Entropic conditions and hedging

June 2007, Vol. 11, p. 197–216 www.edpsciences.org/ps

DOI: 10.1051/ps:2007015

ENTROPIC CONDITIONS AND HEDGING

Samuel Njoh

Abstract. In many markets, especially in energy markets, electricity markets for instance, the deten- tion of the physical asset is quite difficult. This is also the case for crude oil as treated by Davis (2000).

So one can identify a good proxy which is an asset (financial or physical) (one)whose the spot price is significantly correlated with the spot price of the underlying (e.g. electicity or crude oil). Generally, the market could become incomplete. We explicit exact hedging strategies for exponential utilities when the risk premium is bounded. Our result is based upon backward stochastic differential equation (BSDE) and a good choice of admissible strategies which allows us to solve our hedging problem.

Mathematics Subject Classification. 90A09, 34A12.

Received February 16, 2006. Revised September 6 and October 12, 2006.

Introduction

The issue of the hedge of an European option admits in the situation of incomplete markets many answers by optimization’s programs. We aim to study this question and exhibit optimal strategies for exponential utilities. On the subject, the research for the elaboration of an exact solution is still open. We intend to solve the problem when the risk premium is bounded. From an economical view, we consider an agent who sells an option quantified by an FT-measurable contingent claim almost surely finite H; with FT the information available up to the dateT,T being the maturity of the option. The agent constitutes a portfolio with an initial wealth and invests in the asset of correlated spot price and in a num´eraire. The agent can also decide not to sell. It seems natural to introduce an utility functionUγ which quantifies the preferences of the investor. Here,

Uγ(x) =−exp(−γx)

withγa strictly positive constant. Consequently our agent maximizes its terminal net wealth by the bias ofUγ. Our presentation of hedging by exponential utility is based upon [4]. We use duality methods and elaborate a good setting for the solving of our optimization problem. First the originality of our work is to introduce several sets of probability and admissible strategies that resume the difficulties and are useful to better overcome them.

Hence, we succeed to well define and characterize the optimal hedging strategy. We have tried here to give a complete setting when the risk premium is bounded. To characterize the optimal solution, we use backward stochastic differential equations.

Keywords and phrases. Stochastic optimization, martingale representation theorem.

1Universit´e de Marne-La-Vall´ee, Cit´e Descartes, 5, Bld Descartes, Champs-Sur-Marne, 77454 Marne-La-Vall´ee Cedex 2, France;

[email protected]

c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.edpsciences.org/psor http://dx.doi.org/10.1051/ps:2007015

By taking the definition of admissible strategies in [4] rather than the one in [8], we give complete proofs of results of [8]. The document is organized as follows. First, the Section 1 will recall some general results about the risk neutral pricing in a general financial model; also we will establish a martingale representation theorem when one changes the probability measure. We also focus on admissible strategies’s space and no arbitrage opportunities. Our setting deals with cross hedging, see also [4]. But the cross hedging problem can be viewed as a constraint on hedging portfolio and henceforth enter in the framework regarded in [8]. Our Section 2 is devoted to the characterization of dual problem. We point out the set of martingale measures which allows us to elaborate the dual problem. The Section 3 is concerned by the resolution of the dual problem using backward stochastic differential equations. Indeed, we do verify that the optimal solution is in the set of measure we have previously defined. The Section 4 solves thoroughly the problem of optimal hedging. We look also at the indifference price when the correlation is perfect. In Section 5 we conclude.

1. Preliminaries

We consider a financial market with two risky assets and a riskless asset, namely a bond. We make the assumption that one of the risky asset cannot be detained in a financial portfolio. Let S^e andS^g be the spot prices processes of the risky assets and let S⁰ ≡1 be the one of the asset without risk. We consider that the physical stock of priceS^ecannot be exchanged in the market, though it is taken as the underlying of contingent claims.

We recall the definition of a martingale measure:

Definition 1.1. A probability measureQis a martingale measure ifQis equivalent toPonFT and the process S^g is a local martingale underQ. LetMe be the set of martingale measures.

