumption at time t; i.e., Zt =³

Zi tt

´

it2I t: Let ½be

a discount factor. If an agent sells a European derivative by the amount C;

20


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and that derivative has a payo®at maturity given Gi T T , themaximumutility

that he or she can attain is

u¤sell (C;p) = supf ¢ t ;Btgt=0;:::;T ¡ 1

EG;P0X T

t=0½tut (Zt)

subject to

Z0 +¢ 0S0+ B0 · C + y

Zi tt + ¢itt S

itt + B

i tt · ¢

iitt¡ j

t¡ j Si tt +R

j Bii tt¡ j

t¡ j

Zi T T · ¢ii T T ¡ j

T¡ j Si T

T + R j B

ii T T ¡ j

T¡ j ¡ Gi T T

for all i T 2 I T ; it 2 I t, j · t and t = 1;::: ;T ¡ 1 where E G;P0 denotes abivariateexpected value, at t = 0, with respect to theprobabilityP inducedby the market existence and the probability G underlying the stochasticevolution of the price process.

Similarly, if the agent decides not to include derivatives in his or herportfolio, the maximumutility that he or she can attain is given by

u¤ (p) = supf ¢ t ;Btgt=0;:::;T ¡ 1

E G;P0X T

t=0½tut (Zt)

subject to

Z0+ ¢ 0S0 +B0 · y

Zi tt + ¢itt S

itt + B

i tt · ¢

iitt¡ j

t¡ j Si tt +R

j Bii tt¡ j

t¡ j

Zi T T · ¢ii T T ¡ j

T ¡ j Si T

T + R j B

ii T T ¡ j

T¡ j

L emma 14 I n the case of random dryness, there is p¤ > 0 such that, forall p < p¤; the utility attained selling the derivative by Cu, is larger than theutility attained if the derivative is not included in the portfolio.

u¤sell (Cu; p) ¸ u¤(p) :

P roof. Let, for a given p; f ¢ t; Btgt=0;:::;T¡ 1 denote the solution of theutility maximization problem with no derivative and f ¢ ut ;B

ut gt=0;:::;T¡ 1 de-

note the solution of theminimization problem that must be solved to ¯ndthe upper bound if statistical arbitrage opportunities are considered (seesection 4.1). Moreover, let

©¢ sellt ;B

sellt

ªt=0;:::;T¡ 1

denote an admissible so-lution of the utility maximization problem when theagent sells one unit of

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the derivative. Now, consider the limit case, when p approaches zero. In

that case, the portfolion

¢ sellt ;Bsellt

o

t=0;:::;T ¡ 1´ f ¢ t +¢

ut ;Bt + B

ut gt=0;:::;T¡ 1

is anadmissible solution of theutility maximization problemwhen the agentsells one unit of derivativeby Cu. The reason is as follows. T he constraintset of theproblemthat must besolved to ¯nd theupper boundis continuousinp: Hence, when p! 0; thesolution of theproblemisf ¢ ut ;B

ut gt=1;::: ;T¡ 1 =©

¢ u0;RtB u0

ªwhere f ¢ u0;B

u0g is thesolution of following problem

½min¢ ;B

¢ S0+ B s.a. ¢ Si T

T + R T B ¸ Gi T T ;8i T

¾:

As C = Cu = ¢ u0S0 + B0; the portfolio f ¢ut ;B

ut gt=1;:::;T ¡ 1 is an ad-

missible solution of the utility maximization problem when one unit of thederivative is being sold. Moreover, it guarantees a positiveexpected utility.9

Hence, theportfolio©

¢ sellt ; Bsellt

ªt=0;:::;T ¡ 1

is alsoadmissible solution for theoptimization problem, when one unit of the derivative is being sold, whichguarantees a higher payo®than the portfolio f ¢ t;B tgt=0;:::;T ¡ 1 : Hence,

u¤sell (Cu; 0) ¸ u¤(0) :

Continuity on p of both u¤sell and u¤ ensure the result.

Remark 15 Notice that the existence of p¤ follows from the continuity of the utilities in p: Furthermore, it is possible to have p¤ = 1. Examples withdi®erent values of p¤ are given in the end of this paper. T he range of values

p < p¤ characterizes what was vaguely described as \ su±ciently incompletemarkets" in the introduction.

T he reservation price for an agent that is selling the option is dē nedas the value of C that makes u¤sell (C;p) = u

¤ (p). L et Ru denote such

reservation price.

T heorem 16 For all p < p¤ we have Ru · Cu:

P roof. T heoptimal utility value;

u¤sell (C;p) = supf ¢ t ;Btgt=0;:::;T ¡ 1

C + y¡ (¢ 0S0 +B0) + EG;P0

X Tt=1

½tut (Zt) ;

9It su±ces to consider Z0 = y; Zi t = 0 and Zi T = ¢ Si T T

+R T B ¡ G i T T

¸ 0:

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is increasing in C: This, together with lemma 14 leads to the result.

