A numerical algorithm for indifference pricing in incomplete markets

A numerical algorithm for indifference pricing in

incomplete markets

Juyoung Lim∗

Department of Mathematics

The University of Texas at Austin

September 8, 2005

Abstract

This paper proposes a numerical algorithm to compute the indiffer-

ence price and risk monitoring strategy of a contingent claim in incom-

plete markets with the exponential preference. Using the duality between

the exponential optimal investment problem and the minimal relative en-

tropy problem, we recast the option writer’s optimal investment problem

as a minimax problem and derive the complete procedure of finding the

solution numerically. The Lagrange multiplier process emerges from the

iterative minimax optimization procedure, and is shown to be connected

to the “delta” of the indifference price. We present the numerical results

of the algorithm with two representative examples, one with the nontrad-

able assets and the other with the stochastic volatility model. The results

show that the algorithm computes not only the indifference price but also

the indifference “delta” very efficiently.

Keywords: incomplete market, exponential utility, indifference pricing, rela-tive entropy, Lagrange multiplier, multinomial tree

∗[email protected]

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1 Introduction

The purpose of this paper is to propose a numerical scheme to compute the indif-ference price and risk monitoring strategies of contingent claims in incompletemarkets with the exponential preference. The indifference price of a contin-gent claim is found by comparing two value functions of the optimal investmentproblems of an investor with and without the random liability (or endowment)of the claim. The framework of indifference pricing, or more widely put, theutility based pricing rule in incomplete markets has been developed by Hodgesand Neuberger (1989) [19], Davis (1997)[10], Kallsen (2002) [22], and Hugonnier,Kramkov, and Schachermayer (2005) [20], to name a few. There has been signif-icant advance in understanding the theoretical implications of the utility basedpricing rule in incomplete markets, using different methodologies.One direction is to work with a diffusion model and derive a partial differentialequation (PDE) for indifference price. Musiela and Zariphopoulou (2004a) [25]considered an incomplete market where there are a tradable asset and a nontradable asset with correlated Brownian motions and derived a nonlinear PDEfor indifference price of a claim. Sircar and Zariphopoulou (2004)[31] considereda stochastic volatility model and showed that the indifference price is boundedby a risk neutral price and the certainty equivalent price.Another direction is to use the duality method of Fritelli (2000)[15] and Delbaenet al.(2002)[11] and to derive a backward stochastic differential equation (BSDE)for the indifference price process. Rouge and El Karoui (2000)[30] studied adiffusion model with a constrained trading opportunity and derived the BSDEfor indifference price. Mania and Schweizer (2004)[23] considered a general semi-martingale filtration, derived the BSDE for the indifference price and studiedthe asymptotic results with respect to risk aversion coefficient.In contrast to the advance in theoretical study there have been only few stud-ies about the numerical implementation of indifference pricing. Grasselli andHurd (2004)[16] proposed a Monte Carlo algorithm for the exponential hedgingproblems which is computationally expensive. Grasselli and Hurd (2005)[17]studied the numerical methods for indifference pricing for stochastic volatilitymodel when the volatility process follows a reciprocal CIR process, and showedthe result for the volatility claims only, leaving out the claims on the stock.

In this paper we propose a multi-nomial tree model of the indifference pricingand derive the backward iterative algorithm on the tree to compute the indif-ference price process and the risk monitoring strategy of the option writer(andthe buyer) in the incomplete market. The arbitrage free pricing in completemarkets, developed by Black and Scholes (1973) [5], Merton (1973) [24], andRoss (1976) [29], was put to best use by the binomial tree model of Cox, Rossand Rubinstein(1979)[8], which remains to be popular to this date. The benefitof working with the tree model is that we get to look at the dynamic nature ofthe indifference pricing in plain mathematics and that the emerging (backward)recursive formulae for the price and optimal hedging portfolio are numerically

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implementable.The dynamic procedure of pricing a contingent claim in incomplete markets hasbeen studied by El Karoui and Quenez (1995) [13], Rouge and El Karoui (2000)[30], Fritelli (2000b) [14], Becherer (2003)[4] and Musiela and Zariphopoulou(2004b)[26]. All these works focused on the economic coherency such as the noarbitrage condition and the semi-group property. Theoretical advance notwith-standing, it is important to develop the practically implementable algorithmssuitable for incomplete markets and this paper seeks to propose a solution.Our algorithm of indifference pricing uses the exponential preference and hasthe following ingredients; the option writer’s (or buyer’s) risk averse coefficientγ > 0, the discrete state-time stochastic dynamics of the financial market inthe form of a (multi period) multi nomial tree, including the price evolutionof the tradable assets along the tree, and a bounded European claim to price.The algorithm flows backward on the tree, conducting a convex optimizationwith a constraint at each time period. We solve the optimization problemby the method of Lagrange multipliers, giving rise to the Lagrange multiplierprocess. We show that the Lagrange multiplier process is a constant multiple(in fact (−γ)) of the writer’s optimal portfolio of the traded assets with theoption, and is the sensitivity of the writer’s value function with respect to theperturbation of the traded assets’ price. The indifference price is the differenceof the logarithms of two value functions (with and without option writing) andtherefore the “delta” of the indifference price emerges out of the algorithm inthe form of the difference of the Lagrange multiplier processes divided by γ.

We provide two thorough examples of numerical implementations, along withthe graphs of numerical results, one with pricing the basket/spread option whena component is non tradable, and the other with pricing the call option whenthe stock price follows the Heston model. You will see how the indifference pricechanges when we vary the parameters such as the risk averse coefficient and thecorrelation between the tradable asset and non tradable ones (or the volatility).The speed and accuracy of the algorithm appears suitable for practical usage.We also performed the numerical experiment of increasing the time period ofthe trees, and found that the algorithm produces a good approximation of boththe indifference price and its“delta” of the diffusion models.

The rest of the paper is organized as follows. In Section 2, we set up the modelof an incomplete market, define the indifference price of a claim, and derive thepricing algorithm in the simplest one period model, which serves as a buildingblock of the multi period version. Next, in Section 3, we develop the indiffer-ence pricing algorithm in the general multi period and multi nomial tree of anincomplete market. In the multiperiod model, the market incompleteness showsup in the value function in the form of the aggregate entropy of the uncertaintyto come later. In section 4 we study in more detail the financial interpretationof the Lagrange multiplier process that surfaces from the algorithm, and dub itthe indifference hedge portfolio. The following section 5 has three subsections,

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the first two with the examples of numerical implementations and the last withthe experiment to see the convergence behavior of the algorithm as the timeperiod of the tree increases. And then we conclude.

2 Indifference pricing in a discrete time model

2.1 General framework of an incomplete market

Consider a discrete time financial market (Ω,F , F, P ) with the actual probabil-ity measure P , the filtration F = ∪T

t=0Ft and F = FT . There are risky assetsand a riskless bond with zero interest rate, all adapted to F and progressivelymeasurable. Suppose the probability space Ω = ω1, . . . , ωN has finitely manystates and the risky assets are bounded almost surely. We specify the assetsS1, . . . , Sd as tradable to indicate that the admissible portfolio of an investorshould be composed of the riskless bond and the tradable assets in an F pre-dictable manner.A self financing strategy of the investor with the initial capital x and θk(t) sharesof Sk for k = 1, . . . , d has the value

Xx,θ(t) = x +t∑

u=1

θ>u ∆Su. (2.1)

where θ = (θ1, . . . , θd)> and ∆St = St − St−1 denotes the vector increment for

each t. We shall denote by x>y =∑d

k=1 xiyi for x, y ∈ Rd and the vectors are

column vectors.

Definition 1 An FT measurable payoff B is said to be replicable if there is apredictable portfolio process θ(u) and an initial wealth x such that

x +

T∑

t=1

θ>u ∆Su = B a.s.

A financial market is complete if every bounded FT measurable payoff is repli-cable. In that case the arbitrage free price of a payoff B is the initial cost ofthe replicating portfolio. If there exist non replicable payoffs then the marketis incomplete and there is no longer a unique arbitrage free price for them. Itis well known that there is in fact an interval of arbitrage free prices for a nonreplicable payoff, bounded by the sub hedge price and the super hedge price.For more details see El Karoui and Quenez (1991) [13], Cvitanic and Karatzas(1993) [9], and Rouge and El Karoui (2000) [30].For instance if there is a non traded asset, the market is incomplete. It iscommon in a commodity or energy market that basket and spread options aretraded while the reference assets are not quite liquid or practically non tradable.Any kinds of restrictions on the trading opportunity give rise to incompletemarkets. Another important example of incomplete markets occurs when theparameters of the tradable asset dynamics are stochastic on their own. In the

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following analysis we assume that the market is free of arbitrage but may beincomplete.

We model the individual risk preference via an exponential utility function

U(x) = −e−γx, γ > 0

and consider two optimal investment problems. One is the classical Mertonmodel of investment problem, namely

V (x, t) = supθ∈Θ

EP [−e−γXx,θ(T )| Xx,θ(t) = x,Ft]. (2.2)

The other is the optimal investment problem of an investor with a randomliability B that is FT measurable. We call it the writer’s maximal utility problemand its value function is given by

V w,B(x, t) = supθ∈Θ

EP [−e−γ(Xx,θ(T )−B)| Xx,θ(t) = x, ,Ft]. (2.3)

The market is not necessarily Markovian, and therefore the value functionsV and V w,B are not a function of x and t alone but a process adapted toF. Yet the notation above should serve as the shorthand ones. We assumethat the admissible set Θ is a collection of the d dimensional F predictableprocesses that are bounded almost surely and Xx,θ(t) was defined in (2.1). Wedo not impose the short sale or budget constraints, while it is possible to adjustthe subsequent analysis to such assumptions. For the simplest exposition ofthe numerical algorithm later, we stick to the assumptions above. Hence Θ isindependent of x in this paper. We suppress the t-dependence of Θ whenever itis evident. For the discussions on the relevant classes of Θ in a continuous timemodel see Rouge and El Karoui(2000)[30], Delbaen et al.(2002)[11] and Maniaand Schweizer(2004)[23].

Definition 2 The process of writer’s indifference price of a European claim Bis defined as the solution hw

t (B) of the following equation.

V (x, t) = V w,B(x + hwt (B), t). (2.4)

Hence the indifference price is the extra wealth that makes the option writerindifferent between writing and not writing the option under the optimal invest-ment. With the exponential utility the indifference price is independent of thewealth x at time t.We stress that h is not a function of t alone but a stochastic process adapted toF. If the market is described by a Markov process of the state variables, thenthe indifference price would be a function of t and the state variables.Clearly we could define the buyer’s indifference price hb(B) of the same claimB analogously and it is related with the writer’s price by

hw(B) = −hb(−B).

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We focus on the writer’s price throughout our analysis of the pricing algorithmand the notation h or h(B) will denote the writer’s indifference price unlessotherwise mentioned. You will find the numerical results of both of them inSection 5.

The two optimal investment problems (2.2) and (2.3) do not have the closed formsolutions. However their dual problems have a simpler structure and give thesemi-closed form solutions. Let us introduce the relative entropy which naturallyshows up as the convex dual function of the exponential utility function.

Definition 3 The relative entropy H(P1|P2) between two probability measuresP1 and P2 on (Ω,F) is defined as

H(P1|P2) =

EP1

[

log(

dP1

dP2

)]

if P1 P2

∞ otherwise

where dP1

dP2is the Radon Nikodym derivative of P1 with respect to P2.

Recall that we fixed the actual measure P that will be used throughout thepaper. For P we also define

• Me: the collection of equivalent martingale measures of P

• Q: the minimal entropy martingale measure (MEMM) of P i.e.

Q = argminM∈MH(M |P ) (2.5)

The set Me is nonempty if the financial market is arbitrage free. Fritelli(2000)[15] showed the existence and uniqueness of the MEMM for an arbitragefree market assuming the traded assets are bounded. Hence the definition aboveis well defined in our setting.