1.1. Duality concept

LetV_T^x,θ be the terminal value of a trading self-financing portfolio process of valuesV^x,θ, withxbeing the initial wealth of the investor. One of our goal on this work is to prove a duality result on the form

sup

θ∈ A(x)

E

−exp −γ

V_T^x,θ−H

=−exp

γ sup

Q∈ Me

E^Q(H)−x− 1 γh(Q|P)

withh(Q|P) denoting the relative entropy ofQwith respect toPwe will precise in the following section. Then the resulting strategy will be optimal for an exponential utility criterion. We notice that the caseH ≡0 recovers a pure investment problem. Beyond the dual problem, we see that one has to minimize the relative entropy minus a penalizing term depending onH. The greatest task perhaps will then be to construct the martingale measure ˆQwhich allows to get out the optimal trading strategy. Indeed it is necessary to prove that ˆQis in a subset of Me. We will also give in the Section 2 some conditions onH for the solving of the dual problem in our approach. First of all we recall results obtained.

1.2. Our results

LetB be a random variable almost surely finite. LetAb(x) andA(x) be two spaces of admissible strategies we define latter in the section. We set

v(x, B) := sup

θ∈ A(x)

E

−e^−γ(^V_T^x,θ−B)

V(x, B) = 1

γln [−v(x, B)]

Mθ:=

QP|V^x,θQ-integrable andh(Q|P)<+∞

. Then we establish that:

• IfE Be^B

<∞and ifθ ∈ Ab(x), then 1

γlnE exp

−γ

V_T^x,θ−B

= sup

Q∈M_θ

E^Q

−VT^x,θ+B − 1

γh(Q|P)

• and if there are ˆθ ∈ A(x) and ˆQ ∈ Me defined in the following section such that 1

γlnE exp

−γ

V_T^x,θˆ−B

=−x+E^Qˆ(B)− 1 γh( ˆQ|P) then

d ˆQ dP =

exp −γ

V_T^x,ˆθ−B Eexp

−γ

V_T^x,ˆθ−B ;

V(x, B) = −x+ sup

Q∈ M_e

E^Q(B)− 1 γh(Q|P)

= 1

γlnE exp

−γ

V_T^x,θˆ−B

. The optimal strategy is

θˆt=− mt

γσ_t^gS_t^g + z_t⁽¹⁾ σ_t^gS_t^g

with the assumption that the risk premium process mt is bounded, (X^B, z) being the solution of a quadratic BSDE we elaborate in Section 3; the processσ_t^g is defined in Section 1.4.

1.3. Some reminders about BSDEs

We recall some facts about backward stochastic differential equations and optimization problems. Let H^p,dT

be the space ofR^d-valued progressively measurable processesζ such that:

E T 0

ζt^pdt

<∞,

andH^∞T (R²) be the space of progressively measurableR²-valued processesζsuch that:

ess sup

Ω×[0,T]

ζt<∞.

1. We assume that the mappingf (also called further a generator) from Ω×[0, T]×R×R²intoRis such thatf(ω, t,0,0) belongs toH²T^,1. Moreover, we set the condition: f is uniformly Lipschitz in (y, z), that is there is a constantλ >0 such that

|f(t, y⁽¹⁾, z⁽¹⁾)−f(t, y⁽²⁾, z⁽²⁾)| ≤λ

|y⁽¹⁾−y⁽²⁾|+|z⁽¹⁾−z⁽²⁾|

for all (y⁽ⁱ⁾, z⁽ⁱ⁾)∈ R×R^d,i= 1,2. Also we impose tof to verifyf(t,0,0)∈ HT^2,1. 2. Then, by a well known result the BSDE

(∗)−dyt=f(t, yt, zt)dt−z_tdWt, yT =φ

withφ ∈ L²(Ω,F,P) admits a unique solution (y, z) inHT^∞,1× HT^2,2. We recall also the so-called comparison theorem:

3. Suppose that the pair (f⁽ⁱ⁾, φ⁽ⁱ⁾),i= 1,2 satisfies the condition in 1.. We assume also that f⁽¹⁾(t, y, z)≥f⁽²⁾(t, y, z), ∀(y, z)∈R×R²,

φ⁽¹⁾ ≥φ⁽²⁾.