T he same applies for thecase when the agent is buying a derivative. Inthat case, if an agent is buying the derivative by C, the maximum utilitythat he or she can attain is

u¤buy (C;p) = supf ¢ t ;Btgt=0;:::;T ¡ 1

EG;P0X T

t=0½tut (Zt)

subject to

Z0+ ¢ 0S0 +B0 · ¡ C + y

Zi tt + ¢itt S

itt + B

i tt · ¢

iitt¡ j

t¡ j Si tt +R

j Bii tt¡ j

t¡ j

Zi T T · ¢ ii T

T ¡ j T¡ j Si T T + R j B i

i T

T ¡ j T¡ j +Gi T T

L emma 17 I n the case of random dryness, there is p¤ > 0 such that, forall p < p¤; theutility attained buying the derivative by Cl, is larger than theutility attained if the derivative is not included in the portfolio.

u¤buy (Cl;p) ¸ u¤ (p) :

P roof. T he proof is analogous to the one in proposition 14Let Rl denote the reservation selling price, i.e., the price such that

u¤ (p) = u¤buy (Rl; p).

T heorem 18 For all p < p¤ we have Rl ¸ Cpl :

P roof. The proof is analogous to the one presented in theorem (16).However, in this case theutility is a decreasingfunction of C; and we obtain

u¤buy (Cl; p) ¸ u¤ (p) ) Rl ¸ Cl:

Several il lustrations are presented in section 7.

6 C omparisons with the L iterature

In what follows we compare our methodology with others in the literature,namely Bernardo and Ledoit (2000) and Cochraneand Sa¶a-Requejo (2000).Cochrane and Sa¶a-Requejo (2000) introduce the notion of \ good deals", orinvestment opportunitieswith high Sharperatios. They show that rulingoutinvestment opportunities with high Sharpe ratios, they can obtain narrower

23


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bounds on securities prices. However, as stressed in Bondarenko (2003), not

all pure arbitrage opportunities qualify as \ good deals". Moreover, for agiven set of parameters wefound out that in order to contain thereservationprices of a risk neutral agent the interval is more broad than the one thatwas obtained in our formulation.

We ¯rst provide a simple example to compare our bounds with purearbitragebounds.

Example 19 Consider a simple two peri ods example, where transactionsare certainly possible at times t = 0 and t = 2: At time t = 1 there aretransactions with a given probability p = 0:65: The initial stock price isS0 = 100and it mayeither increase in eachperiod witha probability 0:55; or

decrease with a probability 0:45: We take and U = 1:2;D = 0:8 and R = 1:1:A call option that matures at time T = 2 with exercise price K = 80 isconsidered. Using pure arbitrage arguments we ¯nd the following range of

variation for the value of the call option

[33:88; 37:69]

Using the notion of statistical arbitrage opportunity, the above range getsnarrower and is given by

[34:31; 35:17];

clearly narrower that the above interval.I f markets were complete (p= 1), the valueof the option would be 34:71.

Also, the reservation price10 for a risk neutral agent is equal to 35:09. Noticethat both intervals include the complete market value and the reservation

price.

We now use the same example to compare our methodology with theone presented by Cochrane and Sa¶a-Requejo (2000). We show that eitherour interval is contained in theirs, or else, their interval do not contain theabove mentioned reservation price.

With theSharpe ratio methodology thelower bound is given by

C¹

= minf mg

E ©

m[S2 ¡ K ;0]+ª

10In this example, the reservation price for an agent who is buying the derivative coin-

cides with that of an agent who is selling it.

24


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subject to

S0 = E [mS2]; m ¸ 0; ¾(m) · h

R2;

where S0 is theinitial priceof the risky asset, and S2 isthepriceof theriskyasset at time t = 211: Theupper bound is

¹C = maxf mg

E©

m[S2 ¡ K ; 0]+ª

subject to

p = E [mS2];m ¸ 0; ¾(m) · h

R2:

Example 20 In order to compare the statistical arbitrage interval with theSharperatio bounds, we must choose the ad-hoc factor h so as to make oneof the limitingbounds to coincide. I f we want the upper bound of the SharpeRatio methodology to coincide with the upper bound obtained with statisticalarbitrage, we must take h = 0:3173: I n that case, the lower bound will be33:88 and the range of variation will be

[33:88; 35:17];

worse than the statistical arbitrage interval.

Alternatively, if wewantthelower boundof theSharpeRatio methodologyto coincide with the lower bound obtained with statistical arbitrage, we musttake h = 0:28359:12 In that case, the upper bound will be 34:49 and the rangeof variation will be

[34:31; 34:49]:

Although this interval is tighter than thestatistical arbitrage interval, it doesnot contain the reservation price for a risk neutral agent.

I n a di®erent paper Bernardo and Ledoit (2000) preclude investments

o®ering high gain-loss ratios to a benchmark investor, somehow analogous11As stressed by Cochraneand Sa¶a-R equejo, in a former paper Hansen and J agannathan

(1991) haveshown that a constraint on thedi scount factor volatility is equivalent to imposean upper limit on t he Sharpe ratio of mean excess return to standard deviation.