Next let us present the duality theorem of the maximal utility problem with anexponential utility for reference purpose.

Proposition 2.1 The optimal investment problem with the future liability B isdual to minimal relative entropy problem with a penalty term as follows:

supθ∈Θ

E[−e−γ(Xx,θ(T )−B)] = −e−γx exp

(

γ supM∈Me

− 1

γH(M |P ) + E

M [B]

)

(2.6)

The duality relation has been established in various settings such as the discretemulti period models and the continuous time semi-martingale models. See Del-baen et al. (2001) [11] for a thorough review of the known variations. We use

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the discrete versions in the derivation of the numerical algorithm to computethe values. Also useful to note is that the duality relation remains valid withthe conditional expectation on both sides, namely,

supθ∈Θ

E[−e−γ(Xx,θ(T )−B)|Ft] = −e−γx exp

(

γ supM∈Me

− 1

γHt(M |P ) + E

M [B|Ft]

)

(2.7)where the conditional entropy is defined by

Ht(M |P ) =

EM[

log(

dMdP

)

|Ft

]

if M P∞ otherwise

(2.8)

and the wealth process Xx,θ(T ) = x +∑T

u=t+1 θ>u ∆Su starts at t, for t ∈0, 1, . . . , T. We can rewrite (2.7) as

supθ∈Θ

E[−e−γ(Xx,θ(T )−B)|Ft] = −e−γx exp

(

− infM∈Me

EM

[

log

(

dM

dP

)

− γB

∣

∣

∣

∣

Ft

])

(2.9)which will be used later.

Now let us rephrase the duality relation (2.6) as follows, keeping in mind thateverything below is valid with conditioning with respect to Ft. First, multiply(−1) to and take the logarithms of (2.6) to get

infθ∈Θ

log E[e−γ(Xx,θ(T )−B)] = −γx − infM∈Me

H(M |P ) − EM [γ · B]

. (2.10)

Recall that Xx,θ(T ) = x + X0,θ(T ) for any θ ∈ Θ and add γ · x to both sides of(2.10) then it becomes

infθ∈Θ

log E[eγ(B−X0,θ(T ))] = − infM∈Me

H(M |P ) − EM [γ · B]

.

Now the value functions of the writer’s optimal investment problem withoutand with the option writing, can be written as

V (x, t) = − exp

(

−γx − infM∈Me

Ht(M |P )

)

and

V B(x + h, t) = − exp

(

−γ(x + h) − infM∈Me

Ht(M |P ) − EM [γ · B|Ft]

)

.

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Definition 4 For an F measurable and bounded payoff B, we define a realvalued functional process

vP,γt (B) := − inf

M∈Me

Ht(M |P ) − EM [γ · B|Ft]

(2.11)

and

vPt (0) := − inf

M∈Me

Ht(M |P ) (2.12)

for t ∈ 0, 1, . . . , T.

The functionals are finite and well defined with the no arbitrage assumption inthe discrete time-state model. It follows from the convexity of Me and the lowersemi continuity of the relative entropy. (See Lemma 1.4.3 part (b) of Dupuisand Ellis (1997) [12]). We immediately get the following from the definition ofthe indifference price.

Proposition 2.2 The writer’s indifference price process ht of B equals to

ht(B) =1

γ

[

vP,γt (B) − vP

t (0)]

(2.13)

for t ∈ 0, 1, . . . , T.

For the functional vP,γt we will drop the superscript P and just write vγ

t (·)unless the base probability measure is different. Note that v(0) is independentof γ. Our numerical algorithm is designed to compute the value of vγ(B) andconsequently the indifference price. The advantage of using (2.13) is twofold.Firstly the functional vγ(·) is the value function of a stochastic control problemwith a running cost H(·|P ) and the final cost −γ ·B, permitting the use of thedynamic programming type algorithm. Secondly the relative entropy has manydesirable properties helpful in analyzing the algorithm.

2.2 The one period model

The purpose of this section is to derive a numerical algorithm to compute thefunctional vγ(B) and the indifference price h(B) in a one period model withdiscrete states.

Suppose that the state space is Ω = ω1, . . . , ωN and the actual measure P onΩN is given by a probability vector

P (ωi) = pi > 0 with

N∑

i=1

pi = 1.

We consider the σ algebra F naturally generated by Ω. Let M be any givenprobability measure on Ω then the relative entropy between P and M equals to

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H(M |P ) =

N∑

i=1

mi log

(

mi

pi

)

(2.14)

if mi > 0 for all i ∈ 1, . . . , N, and is ∞ otherwise.

Let us characterize Me and Q associated with the market (Ω,F , P ). Recall theriskless interest rate is zero for simplicity. Let Sk(0) be the initial price of thek th tradable asset and denote its vector increment by

∆sik = Sk(ωi) − Sk(0)

for i ∈ 1, . . . , N and k ∈ 1, . . . , d. We make two standard assumptions.

Assumption 2.1 None of the tradable assets is redundant. i.e. the incrementmatrix [∆sik] has rank d.

This implies N ≥ d and that the covariance matrix of the asset increments withany equivalent measure is positive definite.

Assumption 2.2 The market is free of arbitrage. Then it implies (∆s1k, . . . , ∆sNk)has both positive and negative elements, for each k ∈ 1, . . . , d.

Then the collection of the equivalent martingale measure on (Ω,F , P ) is givenby

Me =

(m1, . . . , mN) ∈ RN+ |

N∑

i=1

mi = 1, andN∑

i=1

mi∆sik = 0, 1 ≤ k ≤ d

.

(2.15)Hence we can express vγ(B), defined in (2.11), as the optimal value of theminimax problem

minλ,µ

maxm

L(λ, µ, m)

where the Lagrangian L(λ, µ, m) is given by

−N∑

i=1

mi

(

log

(

mi

pi

)

− γbi

)

+ λ>(

N∑

i=1

mi∆sik

)

+ µ

(

N∑

i=1

mi − 1

)

(2.16)with bi := B(ωi). Note that λ = (λ1, . . . , λd)

> and µ are the Lagrange multi-pliers.

Finding a measure with the minimal relative entropy with moment constraints isa classical problem in information theory and statistical mechanics (See Coverand Thomas (1991) [7] and Jaynes (1957) [21]). See Cherny and Maslov (2004)

9

[6] for the common features of the minimal entropy problems arising in thedifferent disciplines.In the context of mathematical finance Avellaneda et al. studied the problemfor minimal entropy calibration of the risk neutral measure and proposed thenumerical procedures to solve it; see Avellaneda et al.(1997) [3], (1998) [1],(2001)[2].The main difference of our algorithm from theirs is that they continue to usethe linear pricing rule with the calibrated risk neutral measure whereas we usethe indifference pricing which is nonlinear for non replicable payoffs.

It is straightforward to find the solution to (2.16). Below we use the notationthat

∆S(i) = (∆S1(ωi), . . . , ∆Sd(ωi))>

for i ∈ 1, . . . , N and λ>∆S(i) =∑d

k=1 λk∆sik. The same goes for the initialprice vector S(0).

Proposition 2.3 For the minimax problem (2.16)

1. the optimal M = (m1, . . . , mN) has the Boltzmann Gibbs form

mi = pi

exp(

γbi + λ>∆S(i))

Z(λ)(2.17)

with the normalizing constant

ZB(λ) :=

N∑

i=1

pi exp(

γbi + λ>∆S(i))

. (2.18)

2. the Lagrange multiplier λ is the minimizer of the function

WB(λ) := log ZB(λ). (2.19)

3. the function WB(λ) is coercive, strictly convex and has a unique mini-mizer.

4. We have vγ(B) = minλ

WB(λ) = −H(M |P ) + EM [γB].

Proof The gradient of the Lagrangian is

∂L

∂mi

= − log

(

mi

pi

)

+ γbi − 1 + µ + λ>∆S(i) = 0 for i ∈ 1, . . . , N.

Using the constraint∑N

i=1 mi = 1 we elimiinate µ variable and obtain (2.17).Plugging in the minimizing measure, the Lagrangian minλ L(λ, m∗) then be-comes log Z(λ) and its critical point satisfies

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0 =1

ZB(λ)

∂ZB

∂λk

=1

ZB(λ)

N∑

i=1

pi exp(

γbi + λ>∆S(i))

∆sik

= EM [∆Sk]

for k = 1, . . . , d. Hence a critical point of WB(λ) gives rise to an equivalentmartingale measure on Ω.

It remains to show that WB(λ) is coercive and strictly convex. From Assumption2.2 (no arbitrage) we see that lim‖λ‖→∞ WB(λ) = ∞, establishing coercivity.To show the strict convexity of W (λ), consider its second partial derivative then

∂2WB

∂λk∂λl

=

∂2ZB

∂λk∂λl

ZB(λ)−

∂ZB

∂λk

∂ZB

∂λl

Z2B(λ)

= CovM [∆Sk∆Sl].

for k, l ∈ 1, . . . , d. Hence the Hessian matrix of WB is positive definite fromAssumption 2.1 (no redundant asset) and we conclude that WB has a uniqueminimizer λ∗.Finally the formulae in 4. are obtained by substituting (2.17) in the Lagrangian.

In particular with the zero payoff B ≡ 0 we obtain the minimal martingalemeasure (MEMM) Q. Let us denote by

Z0(λ) =

N∑

i=1

pi exp(

λ>∆S(i))

and W0(λ) = log Z0(λ) for λ ∈ Rd. (2.20)

Corollary 2.1 The relative entropy between the MEMM Q and the actual mea-sure P is given by

H(Q|P ) = −minλ

W0(λ). (2.21)

Corollary 2.2 The indifference price of B equals to

h =1

γ[min

λWB(λ) − min

λW0(λ)] (2.22)

Based on this we have the following algorithm to compute the indifference priceof B in the one period model:

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1. Construct a finite state space Ω calibrated to a model of financial market.(For instance a diffusion model or a time series model.)

2. Compute the increment matrix ∆sik 1≤i≤N

1≤k≤d

of the tradable assets.

3. Find the MEMM q1, . . . , qN

(a) Using a gradient based optimization routine, minimize the convexfunction W0(λ).

(b) With the Lagrangian multiplier λ the MEMM is given by qi =pi exp(λ>∆S(i))/Z0(λ).

4. For a given claim to price and the risk aversion coefficient γ, compute therisk adjusted payoff vector γ · biN

i=1.

5. Using a gradient based optimization routine, minimize the convex functionWB(λ).

6. Output the indifference price h(B) = 1γ[WB(λ) − W0(λ)], each evaluated

at its own Lagrange multiplier.

Observe that the steps (1) through (3) of finding the MEMM, are independentof the payoff B. Extra computations for each payoff are limited to minimizingWB(λ). We will extend this to a more realistic multi period model in the nextsection, but the key ingredients are summarized here. The indifference priceof a non replicable payoff is found by solving a minimax problem associatedwith the dual formulation. The Lagrange multipliers λ are found by minimizingthe convex function WB(·) and W0(·) respectively, and has a natural financialinterpretation as follows.

Theorem 2.1 Let λ0 and λγB be the minimizers of W0(λ) and WB(λ), respec-

tively, and setη := λγ

B − λ0.

Then the sensitivity of the indifference price to the perturbation of the initialprice of the k th tradable asset equals to ηk divided by γ and multiplied by (−1).

∂h

∂Sk(0)= − 1

γηk , for k ∈ 1, . . . , d. (2.23)

Proof In the Lagrangian (2.16)to compute vγ(B), the constraint∑

i=1 mi∆sik =0 is equivalent to

∑

i=1 miSk(ωi) = Sk(0). Hence, perturbing Sk(0) amounts toperturbing the constraint. It follows from the Kuhn Tucker theorem that theLagrange multiplier is the sensitivity vector (See for instance Theorem 5.2.16 of

Peressini et al. (1988) [28]), i.e. ∂vγ(B)∂Sk(0) = λγ

B and ∂v(0)∂Sk(0) = λ0. Therefore we

conclude that

∂h

∂Sk(0)= − 1

γ

[

∂vγ(B)

∂Sk(0)− ∂v(0)

∂Sk(0)

]

= − 1

γ[λγ

B − λ0]

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completing the proof.