Let (y⁽ⁱ⁾, z⁽ⁱ⁾),i= 1,2 denote the solutions of the BSDE (∗) with the parameters (f, φ) to be replaced by the pair (f⁽ⁱ⁾, φ⁽ⁱ⁾),i= 1,2 respectively. Then we get

∀t, yt⁽¹⁾≥y_t⁽²⁾ P−a.s. and φ⁽¹⁾ ≥φ⁽²⁾, φ⁽ⁱ⁾∈ L²(P), i= 1,2.

4. We enounce some results of [14] about quadratic generators and BSDE. That is f to satisfy

∀(t, y, z) ∈[0, T]×R×R² |f(t, y, z)| ≤λ₁+λ₂|y|+λ₃(|y|)z²

withλ₁,λ₂constants andλ₃to be a continuous increasing nonnegative function. LetφinL^∞(P). Then (∗) has a unique solution (y, z) inHT^∞,1× H²T^,2withycontinuous mapping. In this case, the comparison principle gives

∀t ∈[0, T] y_t⁽¹⁾≥y_t⁽²⁾ P−a.s.,

the assumptions on (f⁽ⁱ⁾, φ⁽ⁱ⁾),i= 1,2 being the same as previously, with the supplementary condition:

φ⁽ⁱ⁾∈ L^∞(P), i= 1,2.

1.4. The market model

Let (Ω,F,F,P) be a filtered probability space, equipped withF= (Ft)_0≤t≤T the associated filtration we will precise. Define Ft^W =σ(Ws⁽¹⁾, Ws⁽²⁾; 0 ≤s≤t) to be a family of sub-σ-algebras fields such that Fs^W ⊆ Ft^W, s ≤ t. Let N denote the P-null subsets of FT^W. Then Ft = σ Ft^W ∪ N

is the augmented filtration of the filtration genrated by the processes W⁽¹⁾ and W⁽²⁾. The filtration F is said to satisfy usual conditions. We notice thatF=FT. The processesW⁽¹⁾ andW⁽²⁾ are twoF-adapted independent standard Brownian motions under the probability measureP. Ftrepresents the information available at timet;Pis the physical probability under which are modeledS^e andS^g. We assume that the processesS^e andS^g do satisfy in the time interval [0, T], the stochastic differential equations:

⎧⎪

⎨

⎪⎩

dS^e_t =S_t^e

µ^e_tdt+σ^e_t

ρtdW_t⁽¹⁾+

1−ρ²_tdW_t⁽²⁾ dS^g_t =S_t^g

µ^g_tdt+σ_t^gdW_t⁽¹⁾

.

The model’s coefficientsµ^e, µ^g, σ^e, σ^g, andρareF-adapted and continuous. We add the following assumption onσ^g:∀t ∈[0, T] σ^g_t = 0 dt-a.s..To ensure the conditions of integrability, we have:

T 0

|µⁱ_tS_tⁱ|dt+ T

0

|σ_tⁱS_tⁱ|²dt <+∞, fori=e, g.

As the cross variation of the two processes is dS^e, S^gt=ρtσ^e_tσ_t^gS_t^eS_t^gdt,ρtindicates the level of the correlation between the pricesS^e andS^g during the interval [t, t+ dt]. We take the level of correlation to be strictly less than one (|ρt|<1, 0≤t≤T).

1.5. Martingale probabilities

Let Q be a probability measure equivalent to P on FT. It is well known that there exists two F-adapted processesmandν such that:

T 0

m²_t+ν_t²

dt <+∞ P-a.s. and dQ

dP = exp T

0

mtdW_t⁽¹⁾−1 2

T 0

m²_tdt+ T

0

νtdW_t⁽²⁾−1 2

T 0

ν_t²dt

. (1.1)

By Girsanov theorem, the processes ˜W⁽¹⁾ and ˜W⁽²⁾ are defined as follows:

d ˜W_t⁽¹⁾= dW_t⁽¹⁾−mtdtand d ˜W_t⁽²⁾= dW_t⁽²⁾−νtdt (1.2) are two independent standard Brownian motions under the probabilityQ.