12In order to get a lower bound higher than 33:88 it is necessary to impose aditionallythat m> 0: I f that were not the case, then the lower bound would only be dē ned for h¸ 0:2980 and would be equal to 33:88:

25


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to the \ good deals" of Cochrane and S¶aa-Requejo. T hecriterion, however,

is di®erent since Bernardo and Ledoit (2000) propose a utility-based ap-proach, as stressed in theIntroduction. In this way, the arbitrage-free rangeof variation for the value of the European derivative is narrower than inthe case of pure arbitrage. Let ~z+ denote the (random) gain and ~z¡ de-note the (random) loss of a given investment opportunity. T he utility of abenchmark agent characterizes a pricing kernel that induces a probabilitymeasure, according to which the expected gain-loss ratio is bounded fromabove

E ( ~z+)

E ( ~z¡ ) · ¹L :

T he fair price is the one that makes the net result of the investment tobe null. In other words, for a benchmark investor, it would correspond tothe pricing kernel that would make

E¡

~z+ ¡ ~z¡ ¢

= 0, E ( ~z+)

E ( ~z¡ ) = 1:

This last equality characterizes the benchmark pricing kernel for a givenutility.

Notice that the fair price constructed in this way coincides with ourdē nition of the reservation price. T herefore, by choosing ¹L larger thanone, the interval built by Bernardo and Ledoit contains by construction the

reservation price of thebenchmark agent.On the other hand, the arbitrary threshold ¹L can be chosen such that

their interval is contained in the statistical arbitrage-free interval.

T he disadvantages, however, are clear. First, the threshold is ad-hoc, just as in the case of Cochrane and S¶aa-Requejo; second, the constructedinterval dependsonthebenchmark investor; and ¯nall y, theonly reservationprice that is contained for sure in that interval, is the reservation price of the benchmark investor. In other words, we cannot guarantee that thereservation price of an arbitrary agent, di®erent from the benchmark, iscontained in that interval.

7 N umerical E xamples

7.1 Upper and L ower B ounds

In this section several numerical examples areprovided in order to illustratethe properties of the upper and lower bounds presented in the previous

26


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sections.

Using numerical examples we can conclude that, for a call option,

Cu · pC1u + (1¡ p)C

0u:

If the Call Option is sold by the expected value of the call, regarding theexistence (or not) of market, there will be an arbitrage opportunity in sta-tistical terms. T he reason is that the agent that sells the call option canbuy a hedging portfolio (in a statistical sense) by an amount smaller thanthe expected value of the call option. As a result, there is an arbitrage op-portunity, becauseheis receiving more for the call option than is paying forthe hedging portfolio.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 133

34

35

36

37

38

39

40

p

Upper Lower

Figure 2: T he Upper and Lower Bounds for a European Call Option withdi®erent value of p in a two period model, with U = 1:2, R = 1:1, D = 0:8;S0 = 100 and K = 80:

However, in what concerns thelower bound, it is not possibleto concludewhether Cl · pC1l +(1¡ p)C

0l or Cl ¸ pC

1l +(1¡ p)C

0l : T hat dependson the

valueof the parameters. Although in thetwo-period simulation in Figure2weseemto havethe former case, thethree period examplein Figure3 seemsto suggest the latter, since the lower bound behaves as a concave functionof p for most of its domain.

27


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

p

Upper Bound

Lower Bound

Figure 3: T he Upper and Lower Bounds for a European Call Option withdi®erent value of p in a three period model, with U = 1:3, R = 1, D = 0:6;S0 = 100 and K = 100:

Finally, we may use Figure 4 to illustrate several features. T he Upper and L ower bounds of an European Call Option for di®erent

values of p and K (K = 80, K = 100, K = 120, K = 140 e K = 160) in athree period model, with U = 1:2; R = 1:1; D = 0:8 and S = 100.

First, let us regard thesituations in this Figure that are related to pure

arbitrage. This includes the value of the derivative under perfectly liquid(p = 1) and perfectly illiquid (p = 0) markets. I n the former case, theuniquevalueof the derivative clearly decreases with theexercise priceK ; asit should. In thelatter, both the upper bound C0u and the lower bound C

0l

also decrease with K : More curiously, however, the spread C0u ¡ C0l has a

non-monotonic behaviour. Obviously this di®erence is null for K less than

the in¯mum value of thestock at maturity and must goto zero asthe strikeapproaches the supremum of the stock's possible values at maturity. In themiddle of this range it attains a maximum. In our numerical example weobserve that the maximum value of the spread is attained for K close to120.

28


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

p

K=80

K=100

K=120

K=140

K=160

Figure 4: The Upper and Lower bounds of an European Call Option fordi®erent values of p and K (K = 80, K = 100, K = 120, K = 140 eK = 160) in a three period model, with U = 1:2; R = 1:1; D = 0:8 andS = 100.

Regarding the statistical arbitrage domain when p 2 (0; 1) ; we noticethat all the above remarks remain true. Figure4 also suggests that, for anygiven p; the spread attains its maximum for the same value of K as before.Noticethat thespread Cu ¡ Cl decreases with p for ¯xed strikeK convergingto zero as p ! 1: Hence, although somehow di®erent from the traditional

dē nition of arbitrage, the notion of statistical arbitrageseems to provideavery nicebridge, for 0 < p< 1, between thetwo extreme cases above (p= 0and p = 1), where the original concept of arbitrage makes sense. This isillustrated in a more direct way in Figure 5.