The sensitivity of indifference price is expected to build a “hedge” portfolio inincomplete market which is discussed in Section 4. Our algorithm computes itwithout any extra work.

Clearly our algorithm easily computes the risk neutral price of the payoff underthe MEMM. It was named as the fair price by Davis (1997) [10] and also asentropy price in Fritelli (2000) [15], and, in terms of our notation, equals to

he :=

N∑

i=1

qibi.

It is also easy to compute the sensitivity of the entropy price with respect tothe perturbation of the price of tradable asset.

Proposition 2.4 (Avellaneda (1998) [1] ) The sensitivity of entropy priceto the perturbation of the initial price of the k th tradable asset is the regressioncoefficient of the payoff on the k the tradable asset under the MEMM.

If we express the regression coefficient in terms of our notation the sensitivityof the entropy price is computed as follows:

∂he

∂Sk(0)=

d∑

j=1

(

H−1)

jkgj.

where H−1 is the inverse of the the Q-covariance matrix H of the incrementmatrix and g is the the Q-covariance of the tradable assets and the payoff B,namely,

Hkl =

N∑

i=1

qi∆sik∆sil and gk =

N∑

i=1

qibi∆sik

for k, l ∈ 1, . . . , d.

3 A multi-period model

In this section we work with a multi period and multi nomial tree model andderive the algorithm to compute the functional vγ(B) and the indifference priceh(B).Consider an N -nominal tree with the time period t ∈ 0, 1, . . . , T. If the tree isnon recombining the number of the final states is NT . If it is recombining likethe familiar binomial tree of Cox, Ross, and Rubinstein (1979) [8], the numberof the final states would be of the order T N−1, far less than NT . In this section

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we’ll be deriving the algorithm for the non recombining tree since it is moregeneral. It is straightforward to adapt the algorithm to a recombining tree aswill be explained in detail in Section 4.

The state space is the set of the final nodes of a T -period and N -nomial tree,namely, Ων = ω1, . . . , ων with ν = NT and let F = ∪T

t=0Ft be the filtrationgenerated by the tree. The actual measure P is given by a probability vectorP (ωi) = pi > 0 for ωi ∈ FT with

∑νi=1 pi = 1.

We shall describe the algorithm in a backward inductive manner, which requiresa reference system for the subtrees within. It is convenient to think of thetree and a probability measure on it as a Markov chain that may be timeinhomogeneous. The following notation is a generalization of Figure 1.

ω0,1

ZZ

ZZp(1, 1) p(1, 2)

ω1,1 ω1,2

@@@

@@@

ω2,1 ω2,2 ω2,3 ω2,4

p(2, 1) p(2, 4)

Figure 1: The indexing of a binomial tree

Notation Of the T -period and N -nomial tree and a probability measure Pon it,

• for each t ∈ 0, 1, . . . , T, the set of the spatial indices is given by It =1, 2, . . . , N t.

• ωt,i denotes an atom of Ft for i ∈ It.

i.e. P[ω(t) = i] = P[ωt,i] > 0 ⇔ i ∈ It.

• P[ω(0) = 1] = 1

• The transition probability is denoted by

p(t + 1, j|t, i) := P[ω(t + 1) = j|ω(t) = i].

• we enforce the tree structure as follows:

for each t ∈ 0, 1, . . . , T − 1 and i ∈ It the spatial indices of the childrennodes are given by

j = i(c) := (i − 1) · N + c , c ∈ 1, . . . , N.

and we set

14

p(t + 1, j | t, i) =

p(t + 1, j) if j = i(c) for some c ∈ 1, . . . , N0 otherwise

(3.1)

with∑N

c=1 p(t + 1, i(c)) = 1.

• we shall simply abbreviate p(t + 1, j |t, i) with j = i(c) by

p(t + 1, i(c)) or p(c) (3.2)

,dropping the right conditioning part (t, i) whenever the conditioning stateωt,i is clear.

• For a positive integer i we define r(i) the unique integer satisfying

i = (r(i) − 1) · N + c (3.3)

with a positive integer c = 1, . . . , N .

Hence the collection of the subsequent nodes of ωt,i’s is ωt+1,i(1), . . . , ωt+1,i(N)and its preceding node is ωt−1,r(i). The probability of a final state is the productof the transition probabilities of the preceding branches, namely

pi = p(ωi) =T−1∏

t=0

p(

T − t, rt(i))

(3.4)

for i ∈ IT where rt(i) = r r · · · r(i) denotes the t fold compositions of findingthe preceding node.Another important note is that we will use the indexing system for all thestochastic processes such as the tradable asset and the payoff.

With this notation, let us identify Me and Q associated with (Ω,F , F, P ). Thecollection Me of the equivalent martingale measures is given by

Me = M = (m1, . . . , mν) ∈ Rν | mi > 0,

ν∑

i=1

mi = 1

with the additional specification that

N∑

c=1

m(t + 1, i(c)) · ∆Sk(t + 1, i(c)) = 0 (3.5)

for k ∈ 1, . . . , d, t ∈ 0, 1, . . . , T − 1 and i ∈ It. This is simply requiring theconditional weights on each subtree form an equivalent martingale measure. Wealso make the following assumptions that are the multi period versions of theassumptions in Section 2.

15

Assumption 3.1 The increment matrices [∆Sk(t+1, i(c))] 1≤k≤d

1≤c≤N

have rank d,

for all t ∈ 0, 1, . . . , T − 1 and i ∈ It.

Assumption 3.2 The market is free of arbitrage. It implies that . (∆Sk(t +1, i(1), . . . , ∆Sk(t + 1, i(N))) ∈ R

N has both positive and negative elements, forall k ∈ 1, . . . , d, t ∈ 0, 1, . . . , T − 1 and i ∈ It.

Now the functional vγ0 (B) equals to

minµ,λ

maxm∈Rν

−∑

i∈IT

mi

[

log

(

mi

pi

)

− γbi

]

+ µ

(

∑

i∈IT

mi − 1

)

+∑

t,i,k

λt,i,k

(

N∑

c=1

m(t + 1, i(c))∆Sk(t + 1, i(c))

)

where the last summation runs over t ∈ 0, 1, . . . , T − 1, i ∈ It and k ∈1, . . . , d, and the solution is given below.

Theorem 3.1 The transition probabilities of the optimal M are found by back-ward induction as follows.

1. Conditioned on FT−1. i.e. for a fixed ωT−1,i,

• the optimal transition probabilities to reach the final states are givenby

m(c) = p(c) · exp(γb(c) + λ>∆S(c))

Z(λ)(3.6)

for c ∈ 1, . . . , N with the normalizing constant

Z(T−1,i)(λ) =N∑

c=1

p(c) · exp(γb(c) + λ>∆S(c)). (3.7)

• The Lagrange multiplier λ is the unique minimizer of the convex func-tion log Z(T−1,i)(λ).

• It holds

vγ(T − 1, i ; B) = minλ

[

log Z(T−1,i)(λ)]

. (3.8)

2. Fix t ∈ 0, 1, . . . , T−2 and conditioned on Ft, i.e. for a fixed ωt,i, supposewe have computed the auxiliary process vγ(u, i; B) for u ∈ t+1, . . . , T−1and i ∈ Iu. Then we have

16

• the optimal transition probabilities to reach the subsequent states aregiven by

m(c) = p(c) · exp[

vγ(t + 1, i(c) ; B) + λ>∆S(t + 1, i(c))]

Z(λ)(3.9)

for c ∈ 1, . . . , N with the normalizing constant

Z(t,i)(λ) =

N∑

c=1

p(c) · exp[

vγ(t + 1, i(c) ; B) + λ>∆S(t + 1, i(c))]

.

(3.10)

• the Lagrange multiplier λ is the unique minimizer of the convex func-tion log Z(t,i)(λ).

• it holds

vγ(t, i ; B) = minλ

[

log Z(t,i)(λ)]

. (3.11)

The difference from the one period model is that the task of finding the opti-mal transition probability is no longer local in nature but takes into accountthe aggregate entropy for the future time periods, represented by the auxiliaryprocess vγ . The proof is found in the Appendix.

Corollary 3.1 The MEMM Q of the multinomial tree (Ω,F , P ) is given by thealgorithm with B ≡ 0.

Corollary 3.2 The indifference price process of B in the multi nomial tree isgiven by

h(t, i; B) =1

γ[vγ(t, i; B) − v0(t, i)]

for all (t, i).

Based on the discussion so far we have the following algorithm for a multinominal tree model.

Input : A T -period and N -nomial tree with the actual probability p(t, i)and the increment matrices ∆Sk(t, i), calibrated to the market.The payoff vector bii∈IT

to price.

17

Output : q(t, i) the MEMM, and h the indifference price.

Set t = Tfor i ∈ 1, 2, . . . , NT

set vγB(T, i) = γB(i) and v0(T, i) = 0.

while (t > 0)set t := t − 1 and i := 1

while (i < N t + 1)minimize the convex function

ZB(λ) =

N∑

c=1

p(c) · exp[

vγB(t + 1, i(c)) + λ>∆S(t + 1, i(c))

]

and set λ∗B equal to the minimizer.

minimize the convex function

Z0(λ) =

N∑

c=1

p(c) · exp[

v0(t + 1, i(c)) + λ>∆S(t + 1, i(c))]

and set λ∗0 equal to the minimizer.

Updatevγ

B(t, i) = log ZB(λ∗B)

v0(t, i) = log Z0(λ∗0)

Set i := i + 1

Output h = 1γ[vγ

0 − v00 ].

We finish this section illuminating the dynamic properties of the emerging quan-tities form the algorithm.

Proposition 3.1 In the setting of Theorem 3.1, the function log Z(t,i)(λ) in

(3.7) and (3.10) has a unique minimizer in Rd for each t ∈ 0, . . . , T − 1 and

i ∈ It.And therefore, the Lagrange multiplier process defined by

λγB(t, i) := argminλ∈Rd log Zγ,B

(t,i)(λ) (3.12)

is well defined.

We show in the Appendix that each of log Z(t,i)(λ) is strictly convex and coercivesimilarly as we did in the proof of Proposition 2.3, using Assumption 3.1 and3.2.

Proposition 3.2 In terms of the Lagrange multiplier process, the auxiliary pro-cess vγ equals to

vγ(t, i; B) = log EP(t,i)

[

exp(

vγ(t + 1) + λγ(t)>∆S(t + 1))]

(3.13)

= log EP(t,i)

[

exp

(

γB +

T−1∑

u=t

λγB(u)>∆Su

)]

(3.14)

18

with the optimal measure M found in Theorem 3.1, for all t ∈ 0, 1, . . . , T − 1and i ∈ It and with vγ(T, i; B) = γB(i).

Observe that (3.13) is precisely the dynamic programming principle of the mul-tiplicative form and (3.14), and is an immediate consequence of (3.7) and (3.10)with the definition of the Lagrange multiplier process. (3.14) comes easily from(3.13) by mathematical induction.

Recall from the one period model that the Lagrange multiplier was the sensi-tivity vector of the functional vγ with respect to the tradable asset prices. Herewe have such a relation dynamically due to the iterative optimizing procedure.

Proposition 3.3 In the setting of Theorem 3.1, the Lagrange multiplier processequals to

−λγB(t, i) =

(

∂vγ(t, i; B)

S1(t), . . . ,

∂vγ(t, i; B)

Sd(t)

)

(3.15)

for each t ∈ 0, 1, . . . , T − 1 and i ∈ It.