As we have in our general model a Brownian filtration, we will use in the sequel only a previsible representation property for aQ-local martingale adapted to the filtration (Ft), whereQis a probability equivalent toP. The change of Brownian motion induced by the change of probability doesn’t allow the use of the representation property for martingales adapted to the filtration (Ft). Nevertheless, we state the following decomposition theorem of such local martingales:

Theorem 1.1. Let Q¯ be a probability equivalent toP. LetW¯ = ( ¯W⁽¹⁾,W¯⁽²⁾)be aQ-standard brownian motion¯ issued from W = (W⁽¹⁾, W⁽²⁾) by Girsanov theorem. Let N be a Q-local martingale process. Then¯ N admits a version such that there exists aR²-valued processh= (˜h⁽¹⁾,˜h⁽²⁾)F-adapted, T

0

(˜h⁽¹⁾t )²+ (˜h⁽²⁾t )²

dt <+∞

P-a.s.:

Nt=N₀+ t

0

˜h⁽¹⁾_s d ¯W_s⁽¹⁾+ t

0

˜h⁽²⁾_s d ¯W_s⁽²⁾, 0≤t≤T.

Moreover, if N is a square integrableQ-matingale, then¯

E^Q¯N, NT < +∞.

The proof is in appendix.

A necesary and sufficient condition for the processS^g to be a local martingale underQis:

µ^g_t+σ_t^gmt= 0 (1.3)

and we get

dS_t^g

S_t^g =σt^gd ˜Wt⁽¹⁾. (1.4)

From now we suppose the equality (1.3) to be satisfied,i.e. mt:=−µ^g_t

σ_t^g, 0≤t≤T.

The condition S^g local martingale under Q fixes the value of m in (1.1). Consequently, the martingale measures onFT are parametrized by a processν. More precisely:

Let (ξt^ν)_0≤t≤T be the real process defined by the stochastic differential equation: dξt^ν = ξ^νtmtdWt⁽¹⁾ + νtdW_t⁽²⁾),ξ₀^ν = 1. We haveP-a.s.

ξ_t^ν= exp t

0

msdW_s⁽¹⁾−1 2

t 0

m²_sds+ t

0

νsdW_s⁽²⁾−1 2

t 0

ν_s²ds

,0≤t≤T.

Define then the set K of F-adapted and real processes ν such that T

0

ν_t²dt < +∞P-a.s. and (ξt^ν)_0≤t≤T is a P-martingale. Therefore, given any ν ∈ K, we can define a probability measure P^ν equivalent to P by dP^ν=ξ_T^νdP. We have the following characterization of the setMe:

Me={P^ν|ν ∈ K}. (1.5)

The new probability measure

P^ν(A) =

A

ξ^ν_TdP, Abelongs toFT,

is such thatP^ν is absolutely continuous onFT with respect toP(P^νP). Moreover, sinceξ^ν_T >0P-a.s., then the two measures are equivalent (P^ν ∼P).

Now we construct our financial portfolio. We have an investor who starts with some initial endowmentx and invests it in the bond and in the proxy asset. Let (Vt)_0≤t≤T be the investor’s wealth process. It will be the hedging portfolio when the investor holds an option. Then, in our model the portfolio is constructed upon the assets of pricesS^g andS⁰≡1. Thus, its value at timetis

Vt=ηtS_t⁰+θtS_t^g=ηt+θtS_t^g;

ηtand θt being respectively the quantities of assets of pricesS⁰ and S^g detained at timet by the investor in the portfolio. We make the assumption that the portfolio is self-financing; that is

Vt=V_t^x,θ=x+ t

0

θsdS_s^g withθ an element of

L(S^g) =

θF-adapted| T

0

θ_t²dS^g, S^gt < +∞P−a.s.

.

We will precise latter the space of admissible strategies. Define a contingent claim to be an FT-measurable random variable.