A third issuedriven byFigure4 is that coexistingderivativesmay restrictmore the no-arbitrage interval. As an example, suppose that derivativeswith K = 120 and K = 140 coexist. When p = 0, the upper bound forK = 140is abovethelower boundwhen K = 120. I norder to avoid arbitrageopportunities, theequilibriumsellingpriceof the K = 120 derivativehas tobe abovethe equilibriumbuying price of the K = 140 derivative. Otherwise

29


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40 60 80 100 120 140 160 180-2

0

2

4

6

8

10

12

14

16

K

C u - C l

p=0

p=1

Figure 5: T he Spread (Cu ¡ Cl) of an European Call Option for di®erent

values of K and p (p = 0;::: ;1 with increments of 0:1) in a three periodmodel, with U = 1:2; R = 1:1; D = 0:8 and S = 100.

one would buy the K = 120 derivative with the proceeds of selling theK = 140 derivative, to get a net positive payo®at maturity at no cost. This implies that the upper bound of the K = 140 derivative should bethe equilibrium selling price of the K = 120 derivative and not necessarilythe bound we derived above. T his situation stresses the limitation thatour bounds were constructed assuming that there was only one derivativethat, aditionally, was not part of the replicating portfolio. I n fact, thepresence of more derivatives may help to complete the market, making theoverlappingarbitrage-free regions not viable. As markets become complete,the arbitrage-free regions shrink to a point, corresponding to the uniquevalue of thederivatives under complete markets.

7.2 Utility and Reservation P rices

In this section we illustrate several aspects related to the determination of the reservation price. In Figure 6 we represent the utility of an agent inthree di®erent situations. Without the derivative; a short position on the

30


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derivative, when the instrument is sold by the statistical arbitrage upper

bound; and a long position on the derivative when the instrument is boughtby the statistical arbitrage lower bound. Notice that for p = 0 the bestsituation is the short position on the derivative and the worst is withouttrading the derivative. T his is consistent with lemma 14 and lemma 17.Notice that there is a value of p such that, for larger probabilities, theutility without trading the derivative is no longer the worst. T hat criticalvalue of p is what we called p¤:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 155

60

65

70

75

80

85

p

U t i l i t y V a l u e

Without DerivativeSelling DerivativeBuying Derivative

Figure 6: Utility value for the following parameters: U = 1:3, D = 0:6,R = 1, S0 = 100, K = 100, ½= 1=R, q = 0:5, y = 50 and T = 3.

Figure 7 represents the statistical arbitrage-free interval together withthe reservation price for a risk neutral agent. By construction, the proba-bility associated to the point where the reservation price coincides with theupper bound, corresponds to the critical probability p¤: Notice from Figure

6that the utility of the position associated to a longposition on thederiva-tiveis always abovetheutility without trading thederivative. T his impliesthat the reservation price is always above the lower bound. Likewise, thefact that the utility of the short position on the derivative goes below theutility without trading thederivative, impliesthat thereservation pricegoesabove theupper bound.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

p

Reservation Selling PriceReservation Buying PriceUpper BoundLower Bound

Figure 7: Statistical Arbitrage free bounds and reservation prices for thefollowing parameters: U = 1:3, D = 0:6, R = 1, S0 = 100, K = 100,

½= 1=r, q = 0:5, y = 40 and T = 3.

32


33/57


34/57

R eferences

[1] Amaro de Matos, J . and P. Ant~ao, 2001, \ Super-Replicating Boundson European Option Prices when the Underlying Asset is I lliquid",Economics Bulletin, 7, 1-7.

[2] Bernardo, A. and Ledoit, O., 2000, Gain, Loss, and Asset Pricing, J ournal of Political Economy, 108, 1, 144-172.

[3] Bondarenko, O., 2003, Statistical Arbitrage and Securities Prices, Re-view of Financial Studies, 16, 875-919.

[4] Cochrane, J . and Sa¶a-Requejo, J ., 2000, Beyond Arbitrage: Good-DealAsset Price Bounds in Incomplete Markets, J ournal of Political Econ-omy, 108, 1, 79-119.

[5] Edirisinghe, C., V. Naik and R. Uppal, 1993, Optimal Replication of Options with Transaction Costs and Trading Restrictions, J ournal of Financial and Quantitative Analysis, 28, 117-138.

[6] El Karoui, N. and M .C. Quenez, 1991, Programation Dynamique et¶evaluation des actifs contingents en march¶e incomplet, Comptes Ren-dues de l' Academy des Sciences de Paris, S¶erie I, 313, 851-854

[7] El Karoui, N. and M .C. Quenez, 1995, Dynamic programming andpricing of contingent claims in an incomplete market, SIA M J ournal of

Control and Optimization, 33, 29-66.

[8] Hansen.L .P. andR. J agannathan.I mplicationsof Security Market Datafor Models of Dynamic Economies. J ournal of Political Economy, 99,225-262, 1991.

[9] Karatzas, I. and S. G. Kou, 1996, On thePricingof Contingent Claimswith Constrained Portfolios, Annals of Applied Probability, 6, 321-369.

[10] Longsta®, F ., 1995, How Much Can Marketability A®ect Security Val-ues?, TheJ ournal of Finance, 50, 1767{1774.

[11] Longsta®, F., 2001, Optimal Portfolio Choice and the Valuation of Illiq-

uid Securities, Review of Financial Studies, 14, 407-431.