We study deeper the financial interpretation of the Lagrange multiplier processin Section 4. Finally let us close Section 3 with a useful summary of the RadonNikodym derivative of the optimal measure M emerging from Theorem 3.1.

Corollary 3.3 In the setting of Theorem 3.1, set

D(t) :=exp

(

vγ(t) + λγB(t − 1)>∆St

)

Z(t−1,·)(λγB(t − 1, ·)) (3.16)

for t ∈ 0, . . . , T − 1, then the Ft conditional Radon Nikodym derivative of theoptimal measure with respect to the actual measure P equals to

dM

dP|t =

t−1∏

u=0

D(u) (3.17)

and the joint one equals to

dM

dP=

T−1∏

u=0

D(u) (3.18)

=exp

(

γB +∑T

u=1 λγB(u − 1)>∆S(u)

)

Z(0,1)(λγB(0))

(3.19)

19

4 Indifference hedging

The crux of arbitrage free pricing theory in the complete market lies in relatingthe arbitrage free price of a contingent claim with the replicating portfolio,i.e. the duality between pricing and hedging. We show that the two importantoperations of pricing and hedging a derivative security are naturally put togetherin the framework of indifference pricing. We assume the multinomial tree modelof the financial market in this section.

4.1 The Lagrange multipliers as the “delta”

An indifference option writer with the risk aversion coefficient γ evaluates theauxiliary function processes vγ

t (B) and vt(0) iteratively, whence the optimalmartingale measure M associated with B and γ and the MEMM Q emerged inTheorem 3.1. We aim to study how the optimal measure relates to the optimalinvestment portfolios for the value functions, which were used in the definition ofthe indifference price. We also study the risk monitoring effect of the emergingportfolios for the hedge purposes of the option writer with the exponential utilitypreference.

Proposition 4.1 In the setting of Theorem 3.1 denote the optimal portfolioprocess by

θγ(t, i ; B) := argminθE(t,i)[eγ(B−X0,θ(T ))] (4.1)

for t ∈ 0, 1, . . . , T − 1 and i ∈ It. Then it is well defined (i.e. the optimalportfolio is unique) and equals to a constant multiple of the Lagrange multiplierprocess (3.12) as follows;

θγ(t, i ; B) = − 1

γλγ

B(t, i) (4.2)

for all t ∈ 0, 1, . . . , T − 1 and i ∈ It.

We prove Proposition 4.1 in the Appendix by the uniqueness of the minimizer ofthe convex function log Z(t,i)(λ) and by invoking the conditional duality relation(2.7) in a backward inductive way. Let us point that (4.2) is the link betweenthe optimal investment problem of the option writer and the optimal measureM emerging from the dual formulation. Putting Theorem 3.1 and Proposition4.1 together we deduce that the optimal measure associated with B is of theBoltzmann Gibbs form with (−γ) times the optimal portfolio for the optionwriter. We stress that λ(t, ·) and consequently θ(t, ·), in a multinomial tree, aresimply computed by means of our algorithm.

Next we look into how the indifference option writer makes use of the optimalportfolios for risk management purposes in the incomplete market. We start by

20

recalling from Proposition 3.3 that λ(t, ·) is the sensitivity vector of vγ(t, ·; B)with respect to S(t). Hence the Lagrange multiplier process should be inter-preted as the “delta” of the functional vγ(B). To study the sensitivity vectorof the indifference price, let us denote by θ0(t, i) the optimal portfolio processwithout the option; i.e.

θ0(t, i) := argminΘEP(t,i)[e

−γX0,θ(T )] (4.3)

for all t ∈ 0, 1, . . . , T − 1 and i ∈ It.

Proposition 4.2 The sensitivity of the indifference price process with respectto the tradable asset price is given by

∂hγ(t, i; B)

∂S(t)= θγ(t, i) − θ0(t, i) (4.4)

Proof Recall from Proposition 2.2 that ht = 1γ[vγ

t − v0t ], and that we have

∂v0t

∂S(t) = 1γθ0(t) by Proposition 3.3 and 4.1 with B ≡ 0. Hence we conclude that

∂ht

∂S(t)=

1

γ

[

∂vγt

∂S(t)− ∂v0

t

∂S(t)

]

= θγ − θ0.

It is interesting to note that the portfolio difference θγ(t, i)−θ0(t, i) is an optimalportfolio process on its own under the MEMM.

Proposition 4.3 Set θ∗(t, i) := θγ(t, i)−θ0(t, i), then it is the optimal portfolioprocess of the value function

infθ∈Θ

EQ[eγ(B−X0,θ(T ))] (4.5)

where Q is the MEMM of the actual measure P .

We prove Proposition 4.3 in the Appendix, by applying Theorem 3.1 to themarket with the measure Q, which implies that the Lagrange multiplier processλQ,γ associated with the payoff B and the measure Q is well defined. And thenwe establish the relation

vQ,γ(t, i; B) := − infM∈Me

H(M |Q)− EM [γ · B]

= vP,γ(t, i; B) − v0(t, i; 0) (4.6)

and

λQ,γ(t, i) = λP,γ(t, i) − λ0(t, i) (4.7)

for all (t, i) and invoke the relation (4.1) to conclude that θ∗(t, i) := θγ(t, i) −θ0(t, i) is the optimal portfolio for (4.5).

21

We clearly see in (4.5) the role of the “delta” of the indifference price in riskmanagement in the incomplete market. The delta, i.e. ∂h

∂S= θ∗, is the optimal

hedge portfolio that minimizes the expected value of the loss from the hedgingerror B − X0,θ(T ), with the loss function being eγ·. We call ∂h

∂S= θ∗ the

indifference hedge portfolio of the claim B (with γ), with an understanding thatthe “hedge” is an imperfect yet optimal one in the sense of (4.5).

Next we study how the option writer prices and hedges the claim in the indiffer-ence pricing framework dynamically and coherently, by looking at the evolutionof the indifference price process and her wealth till the expiry.

Theorem 4.1 The indifference price process satisfies

h(t, i) =1

γlog inf

θ∈RdE

Q

(t,i)

[

eγ(h(t+1)−θ>∆St+1)]

(4.8)

In particular with the indifference hedge portfolio θ∗ it holds

h(t, i) = θ∗>(t)S(t) +1

γlog E

Q

(t,i)[eγ(h(t+1)−θ∗>(t)S(t+1))] (4.9)

for t ∈ 0, 1, . . . , T − 1 with h(T, i) = B(ωi).

Proof The equation (4.6) implies that

h(t, i) =1

γvQ,γ(t, i; B)

for each (t, i). If we apply Proposition 3.2 to the market with the measure Qthen the formula (3.13) implies

h(t, i) =1

γlog inf

θ∈RdE

Q[

eγ·h(t+1)+λQ,γ(t,i)>∆S(t+1)]

. (4.10)

We also apply Proposition 4.1 to the market with the measure Q then deducethat the indifference hedge portfolio θ∗ is a constant multiple of λQ,γ as

λQ,γ(t, i) = −γθ∗(t, i).

Substituting this to (4.10) we obtain (4.8) and then (4.9) follows immediately.

Let us make remarks on the financial implications. Firstly (4.8) is the dynamicprogramming principle for the indifference price process h(t) in a multiplicativeform. What the investor can do best from time t+1 till the expiry of the optionis summarized in h(t + 1, ·). Hence the sequential procedure of the indifferencepricing and hedging is decomposed to finding the surrogate payoff h(t + 1)and then solving the indifference hedging problem with the target payoff beingh(t + 1).

22

Secondly we see in (4.9) how the investor decomposes the payoff h(t+1) into thesum of the replicable and non replicable parts, and price them differently. Theprice charged to the buyer is the summation of the the cost of the indifferencehedge portfolio θ∗>St and the certainty equivalent of h(t + 1) − θ∗>St+1. Thecertainty equivalent part is the monetary compensation for the risk of mishedge,which is positive if and only if the market is incomplete and is an increasingfunction with respect to γ. It is not the static type certainty equivalent thatoften appears in the insurance literature. The option writer takes the bestadvantage of the financial market in terms of her exponential preference, asseen in (4.5), and the emerging indifference hedge portfolio allows the dynamiccombination of arbitrage free pricing for the replicable part and the certaintyequivalent pricing for the remaining risk.Clearly she bears the risk that the certainty equivalent be less than the realizedvalue of h(t + 1) − θ∗>St+1.

Next we wish to look deeper into the evolution of the wealth and expected utilityprocess of the option writer. For a given R

d-valued F predictable process θ wedefine the hedge error process by

ε(θ)t := ∆ht − θ>t ∆St

for t ∈ 1, . . . , T.

Lemma 4.1 For any bounded and predictable Rd valued process θ it holds

EQt [eγε(θ)t+1] ≥ 1.

In particular the indifference hedge portfolio θ∗ satisfies

EQt [eγε(θ∗)t+1 ] = 1.

i.e. the process eγε(θ∗)t is Q-martingale.

Proof The equation (4.8) implies

eγh(t,i) ≤ EQ

(t,i)

[

eγ(h(t+1)−θ>∆St+1)]

(4.11)

where the equality holds if and only if θ = θ∗(t, i). By dividing (4.11) by eγh(t,i)

we obtain the desired results.

Proposition 4.4 For a given portfolio process θ ∈ Θ the utility adjusted hedgeerror, namely,

−e−γ(Xh(0),θ(t)−h(t))

is a Q-super-martingale.It is a Q-martingale if and only if θ is the indifference hedge portfolio.

23

Proof Set Z(θ)t := [−e−γ(Xh(0),θ(t)−h(t))] then it is equal to

Z(θ)t = − exp

(

−γ

(

h(0) − h(t) +

t∑

u=1

θu∆Su

))

= −t∏

u=1

eγε(θ)u

Consider the conditional expectation of the one step ahead, then

EQ[Z(θ)t+1|Ft] = −

(

t∏

u=1

eγε(θ)u

)

· EQ[eγε(θ)t+1|Ft]

≤ −t∏

u=1

eγε(θ)u = Z(θ)t.

where the inequality comes from Lemma 4.1. The equality holds if and only ifθ = θ∗, finishing the proof.

With the indifference hedge portfolio θ∗ the wealth of the indifference optionwriter Xh(0),θ∗

(t) evolves in such a way that the utility from the residual wealthXh(0),θ∗

(t) − h(t) is a Q-martingale, apparently reflecting that the expectedhedge error is being minimized from the exponential utility perspective.Next we consider the property of the indifference price process on its own.

Theorem 4.2 The indifference price process h(t) and the hedge error processε(θ∗)t is a super-martingale under each M ∈ Me.

The proof is found in the Appendix.

Corollary 4.1 The expected hedge error EMt [ε(θ∗)t+1] is decreasing with each

M ∈ Me.

Corollary 4.1 follows immediately from Theorem 4.2 since the wealth process∑t

u=1 θ(u)>∆S(u) with any predictable θ is a martingale under each M ∈ Me.Corollary 4.1 indicates that the closer the expiry approaches the less is theextent of hedge error on average.Before closing the subsection let us cite Stoikov and Zariphopoulou(2005) [32]that studied the optimal portfolio of the exponential preference investor in acontinuous time diffusion model. Becherer(2003)[4] also studied the risk moni-toring strategies in the continuous time model with the exponential utility.

4.2 Relation between indifference pricing and arbitrage

free pricing

Proposition 4.5 For a replicable payoff the indifference price coincides withthe arbitrage free price.

24

Proof For a replicable payoff B with x +∑T

t=1 θ(t)>∆S(t) = B a.s., thearbitrage free price is x. The replicating portfolio θ is certainly the indifferencehedge portfolio with x = h.

The indifference price shares many properties of the arbitrage free pricing such aslinearity for the replicable risk and the time consistency via semigroup property.

Proposition 4.6 For a payoff B = B1 + B2 with B1 replicable it holds

h(B) = h(B1) + h(B2).