To evaluate and hedge the contingent claimH by an exponential utility function, it is necessary to compute the quantities

v(x+p, H) = sup

θ

E

−e^−γ(^VT^x+p,θ−H)

and v(x,0) = sup

θ

E

−e^−γVTx,θ ,

wherepis the option’s price asxis the initial endowment when the investors plans not to sell the option.

Hence we must choose at the beginning the space of admissible strategies Θ(x) = A(x), upon which a supremum is attained.

1.6. The set of admissible strategies

• to avoid arbitrage opportunities on the financial market{S⁰, S^g};

• to ensure the existence ofEe^−γVTx,θ andE

e^−γ(^V_T^x,θ−H)

;

• to guarantee the existence of admissible optimal solutions to the problems (PH) sup

θ

E

−e^−γ(^V_T^x,θ−H)

and (P0) sup

θ

E

−e^−γVTx,θ .

The concept of hedging strategies is introduced in order to allow the solution of the contingent claim valuation problem. Letting

Ab(x) :=

θ ∈ L(S^g) : ∃aθ,x ∈ R, Vt^x,θ≥ aθ,x P-a.s.∀t,0≤t≤T

, we assume that

Me=∅.

Henceforth, forθ ∈ Ab(x),V^x,θ is lower bounded. MoreoverV^x,θ is aQ-local martingale for allQelement of Me. FinallyV^x,θ is aQ-supermartingale according to Fatou’s lemma.

Proposition 1.1. There are no arbitrage opportunities on the financial market {S⁰, S^g} when the strategies are in Ab(x).

Proof. Indeed, assume that there are arbitrage opportunities for a hedging strategyθ ∈ Ab(x): that is there existx ∈ Randθ ∈ Ab(x):

x≤0 andV_T^x,θ ≥0P-a.s., P(VT^x,θ>0)>0.

θ ∈ Ab(x) impliesV^x,θQ-supermartingale; E^Q V_T^x,θ

≤x≤0;

dQdPV_T^x,θ≥0P-a.s. implies E^Q V_T^x,θ

≥0. Hence E^Q V_T^x,θ

= 0;

V_T^x,θ≥0P-a.s.,P(VT^x,θ>0)>0 impliesE^Q V_T^x,θ

=E^Q

V_T^x,θ1_{Vx,θ T >0}

>0.

This is impossible. Hence there are no arbitrage opportunities on the market{S⁰, S^g} forθ ∈ Ab(x).

We choose a space of admissible strategies which contains the strategiesθsuch that the wealth processV^x,θ is not necessarily lower bounded and which could guarantee the existence of an optimal solution in that space.

We then define the set:

A(x) :=

θ ∈ L(S^g)| ∃(θn)n ∈ Ab(x) such that e^−γ(^V_T^x,θn−H)^L→^1(P)e^−γ(^V_T^x,θ−H) .

We get in particular ∀θ ∈ A(x), exp−γ

V_T^x,θ−H

∈ L¹(P). We need e^γH ∈ L¹(P); this hypothesis holds for instance whenH is a bounded random variable.

The set of admissible strategies for a given initial wealth x is A(x). It contains Ab(x) as well as some strategiesθsuch thatV^x,θ is non necessarily lower bounded.

2. Formulation of the dual problem

Leth(Q|P) be the relative entropy of a probability measureQwith respect to P;

h(Q|P) =

⎧⎨

⎩ E dQ

dPln dQ

dP

if QP

+∞ otherwise.

LetB be a random variable.

We denote byE^Q(·) the expectation operator with respect to the probability measureQ.

Let us define

M^B :=

QP|h(Q|P)<+∞andE^Q(B)<+∞ .