[12] Longsta®, F., 2004, T heFlight-to-Liquidity Premium in U.S. TreasuryBond Prices, J ournal of Business, 77, 3.

[13] Mas-Colell, A., Whiston, M. and Green, J ., 1995, Microeconomic T he-ory, Oxford University Press.

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A Some P roofs on the Solution of the U pper B ound

for Statistical A rbitrage Opportunities

A .1 P roof of theorem 10

Proof. For any given path m 2 +it;i T let ¸(i T ;T);m(it ;t)

be the dual variableassociated with the superreplication constraint

Epxt;¶ Ti t

hX

s=t;:::;t0¡ 1pxt; t0

¡xt;t0 = s

¢h¢ iss S

it0 +R

t0¡ sB iss

i i¸ Gi T T

with ik 2 Iik+1k and k = t;: :: ;T ¡ 1: Let n

i Tt be the number of nodes that

are predecessors of node i T at time t; where ni Tt is given by

ni Tt = minf T ¡ (i T ¡ 1) ;i T ¡ 1;T ¡ tg+ 1

At each node it; that is a predecessor of i T ; there are #³

+it ;i T

´paths to

reach i T : For any given path m 2 +i t ;it0

let ®(it0;t

0);m

(it ;t) be the dual variable

associated with the self-̄ nancing constraints

Epx

t;¶t0

i t

h¢ ixx

t; t0Sit0 + R

t0¡ xB ixxt; t0

i¡

³¢

i t0t0 S

it0t0 +B

i t0t0

´¸ 0:

Considering ni t0t be thenumber of nodes that are predecessors of node it0 at

time t we have

nit0t = min

©t0¡ (it0 ¡ 1) ;i t0 ¡ 1;t0¡ tª +1

At each node it that is a predecessor of it0 there are #³

+i t ;it0

´.

T he problem that must be solved in order to ¯nd the upper bound of the range of variation of the arbitrage-free value of an European derivative

is a linear programming problem. Its dual problem is

min¸i T

T +1X

j=1

¸ i T Gi T T

where ¸ i T is the sum of thedual variables associated with the positive ex-pected payo®constraints that have the right member equal to Gi T T ; i.e.,

¸i T = T¡ 1X

t=0

X

f i t:it2I tg

X

nm2- +it ;i T

o¸(i T ;T);m(it ;t) S

i T T

35


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37/57


38/57

Summing up all the constraints that concern ¢ ik with ik 2 I k the term

associated with and the term associated with Si T is

pxt; T (xt;T = k)X

f it :i t2I tg

X

nm2- +

it ;i T

o¸(i T ;T);m(i t ;t)

Si T T

As,

T¡ 1X

k=t

pxt;T (xt;T = k) = 1;

summing for all k ¸ t, the term associated with Si T T that is multiplying bypxt;T (xt;T = :) is

X

f i t :it2I tg

Xn

m2- +i t ;i T

o¸(i T ;T);m(it;t)

Si T T

Hence, summing up over all constraints associated with primal variables¢ 's we have that the terms in ¸ associated with Si T T are

T¡ 1X

t=0

X

f it:i t2I tg

X

nm2- +it ;i T

o¸(i T ;T);m(it ;t)

Si T T

Therefore, the sum over all S i T T is

X

f i T :i T 2I T g

T¡ 1X

t=0

X

f it :i t2 I tg

X

nm2- +

i t ;i T

o¸(i T ;T);m(it ;t)

Si T T = S0

Still considering only the constraints associated with primal variables¢ 's; in what follows we describe the terms in ®. For a given Si tt ; the termsin ®are

¡X

s


39/57

As,

t¡ 1X

s=k

pxk ;t (xk;t = s) = 1

the above equations sum up to zero. Summing up over all Si tt a zero willalsobeobtained. Hence, if all dual constraints that concern Sitt are summedup, the following relation is obtained:

S0 =X

f i T :i T 2 I T g

T¡ 1X

t=0

X

f i t:it2I tg

X

nm2- +i t ;i T

o¸(i T ;T);m(i t;t)

Si T T (4)

Now, proceedingin a similar way but considering thedual constraints asso-ciated with B¶s: Because the right member of the constraints is equal to 0,excepting the one associated B0; we multiply each constraint by a constant.

The constraint associated with the variable Bik is multiplied by Rk: Then,

all theconstraints associated with B¶s are summed up, and

X

f i T :i T 2I T g

T¡ 1X

t=0

X

f it :it2 Itg

X

nm2- +

i t ;i T

oq(i T ;T);m(i t;t)

= 1

where

q(

i T ;T);m(i t ;t) = R T¡ t ¸

(i T ;T

);m(it;t) :

Denoting,

qi T =

T¡ 1X

t=0

X

f i t:it2I tg

X

nm2- +

it ;i T

oq(i T ;T);m(it ;t)

Si T T

equation (4) can be written as

S0 = 1

R T

X

f i T :i T 2 I Tg

qi T Si T T

withX

f i T :i T 2I T g

qi T = 1

39


40/57

A .2 P roof of theorem 10 with T =3

Proof. As theproblemsketched in the exampleof section to obtain the up-per boundis alinear programmingproblem, consideringSitt = U

t¡ (i t¡ 1)Di t ¡ 1;its dual is written as follows:

min½¸(i T ; T );m( it ;t )