Proof Suppose B1 = x1 +∑T

t=1 θ1(t)>∆S(t) a.s. then h(B1) = x1 from

Proposition 4.5. The primal formulation of h(B) is given by

exp γh(B) = infθ∈Θ

EQ[

eγ(B1+B2−PT

t=1 θ(t)>∆S(t))]

= infθ∈Θ

EQ[

eγ(x1+PT

t=1 θ1(t)>∆S(t)+B2−

PTt=1 θ(t)>∆S(t))

]

= eγx1 infθ∈Θ

EQ[

eγ(B2−PT

t=1 θ(t)>∆S(t))]

= eγx1eh(B2)

by renaming θ := θ − θ1. We conclude h(B) = x1 + h(B2).

Proposition 4.7 The indifference price has the semigroup property, i.e.

h(t,i)(BT ) = h(t,i)(h(s,·)(BT ))

for any 0 ≤ t < s ≤ T and i ∈ It.

This is straightforward from (4.8) in Theorem 4.1. An important characteristicof our algorithm is that it prices the replicable payoffs and the non replicableones consistently. If a practitioner has a model with several assets and wishesto price various derivative securities in it, she can consistently price all the se-curities with our algorithm since the algorithm picks up the replicable payoffsautomatically, prices them by arbitrage free pricing, and computes the repli-cating portfolio, which coincides with the indifference hedge portfolio for thereplicable securities. Of course the algorithm prices the non replicable secu-rities by the dynamic combination of arbitrage free pricing and the certaintyequivalent pricing, producing the indifference hedge portfolios associated withthem.

Proposition 4.8 For a payoff B we have the following ordering relation.

EQ[B] ≤ h(B) ≤ 1

γlog E

Q[eγB] (4.12)

This is a direct outcome of Jensen’s inequality, applied backward iteratively to(4.8), and we omit the proof.

25

5 Numerical implementation

In this section we demonstrate the procedure of numerical implementation ofthe algorithm of indifference pricing outlined in Section 3. We selected tworepresentative examples of incomplete markets, one with the non tradable assetand the other being the stock market with stochastic volatility. With eachexample, we use a diffusion model as the limiting model and show how to builda multi nomial tree to approximate the diffusion.Obviously there are infinite possibilities of how to “calibrate” a multinomial treeto the diffusion. We simply suggest a tried and true way to build a recombining4 nomial tree calibrated to the 2 dimensional diffusion model, with no claimfor the highest efficiency. However the result is pretty satisfactory, in terms ofspeed and accuracy. The number of final states of the tree with T period growswith the order of T 3, which is manageable by most desktop PCs. Each time stepis composed of the gradient based minimization of a convex function, which isdone extremely fast. With T = 60,which is more than enough for most practicalpurposes, the indifference prices in both examples that follow where computedin about one tenth of a second with 2.2Ghz AMD Opteron processors. Thisincludes the time to compute the indifference hedge portfolio.After we build a tree, we specialize the algorithm to each example and show thenumerical results.

5.1 A market with non tradable asset

We consider a simple model of an incomplete market with a tradable risky assetS,a non tradable risky asset Y and a tradable riskless bond. The benchmarkcontinuous time model is a two dimensional diffusion model

dS(t) = S(t) (µ dt + σ dW1(t)) (5.1)

dY (t) = Y (t) (b dt + a dW2(t)) (5.2)

with a two dimensional standard Brownian motion (W1, W2) and a constantcorrelation E

P [W1(t) · W2(t)] = ρ · t. For simplicity we assume the risklessinterest rate is zero.

We construct a 4 nomial tree that approximates the diffusion (5.1) as follows.With the initial price S and Y the four possible states ω1, ω2, ω3, ω4 after onetime increment ∆t are given by

S(ω1) = S · u, Y (ω1) = Y · f1, S(ω2) = S · u, Y (ω2) = Y · f2 (5.3)

S(ω3) = S · d, Y (ω3) = Y · f3, S(ω4) = S · d, Y (ω4) = Y · f4 (5.4)

with the multiplicative factors

26

u = 1 + µ∆t + σ√

∆t, d = 1 + µ∆t − σ√

∆t

f1 = 1 + b∆t + a√

∆t(ρ + ρ ), f2 = 1 + b∆t + a√

∆t(ρ − ρ )

f3 = 1 + b∆t + a√

∆t(−ρ + ρ ), f4 = 1 + b∆t + a√

∆t(−ρ − ρ ).

The actual probability of each state is 14 . It is straightforward to show that the

MEMM is

P (ω1) =p

2, P (ω2) =

p

2, P (ω3) =

1 − p

2, P (ω4) =

1 − p

2(5.5)

with p = (1 − d)/(u − d). Note that the MEMM is constant throughout thetree and this simplifies the implementation, which is characteristic of the in-complete market where the non tradable factor does not affect the dynamics ofthe tradable asset explicitly.The marginal distribution of S under the MEMM is the risk neutral distributionof S alone and the marginal distribution of Y under the MEMM is identical tothe actual distribution, as was pointed out by Musiela and Zariphopoulou (2004)[26]. A stochastic volatility model is a counterexample to this and has a statedependent MEMM. Following is a schematic picture of our tree.