¾;

(

®(i t0;t0);m( i t ;t )

)³

¸(1;3)(0;0) + ¸

(1;3)(1;1) +¸

(1;3)(1;2)

´G13

+³

¸(2;3);1(0;0) + ¸

(2;3);3(0;0) + ¸

(2;3);3(0;0) +¸

(2;3);1(1;1) +¸

(2;3);2(1;1) +¸

(2;3)(2;1) +¸

(2;3)(1;2) + ¸

(2;3)(2;2)

´G23

+³

¸(3;3);1(0;0) + ¸

(3;3);2(0;0) + ¸

(3;3);3(0;0) +¸

(3;3)(1;1) + ¸

(3;3);1(2;1) + ¸

(3;3);2(2;1) +¸

(3;3)(2;2) + ¸

(3;3)(3;2)

´G33

+³

¸(4;3)

(0;0)+ ¸

(4;3)

(2;1)+ ¸

(4;3)

(3;2)

´G43

subject to the non-negativity constraints of the dual variables

¸(i T ;T);m(i t;t)

¸ 0;

for all i t 2 Ii Tt ; i T 2 I T and t = 0; 1 and 2;

®(i t0;t

0);m

(i t;t) ¸ 0;

for all i t 2 Ii t0t ; it0 2 I t0 and t

0= 0; 1 and 2; and subject to twelve equalityconstraints, each oneassociated with a variable of the primal problem. The

constraint associated with ¢ 0 is given by

(1¡ p)2"

¸(1;3)(0;0)

S13 + P

m=1;2;3¸ (2;3);m(0;0)

S23 + P

m=1;2;3¸ (3;3);m(0;0)

S33 + ¸(4;3)(0;0)

S43

#

+®(1;1)(0;0)S

11 + ®

(2;1)(0;0)S

21+

+(1¡ p)h

®(1;2)(0;0)S

12 +

³®(2;2);1(0;0) +®

(2;2);2(0;0)

´S23 + ®

(3;2)(0;0)S

32

i= S0

(5)

The constraint associated with B0 is given by

(1¡ p)2R3

"

¸(1;3)(0;0) +

Pm=1;2;3

¸(2;3);m(0;0) +

Pm=1;2;3

¸(3;3);m(0;0) + ¸

(4;3)(0;0)

#

+R³

®(1;1)(0;0) + ®

(2;1)(0;0)

´+ (1¡ p) R2

h®(1;2)(0;0) + ®

(2;2);1(0;0) + ®

(1;2);2(0;0) +®

(3;2)(0;0)

i= 1

(6)

40


41/57

The constraint associated with ¢ 11 is given by

(1¡ p)

"¸(1;3)(1;1)S

13 +

P

m=1;2¸(2;3);m(1;1) S

23 + ¸

(3;3)(1;1)S

33

#+

+p(1¡ p)

"

¸(1;3)(0;0)S

13 +

P

m=1;2¸(2;3);m(0;0) S

23 + ¸

(3;3);1(0;0) S

33

#

+

¡ ®(1;1)(0;0)S

11 + p

h®(1;2)(0;0)S

12 + ®

(2;2);1(0;0) S

22

i+ ®

(1;2)(1;1)S

12 + ®

(2;2)(1;1)S

22 = 0

(7)


(1¡ p) R2"

¸ (1;3)(1;1)

+ P

m=1;2¸ (2;3);m(1;1)

+¸ (3;3)(1;1)

#+

+p(1¡ p) R2

"

¸(1;3)(0;0) +

P

m=1;2¸(2;3);m(0;0) +¸

(3;3);1(0;0)

#

+

¡ ®(1;1)(0;0) + pR

h®(1;2)(0;0) + ®

(2;2);1(0;0)

i+ R

h®(1;2)(1;1) + ®

(2;2)(1;1)

i= 0

(8)


(1¡ p)"

¸(2;3);3(1;1) S

23 +

P

m=1;2¸(3;3);m(1;1) S

33 + ¸

(4;3)(1;1)S

43

#+

+p(1¡ p)

"

¸(2;3);3(0;0) S

23 +

P

m=2;3¸(3;3);m(0;0) S

33 + ¸

(4;3)(0;0)S

43

#

+

¡ ®(2;1)(0;0)S

21 + p

h®(2;2);2(0;0) S

22 + ®

(3;2)(0;0)S

32

i+ ®

(2;2)(2;1)S

22 + ®

(3;2)(2;1)S

32 = 0

(9)

41


42/57


(1 ¡ p) R2"

¸(2;3);3(1;1) +

P

m=1;2¸(3;3);m(1;1) + ¸

(4;3)(1;1)

#+

+p(1¡ p) R2

"

¸(2;3);3(0;0) +

P

m=2;3¸(3;3);m(0;0) +¸

(4;3)(0;0)

#

+

¡ ®(2;1)(0;0) + pR

h®(2;2);2(0;0) + ®

(3;2)(0;0)

i+ R

h®(2;2)(2;1) + ®

(3;2)(2;1)

i= 0

(10)