Figure 2: A recombining tree of two correlated assets

Su2, Y f21

Su2, Y f1f2

Su2, Y f22

Sud, Y f1f3

Sud, Y f2f3

Sd2, Y f23

Sud, Y f1f4

Sud, Y f2f4

Sd2, Y f3f4

Sd2, Y f24

Su, Y f1

Su, Y f2

Sd, Y f3

Sd, Y f4

S, Y

cccccccc

````````

########

aaaaaaaaaa

@@@@@@@@@@

hhhhhhhhhZZZZZZZZZ

!!!!!!!!!

aaaaaaaaa

""""""""""

((((((((

(

hhhhhhhhh

27

We implemented the following algorithm.

Input : Initial price S(0), Y (0), multiplicative factors u, d, f1, f2, f3, f4 andpayoff B = B(ωi).

Output : Indifference price h and indifference hedge portfolio θ.

Set t = T (set the final values)for all 4 tuples (q1, q2, q3, q4) of nonnegative integers

with q1 + q2 + q3 + q4 = T (regardless of the order)

SetS = S(0) · uq1+q2 · dq3+q4

Y = Y (0) · f1q1 · f2

q2 · f3q3 · f4

q4

V (q1, q2, q3, q4) = γ · B(S, Y ).

while (t > 0)set t := t − 1

For all 4 tuples (q1, q2, q3, q4) with q1 + q2 + q3 + q4 = t(nonnegative integers, regardless of the order)Set

S = S(0) · uq1+q2 · dq3+q4

Y = Y (0) · f1q1 · f2

q2 · f3q3 · f4

q4

Minimize the convex function

Z(λ) =p

2· eλS(u−1)

[

eV (q1+1,q2,q3,q4) + eV (q1,q2+1,q3,q4)]

+1 − p

2· eλS(d−1)

[

eV (q1,q2,q3+1,q4) + eV (q1,q2,q3,q4+1)]

let λ∗ denote the minimizer.Update V (q1, q2, q3, q4) = log Z(λ∗).

Output h = 1γV and θ = − 1

γλ∗.

The algorithm above computes the writer’s price. T he buyer’s price is computedby setting the final value as W = −γB and the outcome as h = − 1

γV .

Let us present a few representative graphs of the numerical results. We usedthe following parameters for (5.1)

S(0) = 10, µ = 0.05, σ = 0.2

Y (0) = 10, b = 0, a = 0.4, ρ = 0.5.

First let us show the indifference prices of basket options and spread options.We computed both the call and put prices, i.e. the basket payoffs are B =max(S(T ) + Y (T )−K, 0) or max(K − S(T )− Y (T ), 0), and the spread payoffsare B = max(S(T ) − Y (T ) − K, 0) or max(K − S(T ) + Y (T ), 0).

28

Time to expiry is one month and we used the number of time steps T = 60.We used γ = 0.5 for Figure 3. The horizontal axis is the strike price. Of thethree lines in each graph, the uppermost one is the writer’s price, and the lower-most one is the buyer’s price. The middle one is the entropy price, shown forcomparison. The pattern of bid/ask spread between the writer’s and buyer’sprices look consistent with the typically observed ones in the market in that thespread is bigger for the more expensive options.

16 18 20 22 240

1

2

3

4

5Basket Call option

16 18 20 22 240

1

2

3

4

5Basket Put option

16 18 20 22 24−0.4

−0.2

0

0.2

0.4Put Call parity of basket option

sellerrisk neutralbuyer

−4 −2 0 2 40

1

2

3

4Spread Call option

−4 −2 0 2 40

1

2

3

4Spread Put option

−4 −2 0 2 4−0.4

−0.2

0

0.2

0.4Put Call parity of spread option

sellerrisk neutralbuyer

Figure 3: Indifference price of basket/spread option

29

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75Basket Call K=20

correlation

Cal

l pric

e

γ=0.5γ=0.2

γ=0.1

risk neutral

Figure 4: Indifference price of basket option with different correlation

30

Next we look at the indifference prices of basket option with various values ofcorrelation ρ between the returns of S and Y . The risk neutral price (entropyprice) of a basket option is naturally an increasing function of correlation. How-ever it is not necessarily the case for the indifference price, depending on themagnitude of the risk aversion coefficient γ.With γ = 0.1 and 0.2 the writer’s indifference price of basket option is increasingwith respect to the correlation. In contrast a higher value γ = 0.5 gives rise toa non monotonic and concave graph of the writer’s indifference price. Recallthat the indifference price has two components, the entropy price for replicablecomponent and the certainty equivalent for non replicable component. Thefirst component is increasing with respect to ρ while the second component isdecreasing with respect to |ρ|. The magnitude of γ determines the sensitivityof the second component with respect to ρ and the combined effect of the twocomponents could be either an increasing function or a non monotonic functionof ρ.

5.2 A stock market with stochastic volatility

In this section we present the numerical results of the indifference pricing algo-rithm applied to a stock market with stochastic volatility. We used the Hestonmodel

dS(t)/S(t) = µ dt +√

V (t)dW1(t) (5.6)

dV (t) = κ(σ −√

V (t))dt + a · V (t)dW2(t)

as the limit continuous time model and constructed a recombining quadrinomialtree to approximate it. We used the idea of Hilliard and Schwartz (1996) [18]that transforms the dynamics (5.6) firstly to a diffusion with a constant volatilitymatrix, and then constructs a recombining 4 nomial tree to approximate thetransformed diffusion in a similar manner as was illustrated in Section 5.1. Theoriginal diffusion (5.6) is approximated by the inverse transform of the tree. Formore detail on tree construction we refer the readers to Nelson and Ramaswamy(1990) [27] and Hilliard and Schwartz (1996) [18]. Once we built the tree, thealgorithm is quite close to the one in Section 5.1 and we skip the description.

We used the following parameters for (5.6).

S(0) = 10, µ = 0.0

V (0) = 0.52, κ = 1, σ = 0.5, a = 0.4.

and different values of the correlation ρ,indicated in the figure. The time toexpiry is one month and the number of time-steps is 60. With these input, wecomputed the indifference prices of the call and put options with strike prices9, 9.5, 10, 10.5, and 11, and then computed the implied volatilities of them by

31

inverting the Black Scholes pricing kernel. The implied volatilities were virtuallyidentical with call and put options. In Figure 5 we show the skew/smile of theindifference prices at across the five strike prices, the middle one being at themoney. The two red lines are the writer’s price and the two blue lines are thebuyer’s price. The highest and lowest prices correspond to γ = 0.5 and theother two to γ = 0.2. The entropy prices lie in between the bid/ask spread ofγ = 0.2, but are omitted from the graph to avoid congestion. Each subfigurehas different correlation indicated in the caption.

9 9.5 10 10.5 110.465

0.47

0.475

0.48

0.485

0.49

0.495ρ=−0.6

9 9.5 10 10.5 110.49

0.495

0.5

0.505ρ=−0.3

9 9.5 10 10.5 110.496

0.497

0.498

0.499

0.5

0.501

0.502

0.503

0.504ρ=0.0

9 9.5 10 10.5 110.502

0.504

0.506

0.508

0.51

0.512

0.514ρ=0.3

Figure 5: The skew/smile of the indifference price

32

Let us make a couple of observation on the skew/smile of the indifference prices.Firstly we note that the negative correlation of Heston model typically give riseto a skew and it carries over to the skew of the indifference prices. (See thetop row of Figure 5.) At the zero correlation (the lower left subfigure) we see arather flat smile, perhaps due to the short time to expiry, and, with the positivecorrelation (ρ = 0.3 at the lower right subfigure) the skew is inverted. By andlarge we see that the indifference prices of the Heston model preserve the shapeof the skew/smile of the risk neutral pricing.Next if we look at the bid/ask spreads then we see that the spread are moreor less parallel around the typical shape of the skew/smile associated with theHeston model. The size of the spread does tend to be slightly larger for thehigher strike price (11 in our figure) but it remains to be seen in future researchwhether it is a general feature, amplified in certain circumstances, or a merenumerical error due to discretization.

5.3 Numerical demonstration of the convergence

In this section we investigate the behavior of the outcome from the algorithmwith increasing the number of time steps. It is natural to expect that the in-difference prices and the portfolios of the multi nomial trees should converge tothe continuous time counterparts, provided that the discrete stochastic dynam-ics converge to the continuous one, as the number of time-steps goes to infinity.To conduct an experimental study we choose the following test problem with aclosed form solution for the indifference prices and hedge portfolios.Suppose the tradable asset S follows the geometric Brownian motion with driftµ and volatility σ and that a non tradable asset Y follows a Gaussian process

dY (t) = b dt + a dW2(t) (5.7)

with the correlation ρ between the two Brownian motions.

Proposition 5.1 The indifference price h of a call option on Y , i.e. the payoffmax(Y (T ) − K, 0), is given by

h =1

γlog[

N(α) + e−γαa√

T+ 12 γ2a2T N(−α + γa

√t)]

(5.8)

with

γ = γ(1 − ρ2)

α =1

a√

T

(

K − Y (0) − (b − ρµ

σa)T

)

and N(x) = 1√2π

∫ x

−∞ e−u2

2 du.

Its first derivative is given by

33

hy(y, t) =1

γ

1a√

Tφ(α) + e−γαa

√T+ 1

2 γ2a2T(

φ(α−γa√

T )

a√

T+ γN(−α + γa

√T ))

N(α) + e−γαa√

T+ 12 γ2a2T N(−α + γa

√t)

(5.9)

where φ(x) = 1√2π

e−u2

2 .

Proof It was shown by Musiela and Zariphopoulou (2004) [25] that the indif-ference price of a claim with payoff g(YT ) equals to

h(y, t) =1

γ(1 − ρ2)ln E

Q[eγ(1−ρ2)g(YT )|Yt = y] (5.10)

where Q is the MEMM, provided that Y is an autonomous diffusion not affectingthe dynamics of S.Under the MEMM Q the process Y (t) is Gaussian with the drift b − ρµ

σa and

the volatility a, and therefore eY (t) is log-normal. Hence we can derive (5.8)and (5.9) like the derivation of the Black Scholes formula.

With this test model, the relation between the indifference hedge portfolioand the derivative of the indifference price is quite simple. Musiela and Za-riphopoulou(2004) [25] showed that the indifference hedge portfolio for a claimon Y alone is given by

π(y, t) = ρa

σhy(y, t)

in dollar amount. On the other hand our tree type algorithm gives the portfolioas θ = −λγ−λ0

γin the number of shares. If our algorithm computes the hedge

portfolio well, it should give

hy ' −1

ρ

σ

a

λγ − λ0

γS

Following are the results from our algorithm as we increase the number of timesteps of the trees. The model parameters used are identical to the values inSection 5.1, with T = 1 month and γ = 0.1. The analytical values from contin-uous time model are listed in the caption of each picture. One word of cautionconcerned with implementation is that you should use a very precise formulafor N(·) if you use small γ 1 to evaluate (5.10).

34

Figure 6: K=10: h = 0.04417, he = 0.044012, ∂h∂y

= 0.487309

20 40 600.0442

0.0442

0.0443

0.0443

0.0444Indifference price

20 40 600.044

0.044

0.0441

0.0441

0.0442

0.0442

0.0443Entropy price

20 40 600.4874

0.4876

0.4878

0.488

0.4882

0.4884

0.4886Indifference hedge portfolio

Figure 7: K=9.9: h = 0.109568, he = 0.1090163, ∂h∂y

= 0.79858

20 40 600.1093

0.1093

0.1094

0.1094

0.1094Indifference price

number of time steps20 40 60

0.109

0.109

0.109

0.109

0.109Entropy price

number of time steps20 40 60

0.7982

0.7984

0.7986

0.7988

0.799

0.7992

0.7994Indifference hedge portfolio

number of time steps

35