¸(1;3)(1;2)U3S0 +¸ (2;3)(1;2)U

2DS0+ p³

¸ (1;3)(1;1)U3S0 +¸ (2;3);1(1;1) U

2DS0´

+p³

¸(1;3)(0;0)U

3S0+ ¸(2;3);1(0;0) U

2DS0

´¡ ®

(1;2)(0;0)U

2S0 ¡ ®(1;2);1(1;1) U

2S0 = 0

(11)


Rh

¸(1;3)(1;2) + ¸

(2;3)(1;2)

i+ pR

h¸(1;3)(1;1) + ¸

(2;3);1(1;1)

i

+pRh

¸(1;3)(0;0)

+ ¸(2;3);1(0;0)

i¡ ®(1;2);1

(0;0) ¡ ®(1;2)

(1;1) = 0

(12)

The constraint associated with ¢

2

2 is given by¸(2;3)(2;2)S

23 + ¸

(3;3)(2;2)S

33 + p

h³¸(2;3);2(1;1) +¸

(2;3)(2;1)

´S23 +

³¸(3;3)(1;1) + ¸

(3;3);1(2;1)

´S33

i

+ph³

¸(2;3);2(0;0) +¸

(2;3);3(0;0)

´S23 +

³¸(3;3);1(0;0) + ¸

(3;3);2(0;0)

´S33

i

¡h

®(2;2);1(0;0) + ®(2;2);2(0;0)

iS22 ¡

h®(2;2)(1;1) + ®

(2;2)(2;1)

iS22 = 0

(13)


R

h

¸

(2;3)

(2;2) + ¸

(3;3)

(2;2)

i

+pR

h

¸

(2;3);2

(1;1) + ¸

(2;3)

(2;1) + ¸

(3;3)

(1;1) + ¸

(3;3);1

(2;1)

i

+pRh

¸(2;3);2(0;0) + ¸

(2;3);3(0;0) + ¸

(3;3);1(0;0) + ¸

(3;3);2(0;0)

i

¡h

®(2;2);1(0;0) +®

(2;2);2(0;0)

i¡

h®(2;2)(1;1) + ®

(2;2)(2;1)

i= 0

(14)

42


43/57


¸(3;3)(3;2)

UD2S0 + ¸(4;3)(3;2)

D3S0+ ph

¸ (3;3);2(2;1)

UD2S0+ ¸(4;3)(2;1)

D3S0i

+ph

¸(3;3);3(0;0)

UD2S0 +¸(4;3)(0;0)

D3S0

i¡ ®(3;2)

(0;0)D2S0 ¡ ®

(3;2)(1;1)

D2S0 = 0

(15)


Rh

¸(3;3)(3;2) + ¸

(4;3)(3;2)

i+ pR

h¸(3;3);2(2;1) +¸

(4;3)(2;1)

i+

+pRh

¸(3;3);3(0;0) + ¸

(4;3)(0;0)

i¡ ®

(3;2)(0;0) ¡ ®

(3;2)(1;1) = 0

(16)

Summing up equations (5), (7), (9), (11),(13) and (15) we obtain

S0 =³

¸(1;3)(0;0)

+ ¸ (1;3)(1;1)

+ ¸ (1;3)(1;2)

´S13+

+

ÃP

m=1;2;3¸ (2;3);m(0;0)

+ P

m=1;2¸ (2;3);m(1;1)

+¸ (2;3)(2;1)

+ ¸(2;3)(1;2)

+ ¸(2;3)(2;2)

!

S23

+

ÃP

m=1;2;3¸(3;3);m(0;0) + ¸

(3;3)(1;1) +

P

m=1;2¸(3;3);m(2;1) + ¸

(3;3)(2;2) + ¸

(3;3)(3;2)

!

S33

+³

¸(4;3)(0;0) +¸

(4;3)(2;1) +¸

(4;3)(3;2)

´S43

(17)

Multiplying equations (8) and (10) by R and equations (12), (14) and(15) by R2 and then summing up with equation (6) weobtain

¸(1;3)(0;0) +

P

m=1;2;3¸(2;3);m(0;0) +

P

m=1;2;3¸(3;3);m(0;0) + ¸

(4;3)(0;0)+

¸(1;3)(1;1) +

P

m=1;2¸(2;3);m(1;1) + ¸

(3;3)(1;1) + ¸

(2;3)(2;1) +

P

m=1;2¸(3;3);m(2;1) +¸

(4;3)(2;1)+

+¸(1;3)(1;2) + ¸

(2;3)(1;2) + ¸

(2;3)(2;2) + ¸

(3;3)(2;2) + ¸

(3;3)(3;2) +¸

(4;3)(3;2) =

1R3

Hence, denoting

q(it0;t

0)

(i t ;t) = R3¸

(it0;t0)

(it ;t)

43


44/57

and

q1 = q(1;3)(0;0)

+ q(1;3)(1;1)