It is interesting to compare the convergence behavior of the indifference priceto that of the binomial tree model of Black Sholes pricing. It is well knownthat the binomial tree model converges rather slowly and exhibits a oscillatorypattern. Yet it is possible to speed it up by taking the average of the outcomesfrom the even and odd number of the time steps. Regarding the arbitrage freepricing on a quadrinomial tree, Hilliard and Schwartz(1996) [18] reported thatthe price of 50/51 (the number of time steps) average is in error by less thanone half cent from the 700/701 answer. This clever idea appears relevant to theindifference pricing, as can be seen from Figure 6.However the oscillatory behavior is more complex for the indifference pricing.Recall that the indifference price of a claim has two components, namely, theentropy price for the replicable portion and the certainty equivalent for thenon hedgeable portion. The entropy price component is computed precisely thesame way as the binomial tree model in our algorithm. On the other hand,the certainty equivalent component should decrease as the number of time-stepincreases, since the more frequent updating of the indifference hedge portfolioshould imply the less expected hedge error. Putting these two parts together,we conclude that the (writer’s) indifference prices from the tree type algorithmshow the oscillatory behavior following the decreasing trend line correspondingto the (dynamic) certainty equivalent of the residual risk.

6 Conclusion

We studied indifference pricing in a discrete time multi-nomial tree model andderived a complete procedure to implement the framework numerically. Thesimplicity of the tree structure and the relative entropy on the tree allowed usto clarify the dynamic programming type characteristic of indifference pricing,leading to an algorithm ready to be coded. The main ingredient of the deriva-tion of the algorithm is recasting the value function of the optimal investmentproblem of an option writer as the minmax problem associated with a coerciveand strictly convex Lagrangian. The beautiful duality between the exponentialoptimal investment problem and the minimal relative entropy measure is longtime known, but this work is the first to use the duality theorem to computethe financial quantities numerically.Particularly interesting is the indifference hedge portfolio that emerged as thedifference of the two Lagrange multiplier process with and without the option,divided by (−γ). In the setting of the multi-nomial tree, the parallel rolesof the Black Scholes “delta” and the indifference hedge portfolio were clearlyillustrated, leaving more questions to be answered, such as quantifying the riskmanagement performance of the indifference hedge portfolio.We presented the detailed examples of numerical implementation with the mar-ket with a non tradable asset and the stochastic volatility model, along withthe numerical results. The implementation is fast enough to be used in practiceand the convergence test showed that the discrete time results converge to thecontinuous time one in an oscillatory manner. Let us reiterate that not only the

36

indifference price but also the indifference hedge portfolio converged excellently.The numerical research and practical implementation of derivative securities inincomplete markets have been rather slow due to the multi dimensional nature ofincomplete markets. This paper proposes a solution to confront such challengesby discretizing state and time by a tree type dynamics. We leave it to futureresearch to generalize the algorithm and make it more computationally efficient.

A The proofs from Section 3

We start the preparation for the proofs by expressing the relative entropy interms of the transition probabilities. Using the notation of Section 3, we have

H(M |P ) =

ν∑

i=1

mi log

(

mi

pi

)

=

ν∑

i=1

T−1∏

t=0

m(

T − t, rt(i))

log

(

∏T−1t=0 m (T − t, rt(i))

∏T−1t=0 m (T − t, rt(i))

)

(A.1)

=

ν∑

i=1

T−1∏

t=0

m(

T − t, rt(i))

T−1∑

t=0

log

(

m (T − t, rt(i))

m (T − t, rt(i))

)

(A.2)

To deal with the multi period characteristic of the problem, we introduce theconditional entropy and the transitional entropy.

Definition 5 For two probability measures P and M on (Ων ,F)

1. the t-conditional relative entropy between P and M is defined as

Ht(M |P ) :=∑

i∈It

m(ωt,i)

N∑

c=1

m(t + 1, i(c)) log

(

m(t + 1, i(c))

p(t + 1, i(c))

)

(A.3)

for t = 0, 1, . . . , T − 1.

2. the (t, i)-transition relative entropy between P and M is defined as

Ht+1|(t,i)(M |P ) :=

N∑

c=1

m(t + 1, i(c)) log

(

m(t + 1, i(c))

p(t + 1, i(c))

)

(A.4)

for i ∈ It and t = 0, 1, . . . , T − 1.

3. the (t, i)-conditional probability of P is defined as

p(t,i)l : =

T−t−1∏

s=0

p(T − s, rs(l))

= P[ω(T ) = l | ω(t) = i ] (A.5)

37

and the (t, i) conditional index set is defined as

I(t,i)T = l ∈ IT | rT−t(l) = i

= l ∈ IT | p(t,i)l > 0.

4. the (t, i)- joint relative entropy between P and M is defined as

H(t,i)(M |P ) =∑

j∈I(t,i)T

m(t,i)j log

(

m(t,i)j

p(t,i)j

)

(A.6)

: This is the joint relative entropy between P and M on the subtree startingfrom the node ωt,i.

We make several observations on the relationships between the quantities definedabove. First, it holds

Ht(M |P ) =∑

i∈It

m(ωt,i)Ht+1|(t,i)(M |P ). (A.7)

Second the (t, i) joint relative entropy H(t,i)(M |P ) only requires the values ofm(s, ·) and p(s, ·) for s ≥ t + 1, regardless of the probabilities of the past. Welist below more of the useful identities.

Lemma A.1 (Theorem 2.2.1 and Theorem 2.5.1 of Cover and Thomas(1991)[7])We have

H(M |P ) =

T−1∑

t=0

Ht(M |P ) (A.8)

Lemma A.2

H(M |P ) =N∑

c=1

m(1, c) log

(

m(1, c)

p(1, c)

)

+N∑

c=1

m(1, c)H(1,c)(M |P ) (A.9)

A more general version of Lemma A.2 is following.

Lemma A.3

T−1∑

t=u

Ht(M |P ) =∑

i∈Iu

m(ωu,i)H(u,i)(M |P ) (A.10)

for u = 0, 1, . . . , T − 1.

Now we are ready to begin the proof of Theorem 3.1.

38

A.1 Proof of Theorem 3.1

We want to solve the minimax problem

minµ,λ

maxM

L(µ, λ, M)

where the Lagrangian L is given by

L(µ, λ, M) = −H(M |P ) +∑

i∈IT

mi · γbi + µ

(

ν∑

i=1

mi − 1

)

+∑

t,i,k

λt,i,k

N∑

c=1

m(t + 1, i(c))∆Sk(t + 1, i(c)).

(A.11)

We aim to decompose the Lagrangian into the sum of “disjoint” Lagrangians.For that purpose, we rewrite the joint entropy as follows, using Lemma A.1 andA.3.

H(M |P ) =

T−1∑

t=0

Ht(M |P )

=

T−1∑

t=0

Nt

∑

i=1

m(ωt,i)

N∑

c=1

m(t + 1, i(c)) log

(

m(t + 1, i(c))

p(t + 1, i(c))

)

= (terms with 1 ≤ t ≤ T − 2)

+

NT−1∑

i=1

m(ωT−1,i)

N∑

c=1

m(T, i(c)) log

(

m(T, i(c))

p(T, i(c))

)

Similarly the second term of (A.11) equals to

∑

i∈IT

mi · γbi =∑

i∈IT−1

m(ωT−1,i)

(

N∑

c=1

m(T, i(c)) · γb(c)

)

.

It implies that the aggregate Lagrangian can rewritten as

L = (terms with 1 ≤ t ≤ T − 2) + LT−1

where the last term LT−1 is given by

LT−1 =∑

i∈IT−1

m(ωT−1,i)·[

−N∑

c=1

m(c)

[

log

(

m(c)

p(c)

)

− γ b(c)

]

+ µ

(

N∑

c=1

m(c) − 1

)

+ λ>(

N∑

c=1

m(c)∆S(c)

)]

39

We used m(c) as a shorthand notation for m(T, i(c)) and the same for p(·) andS(·). The Lagrange multipliers µ and λ have the hidden subscripts (T − 1, i).Note that, for each i ∈ IT−1, the terms in the bracket is precisely the Lagrangianwe had optimized in the one period model in Section 2.2, and more importantly,that the Lagrangians are “disjoint” of each other and therefore we can solve eachof them individually. Hence we conclude, from Proposition 2.3 in Section 2.2,that the final transition probabilities are given by the formula (3.6) as was tobe shown.Furthermore, if we substitute the transition probability of the optimal measure,then the value of LT−1 is given by the convex combination of the minimum ofthe convex function (2.19) given in Proposition 2.3 in Section 2.2; namely,

LT−1 =∑

i∈IT−1m(ωT−1,i) · minλ

[

log(

∑Nc=1 p(c)eγb(c)+λ>S(c)

)]

=∑

i∈IT−1m(ωT−1,i) · vγ(T − 1, i; B)

where vγ(T, ·; B) is the auxiliary process. Hence the aggregate Lagrangianequals to

L(µ, λ, M) = −T−2∑

t=0

Ht(M |P ) +∑

i∈IT−1

m(ωT−1,i) · vγ(T − 1, i; B)

+(the Lagrange multiplier terms)

To show the general formula (3.9) for all t by induction argument, fix a timeperiod u ∈ 0, . . . , T − 1 and suppose that we have found the optimal measurem(t, ·) up to t ≥ u + 1 by the formula (3.9) and that

vγ(t, i; B) = −H(t,i)(M |P ) + EM(t,i)[γB] (A.12)

for t ≥ u + 1 and i ∈ It.Now we want to show that m(u, ·) is also of the form (3.9). To do so wedecompose the joint entropy, again by Lemma A.1 and A.3, into

H(M |P ) =

u−2∑

t=0

Ht(M |P ) + Hu−1(M |P ) +

T−1∑

t=u

Ht(M |P )

= (terms with 1 ≤ t ≤ u − 2)

+∑

i∈Iu−1

m(ωu−1,i) ·[

N∑

c=1

m(u, i(c)) log

(

m(u, i(c))

p(u, i(c))

)

]

+∑

i∈Iu−1

m(ωu−1,i) ·[

N∑

c=1

m(u, i(c))H(u,i(c))(M |P )

]

(A.13)

40

The last line comes from

T−1∑

t=u

Ht(M |P ) =∑

j∈Iu

m(ωu,j)H(u,j)(M |P )

=∑

j∈Iu

m(ωu−1,r(j))m(u, j)H(u,j)(M |P )

=∑

i∈Iu−1

m(ωu−1,i)

N∑

c=1

m(u, i(c))H(u,i(c))(M |P )

Using (A.13) and (A.12) we decompose the aggregate Lagrangian (A.11) into

L = (terms with 1 ≤ t ≤ u − 2) + Lu−1

where the last term Lu−1 is given by

Lu−1 =∑

i∈Iu−1

m(ωu−1,i) ·[

−N∑

c=1

m(c)

[

log

(

m(c)

p(c)

)

− vγ(t + 1, i(c); B)

]

+µ

(

N∑

c=1

m(c) − 1

)

+ λ

(

N∑

c=1

m(c)∆S(c)

)]

Observe that we have the new term vγ(t + 1, ·). From the induction hypothesisthe values vγ(t+1, ·) have been found, regardless of the values of m(u, ·). Henceeach term in Lu−1 has precisely the same form as the Lagrangian (2.16) in theone period model in Section 2.3, where vγ plays the role of γ ·B of Section 2.3.From Proposition 2.3 with B = 1

γvγ(t + 1, ·), we conclude that m(u, ·) is given

by (3.9) and the proof of Theorem 3.1 is complete.

Let us finish with the proofs for the lemmas.

Proof of LemmaA.2: We use the notation of Section3. For each c = 1, . . . , Nset

Dc = i ∈ IT | rT−1(i) = c.This is simply the collection of the indices at time T whose past state at time1 was the c th node. Clearly we have

IT = ∪Nc=1Dc

where the sets are disjoint. The joint entropy is

41

H(M |P ) =NT

∑

i=1

T−1∏

t=0

m(

T − t, rt(i))

T−1∑

t=0

log

(

m (T − t, rt(i))

m (T − t, rt(i))

)

=

N∑

c=1

∑

i∈Dc∩IT

T−1∏

t=0

m(

T − t, rt(i))

T−1∑

t=0

log

(

m (T − t, rt(i))

m (T − t, rt(i))

)

For a fixed 1 ≤ c ≤ N and i ∈ Dc ∩ IT we have

m(1,c)i := M [ω(T ) = i | ω(1) = c] =

T−2∏

t=0

m(

T − t, rt(i))

and

mi := M [ω(T ) = i] = m(1, c) · m(1,c)i .

due to the nested structure of a tree, which imply

mi log

(

mi

pi

)

= m(1, c) · m(1,c)i log

(

m(1, c) · m(1,c)i

p(1, c) · p(1,c)i

)

= m(1, c) · m(1,c)i

(

log

(

m(1, c)

p(1, c)

)

+ log

(

m(1,c)i

p(1,c)i

))

= m(1, c) · m(1,c)i log

(

m(1, c)

p(1, c)

)

+ m(1, c) · m(1,c)i log

(

m(1,c)i

p(1,c)i

)

.

Hence the joint entropy is

H(M |P ) =N∑

c=1

∑

i∈Dc∩IT

m(1, c) · m(1,c)i log

(

m(1, c)

p(1, c)

)

+N∑

c=1

∑

i∈Dc∩IT

m(1, c) · m(1,c)i log

(

m(1,c)i

p(1,c)i

)

=

N∑

c=1

m(1, c) log

(

m(1, c)

p(1, c)

)

+

N∑

c=1

m(1, c)∑

i∈Dc∩IT

m(1,c)i log

(

m(1,c)i

p(1,c)i

)

=

N∑

c=1

m(1, c) log

(

m(1, c)

p(1, c)

)

+

N∑

c=1

m(1, c)H(i,c)(M |P )

completing the proof.

Proof of LemmaA.3 : Straightforward from Lemma A.2.

42

A.2 Proof of Proposition 3.1

Each of log Z(T−1,i)(λ) is strictly convex and coercive from Proposition 2.3, andtherefore has a unique minimizer.Suppose the statement holds for log Z(u,·) for u ≥ t + 1 to use the backward

induction. Now fix (t, i) and consider the function W : Rd → R defined by

W (λ) := log Z(t,i)(λ) = log

[

N∑

c=1

p(t + 1, i(c)) exp(

vγ(t + 1, i(c)) + λ>∆S(t + 1, i(c)))

]

.

Assumption 3.2 (no arbitrage) implies that the increment vectors ∆Sk haveboth the positive and negative entries, for each k ∈ 1, . . . , d and therefore wehave lim‖λ‖→∞ |W (λ)| = ∞.To show the strict convexity we consider the second partial derivatives thenthey are given by

∂W (λ)

∂λk∂λl

=

∂2Z(t,i)(λ)

∂λk∂λl

Z(t,i)(λ)−

∂Z(t,i)

∂λk

∂Z(t,i)

∂λl

Z2(t,i)(λ)

=N∑

c=1

m(t + 1, i(c))∆Sk(t + 1, i(c))∆Sl(t + 1, i(c))

−(

N∑

c=1

m(t + 1, i(c))∆Sk(t + 1, i(c))

)(

N∑

c=1

m(t + 1, i(c))∆Sl(t + 1, i(c))

)

= CovM [∆Sk(t + 1)∆Sl(t + 1)]

for k, l ∈ 1, . . . , d. Assumption 3.1 (no redundant asset) implies that thecovariance matrix of the tradable asset increments under any equivalent measureis strictly positive definite. We conclude that W (λ) is strictly convex and thatit has a unique minimizer in R

d. This completes the proof by induction.

A.3 Proof of proposition 3.3

First we observe from (3.13) in Proposition 3.2 that vγ(t, i; B) is independentof ∆S(u)|u = 0, . . . , t − 1, for each t ∈ 0, . . . , T. Next, if we rewrite (3.13)for the multi nomial tree notation it becomes

vγ(t, i; b) = minµ,λ

maxm

−N∑

c=1

m(t + 1, i(c))

[

log

(

m(t + 1, i(c))

p(t + 1, i(c))

)

− vγ(t + 1, i(c); B)

]

+µ

(

N∑

c=1

m(t + 1, i(c)) − 1

)

+ λ>(

N∑

c=1

m(t + 1, i(c))∆S(t + 1, i(c))

)

43

which is identical to the minimax problem (2.16) with γb replaced with vγ(t+1).Since vγ(t+1) is independent of S(t) ceteris paribus, we can think of it as a oneperiod problem, use Theorem 2.1 and conclude that

∂vγ(t, i; B)

∂S(t)= −λγ

B(t, i) ∈ Rd

.

A.4 Proof of Corollary 3.3

We have established in Proposition 3.1 the equation (3.16), and it remains toshow (3.17) and (3.19). Fix t ∈ 1, . . . , T and i ∈ It then it holds

m(ωt,i)

p(ωt,i)=

t−1∏

s=0

m(t − s, rs(i))

p(t − s, rs(i)). (A.14)

The function r was defined in (3.3). From (3.6) and (3.9) in Proposition 3.1 wehave

m(u + 1, i(c))

p(u + 1, i(c))=

exp(

vγ(u + 1, i(c)) + λγ(u, i)>∆S(u + 1))

Z(u,i)(λγ(s, i))

for u ∈ 0, . . . , T − 1 and i ∈ Iu, which allows us to rewrite (A.14) as

m(ωt,i)

p(ωt,i)=

t−1∏

s=0

exp(

vγ(t − s, rs(i)) + λγ(t − s − 1, rs−1(i))>∆S(t − s))

Z(t−s−1,rs−1(i))(λγ(t − s − 1, rs−1(i)))(A.15)

=t−1∏

s=0

D(t − s, rs(i))

establishing (3.17).If we continue to simplify (A.15) then it becomes

m(ωt,i)

p(ωt,i)= exp

(

t∑

s=1

λγ(t − s, rs(i))>∆S(t − s + 1)

)

·

t−1∏

s=0

exp (vγ(t − s, rs(i)))

Z(t−s−1,rs−1(i))(λγ(t − s − 1, rs−1(i)))(A.16)

Since we have

exp(vγ(u, i)) = Z(u,i)(λγB(u, i))

from (3.11) in Proposition 3.1, the equation (A.16) telescopes to

t−1∏

s=0

exp (vγ(t − s, rs(i)))

Z(t−s−1,rs−1(i))(λγ(t − s − 1, rs−1(i)))=

exp(vγ(t, i))

Z(0,1)(λγ(0, 1)).

44

And we conclude that

m(ωt,i)

p(ωt,i)=

exp(

vγ(t, i) +∑t

s=1 λγ(t − s, rs(i))>∆S(t − s + 1))

Z(0,1)(λγ(0, 1))

In particular for t = T it holds

m(ωT,i)

p(ωT,i)=

exp(

γ · B(T, i) +∑T

s=1 λγ(T − s, rs(i))>∆S(T − s + 1))

Z(0,1)(λγ(0, 1))

finishing the proof.

B Proofs from Section 4

B.1 Proof of Proposition 4.1

We prove Proposition 4.1 by the uniqueness of the minimizer of the convexfunction log Z(t,i)(λ) and by invoking the conditional duality relation (2.7) in abackward inductive way.

Set t = T − 1 and fix i ∈ It−1. If we substitute θ(T − 1, i) and λ(T − 1, i) (wedrop γ and B dependence in the proof) then the conditional duality relationreads

log EP [eγ(B−θ(T−1,i)>∆S(T ))] = −vγ(T − 1, i ; B)

= log Z(T−1,i)(λ(T − 1, i)).

In our setting of a multi nomial tree this can be rewritten as

log

N∑

c=1

p(T, i(c)) · exp(

γB(c) − γθ(T − 1, i)>∆S(T ))

= logN∑

c=1

p(T, i(c)) · exp(

γB(c) + λ(T − 1, i)>∆S(T ))

implying

log Z(T−1,i)(−γθ(T − 1, i)) = log Z(T−1,i)(λ(T − 1, i)).

Since the convex function log Z(T−1,i)(λ) had a unique minimizer λ(T − 1, i) in

Rd, we conclude that λ(T − 1, i) = −γ θ(T − 1, i).

45

Now fix t ∈ 0, . . . , T − 2 and i ∈ It and assume that the result has beenproved for u ∈ t + 1, . . . , T − 1 and i ∈ Iu to finish the proof by induction.The conditional duality relation, substituted with θ(u, i) and λ(u, i) for u ≥ treads

log EP(t,i)[e

γ(B−Pt−1

u=t θ(u)>∆Su)]

= −vγ(t, i ; B)

= log Z(t,i)(λ(t, i))

= log

N∑

c=1

p(t + 1, i(c)) exp(

−vγ(t + 1, i(c) ; B) + λ(t, i)>∆St

)

(B.1)

From the induction hypothesis we have −γθ(u, ·) = λ(u, ·) for u ≥ t+1 and canrewrite the first equation as

log EP(t,i)[e

γ(B−Pt−1

u=t θ(u)>∆Su)]

= log EP(t,i)[e

γB+PT−1

u=t+1 λ(u,·)>∆Su−γθ(t,i)>∆S(t)]

= log EP(t,i)[e

−vγ(t+1,· ;B)−γθ(t,i)>∆S(t)] (B.2)

where (B.2) comes from (3.14) in Proposition 3.2 and the law of the iteratedexpectations.Comparing (B.1) to (B.2) we see that

log Z(t,i)(λ(t, i)) = log Z(t,i)(−γθ(t, i))

and conclude that λ(t, i) = −γθ(t, i) since λ(t, i) is the unique minimizer oflog Z(t,i)(λ). This finishes the proof by mathematical induction.

B.2 Proof of Proposition 4.3

Define the auxiliary function vQ,γt (B) as

vQ,γt (B) := inf

M∈Me

H(M |Q) − EM [γ · B|Ft]

then, from Theorem 3.1 applied to the market with the measure Q, we havethe results (3.6) through (3.11) with the measure p(·) replaced with q(·). Let usdenote the associated Lagrange multiplier process by λQ,γ(t).On the other hand, recall from Corollary 3.1, that the MEMM Q is the optimalmeasure associated with the zero payoff, i.e.

q(t + 1, i(c)) = p(t + 1, i(c))exp

(

v(t + 1, i(c); 0) + λ0(t, i)>∆S(t + 1, i(c)))

Z0(t,i)(λ

0(t, i))

(B.3)

46

for all (t, i), where v(t, · ; 0) and the Lagrange multiplier process λ0(t, ·) are theoutcome of the algorithm of Theorem 3.1 with B ≡ 0.

In order to show that θ∗ = θγ − θ0 is the unique optimal portfolio for the valuefunction (4.5), we have to show that

λQ,γ(t, i) = λP,γ(t, i) − λ0(t, i). (B.4)

Then Proposition 4.1 implies the desired conclusion. The rest of the proof isdevoted to establishing (B.4).For each (t, i) the Q Lagrange multiplier process λQ,γ(t, i) is the unique mini-mizer of the convex function W : R

d → R defined by

W (λ) :=N∑

c=1

q(c) exp(

vQ,γ(t + 1, i(c)) + λ>∆S(t + 1, i(c)))

. (B.5)

The necessary and sufficient condition for a point λ ∈ Rd to be the unique

critical point of W is that it should satisfy ∇λW (λ) = 0, or equivalently,

N∑

c=1

q(t+1, i(c))∆S(t+1, i(c)) exp(

vQ,γ(t + 1, i(c)) + λ>∆S(t + 1, i(c)))

= ~0 ∈ Rd.

(B.6)To show that λ = λP,γ − λ0 solves the equation (B.6), we express the left handside of (B.6) substituting (B.3) for q(·), as follows.

N∑

c=1

q(t + 1, i(c))∆S(t + 1, i(c)) exp(

vQ,γ(t + 1, i(c)) + λ>∆S(t + 1, i(c)))

=

N∑

c=1

p(t + 1, i(c))∆S(t + 1, i(c))exp

(

v(t + 1, i(c); 0) + λ0(t, i)>∆S(t + 1, i(c)))

Z0(t,i)(λ

0(t, i))

× exp(

vQ,γ(t + 1, i(c)) + λ>∆S(t + 1, i(c)))

=1

Z0(t,i)(λ

0(t, i))

N∑

c=1

p(t + 1, i(c))∆S(t + 1, i(c))

× exp(

v(t + 1, i(c); 0) + vQ,γ(t + 1, i(c); B))

× exp(

(λ0(t, i) + λ)>∆S(t + 1, i(c)))

. (B.7)

We claim that

vP,γ(t, i; B) = vQ,γ(t, i; B) + v0(t, i; 0) (B.8)

for all (t, i). An entropy identity

47

Ht(M |P ) = Ht(M |Q) + Ht(Q|P ) (B.9)

for any equivalent probability measures P, Q and M , implies

vP,γ(t, i; B) = − infM∈Me

Ht(M |P ) − EM [γ · B]

= − infM∈Me

Ht(M |Q) + Ht(Q|P ) − EM [γ · B]

= − infM∈Me

Ht(M |Q) − EM [γ · B] − Ht(Q|P )

= vQ,γ(t, i; B) + v0(t, i; 0),

establishing (B.8). Now the equation (B.7) substituted with λ = λP,γ(t, i) −λ0(t, i) becomes

1

Z0(t,i)(λ

0(t, i))

N∑

c=1

p(t + 1, i(c))∆S(t + 1, i(c))

× exp(

vP,γ(t + 1, i(c); B) + λP,γ(t + 1, i(c))>∆S(t + 1, i(c)))

, which is zero, since λP,γ(t + 1, i(c)) is the only critical point of the convexfunction log Z(t,i)(λ). From the uniqueness of the minimizer of W in (B.5) weconclude that λQ,γ(t, i) = λP,γ(t, i) − λ0(t, i).

B.3 Proof of Theorem 4.2

We claim the following lemma.

Lemma B.1 For a given M ′ ∈ Me, i.e. any equivalent martingale measure,define a functional process

vM ′,γt := − inf

M∈MeHt(M |P ′) − E

M [γ · B|Ft] (B.10)

for t ∈ 0, . . . , T.

Then all the formulae in Theorem 3.1 and Proposition 3.2 are valid for vM ′γt

with P replaced with M ′.

Furthermore, the process vM ′,γt is M ′-super-martingale.

The first claim that Theorem 3.1 and Proposition 3.2 be valid for M ′ holdsobviously since there was no restriction on P in any of the proofs other thanP being a probability measure on the tree. The second claim of the super-martingale property will be proved after the proof of Theorem 4.2.

Recall that

48

γ · h(t) = vQ,γt − v0

t

= − infM∈Me

Ht(M |Q) − EM [γ · B|Ft] + Ht(Q|P ) (B.11)

Fix an equivalent martingale measure M ′ ∈ Me, then we have

Ht(M |Q) = Ht(M |M ′) + Ht(M′|Q).

Hence we can rewrite (B.11) as

γh(t) = − infM∈Me

Ht(M |M ′) − EM [γ · B|Ft] − Ht(M

′|Q) + Ht(Q|P )

= vM ′,γt − Ht(M

′|Q) + Ht(Q|P ) (B.12)

Of the three terms in (B.12) we will show that the latter two terms are M ′-martingale. Then from Lemma B.1 the proof of Theorem 4.2 is complete.The second term is

Ht(M′|Q) = E

M ′

[

log

(

dM ′

dQ

)

|Ft

]

and therefore is a M ′-martingale. Last, the third term is

Ht(Q|P ) = EQ

[

log

(

dQ

dP

)

|Ft

]

which is obviously a Q-martingale. To see its property under the measure M ′,use Corollary 3.3 with the zero payoff B ≡ 0 then the Radon Nikodym derivativeof Q with respect to P equals to

dQ

dP=

exp(

∑Tu=1 λ0(u − 1)>∆S(u)

)

Z(0,1)(λ0(0))

where Z(0,1)(λ0(0)) = exp(v0(0)) is the normalizing constant. Hence the relative

entropy between Q and P is given by

Ht(Q|P ) = EQ

[

T∑

u=1

λ0(u − 1)>∆S(u) − log Z(0,1)(λ0(0))

∣

∣

∣

∣

Ft

]

=

t∑

u=1

λ0(u − 1)>∆S(u) − log Z(0,1)(λ0(0)) (B.13)

because S(t) is a Q-martingale. We see that (B.13) is an M ′-martingale sinceany measure in Me has S(t) as a martingale under it and the term log Z(0,1)(λ

0(0)) =v0(0) is F0-measurable and therefore a constant in our model. This finishes theproof that (B.11) is an M ′-super-martingale.

49

Proof of Lemma B.1 In order to show that

vM ′,γt = − inf

M∈MeHt(M |M ′) − E

M [γ · B|Ft]

is an M ′-super-martingale, we apply Proposition 3.2 to the market with themeasure M ′ and obtain

vM ′,γ(t, i; B) = log EM ′

(t,i)

[

exp(

vM ′,γ(t + 1) + λM ′,γ(t)>∆S(t + 1))]

(B.14)

where the Lagrange multiplier process λM ′,γ(t) is defined by (3.12) with thebase measure M ′.Now set an auxiliary process

Y (t) := vM ′,γ(t) +

t∑

u=1

λM ′,γ(u − 1)>∆S(u)

then

eY (t) = evM′,γ(t)t∏

u=1

exp(

λM ′,γ(u − 1)>∆S(u))

.

and therefore

EM ′

[eY (t+1)| Ft] = EM ′

[evM′,γ(t+1)+λM′,γ(t)>∆S(t+1)| Ft]

t∏

u=1

exp(

λM ′,γ(u − 1)>∆S(u))

Now the equation (B.14) implies that

evM′,γ(t) = EM ′

[evM′,γ(t+1)+λM′,γ(t)>∆S(t+1)| Ft]

and therefore we conclude that

EM ′

[eY (t+1)| Ft] = evM′,γ(t)t∏

u=1

exp(

λM ′,γ(u − 1)>∆S(u))

= eY (t).

We have just shown that eY (t) is an M ′-martingale, and consequently, Y (t) isan M ′-super-martingale by Jensen’s inequality. Hence

vM ′,γ(t) = Y (t) −t∑

u=1

λM ′,γ(u − 1)>∆S(u)

is an M ′-super-martingale since the latter summation is a martingale under anyequivalent martingale measure.

50

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