+ q(1;3)(1;2)

q2 = P

m=1;2;3q(2;3);m(0;0) +

P

m=1;2q(2;3);m(1;1) + q

(2;3)(2;1) + q

(2;3)(1;2) + q

(2;3)(2;2)

q3 = P

m=1;2;3q(3;3);m(0;0) + q

(3;3)(1;1) +

P

m=1;2q(3;3);m(2;1) + q

(3;3)(2;2) + q

(3;3)(3;2)

q4 = q(4;3)(0;0) + q(4;3)(2;1) + q

(4;3)(3;2)

we can rewrite (17) as

S0 = 1R3

£q1S13 + q

2S23 + q3S33 + q

4S43¤

with qi

are dē ed above and q1

+q2

+q3

+q4

= 1: Taking into consideration equation (10), equation (9) can be rewrittenas

S12 = 1

R

h¼(1;3)(1;2)

S13 + ¼(2;3)(1;2)

S23

i

where

¼(1;3)(1;2)

= ¸

(1;3)

(1;2)+p¸

(1;3)

(1;1)+p̧

(1;3)

(0;0)

¸(1;3)

(1;2)+p¸

(1;3)

(1;1)+p¸

(1;3)

(0;0)+¸

(2;3)

(1;2)+p¸

(2;3);1

(1;1) +p¸

(2;3);1

(0;0)

¼(2;3)(1;2) =

¸(2;3)

(1;2)+p¸

(2;3);1

(1;1) +p̧

(2;3) ;1

(0;0)

¸(1;3)

(1;2)+p¸

(1;3)

(1;1)+p¸

(1;3)

(0;0)+¸

(2;3)

(1;2)+p¸

(2;3);1

(1;1) +p¸

(2;3);1

(0;0)

and ¼(1;3)(1;2) + ¼

(2;3)(1;2) = 1: Proceeding in an analogous way with constraints

(13), (14), (15) and (16) we obtain

S22 = 1

R

h¼(2;3)(2;2)S

23 + ¼

(3;3)(2;2)S

33

i

where

¼(2;3)(2;2) =

¸(2;3)(2;2)

+p³

¸(2;3);2(1;1)

+¸ (2;3)(2;1)

+¸ (2;3);2(0;0)

+¸(2;3) ;3(0;0)

´

¸ (2;3)(2;2)

+p³

¸ (2;3);2(1;1)

+¸ (2;3)(2;1)

+¸(2;3) ;2(0;0)

+¸ (2;3);3(0;0)

´+¸ (3;3)

(2;2)+p

³¸ (3;3)(1;1)

+¸(3;3);1(2;1)

+¸ (3;3);1(0;0)

+¸ (3;3);2(0;0)

´

¼(3;3)(2;2) =

¸(3;3)(2;2)

+p³

¸(3;3)(1;1)

+¸ (3;3);1(2;1)

+¸ (3;3);1(0;0)

+¸(3;3) ;2(0;0)

´

¸ (2;3)

(2;2)

+p³

¸ (2;3);2

(1;1)

+¸ (2;3)

(2;1)

+¸(2;3) ;2

(0;0)

+¸ (2;3);3

(0;0)

´+¸ (3;3)

(2;2)

+p³

¸ (3;3)

(1;1)

+¸(3;3);1

(2;1)

+¸ (3;3);1

(0;0)

+¸ (3;3);2

(0;0)

´

with ¼(2;3)(2;2) + ¼

(3;3)(2;2) = 1; and

S32 = 1

R

h¼(3;3)(3;2)

S33 + ¼(4;3)(3;2)

S43

i

44


45/57

where

¼(3;3)(3;2)

= ¸ (3;3)

(3;2)+p³ ¸ (3;3);2

(2;1) +¸ (3;3);3

(0;0)´

¸(3;3)

(3;2)+p

³¸(3;3);2

(2;1) +¸

(3;3);3

(0;0)

´+¸

(4;3)

(3;2)+p

³¸(4;3)

(2;1)+¸

(4;3)

(0;0)

´

¼(4;3)(3;2)

= ¸

(4;3)

(3;2)+p

³¸(4;3)

(2;1)+¸

(4;3)

(0;0)

´

¸(3;3)

(3;2)+p

³¸(3;3);2

(2;1) +¸

(3;3);3

(0;0)

´+¸

(4;3)

(3;2)+p

³¸(4;3)

(2;1)+¸

(4;3)

(0;0)

´

with ¼(3;3)(3;2)

+ ¼(4;3)(3;2)

= 1:

Moreover, using equations (7), (8), (9) and (10) we have

S11 = 1

R2

hµ(1;3)(1;1)S

13 + µ

(2;3)(1;1)S

23 + µ

(3;3)(1;1)S

33

i+

1

R

h"(1;2)(1;1)S

12 + "

(2;2)(1;1)S

22

i

where

µ(1;3)(1;1)

= R2

h(1¡ p)¸ (1;3)(1;1)+p(1¡ p)¸ (1;3)(0;0)

i

£

µ(2;3)(1;1) =

R2

"

(1¡ p) P

m=1;2¸(2;3);m

(1;1) +p(1¡ p)

P

m=1;2¸(2;3);m

(0;0)

#

£

µ(3;3)(1;1)

= R2

h(1¡ p)¸

(3;3)

(1;1)+p(1¡ p)¸

(3;3);1

(0;0)

i

£

"(1;2)(1;1) =

Rh

p®(1;2)(0;0)

+®(1;2)(1;1)

i

£ ; "(2;2)(1;1) =

Rh

p®(2;2);1(0;0)

+®(2;2)(1;1)

dry markets and statistical arbitrage bounds for european derivatives

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