We will show the following lemma which will be useful to formulate the dual problem of supθE

−e^−γ(^V_T^x,θ−H) : Lemma 2.1. ForB random variable almost surely finite,

ln(Ee^B) = sup

Q∈ M^B

E^Q(B)−h(Q|P)

. (2.1)

Moreover ifQ ∈ M^B, we getln(Ee^B) =E^Q(B)−h(Q|P)if and only if E Be^B

<∞anddQ dP = e^B

Ee^B· Proof. If we takeQ ∈ M^B, thenE^Q(B)−h(Q|P) =E^Q

ln e^B

dQ/dP

. Hence, by Jensen’s inequality, we have, forQ ∈ M^B,

E^Q(B)−h(Q|P) =E^Q

ln e^B

dQ/dP

= lnE

e^BI{^dQdP>0}

≤ln(Ee^B),

whereIA(ω) is the characteristic function of the setA; that isIA(ω) = 1 ifω belongs toA, or 0 otherwise.

But, sincex−→lnxis stictly concave, there is equality only if dQ/dP^eB = constantQ-a.s.

Hence ^dQ_dP =_Ee^eBB a.s., and there is uniqueness if the supremum is attained. It is the case ifB is almost surely finite.

To show the equality: ln(Ee^B) = sup

Q∈ M^B

E^Q(B)−h(Q|P)

, let us introduce the sequence of probability measuresQn defined by

dQn

dP = e^BI_{|B|≤n}

E e^BI_{|B|≤n}·

Since|B|<∞a.s.,Qn is well defined for a sufficiently largenand Qn ∈ M^B. We obtainE^Qn(B)−h(Qn|P) = lnE e^BI_{|B|≤n}

.

By monotone convergence theorem, we have the equality; whence the desired result.

We recall that 0≤t≤T, ξ^ν_t = exp

t 0

msdW_s⁽¹⁾−1 2

t 0

m²_sds+ t

0

νsdW_s⁽²⁾−1 2

t 0

ν_s²ds

.

We recall also the characterization of the set of equivalent martingale measures Me. Indeed, we have in view to formulate the dual problem whose the optimal solution is in a subset ofMeunder certain conditions we will precise in the sequel. Then

Me=

P^ν|dP^ν

dP =ξ_T^ν, νF −adapted such that T

0

m²_t+ν_t²

dt+∞a.s. and ξ^νP−martingale

. We get

ln dP^ν

dP

= T

0

mtdW_t⁽¹⁾−1 2

T 0

m²_tdt+ T

0

νtdW_t⁽²⁾−1 2

T 0

ν_t²dt.

Under the martingale measureP^ν, ln

dP^ν dP

= T

0

mtd ˜Wt⁽¹⁾+1 2

T 0

m²tdt+ T

0

νtd ˜Wt⁽²⁾+1 2

T 0

νt²dt. (2.2)

We define the process Mt := ξ_t^ν =E dP^ν

dP|Ft

which will be frequently used in the proofs throughout this section and in the following.

We define now the set

Me:=

Q=P^ν|E^Pν T

0

m²_t+ν_t²

dt <+∞

and we assume that

Me=∅.

The subsetMe ofMeis the space upon which we want to find the solution of the dual problem that remains to define. First we characterizeMe by showing notably that it is the set of probabilities with finite entropy.

We have the following theorem:

Theorem 2.1. A characterization of the set Me is

Me ={Q=P^ν|h(P^ν|P)<+∞}. Moreover we have, for allP^ν ∈ Me,

h(P^ν|P) =1 2E^Pν

T 0

m²_t+ν_t²

dt. (2.3)

Proof. IfP^ν ∈ Me, E^Pν T

0

m²_t+ν_t²

dt <+∞. Hence the processes .

0

msd ˜W_s⁽¹⁾ and .

0

νsd ˜W_s⁽²⁾ are mar- tingales under the martingale measureP^ν.

But, according to (2.2), we have h(P^ν|P) =E^PνlndP^ν

dP =E^Pν T

0

mtd ˜W_t⁽¹⁾+1 2

T 0

m²_tdt+ T

0

νtd ˜W_t⁽²⁾+1 2

T 0

ν_t²dt

. Henceforth h(P^ν|P) =¹₂E^PνT

0 m²t+ν²t

dt <+∞.

Conversely, there is an equivalence between the assertions:

1. h(P^ν|P)<+∞, 2. E(MTlnMT)<+∞.

Indeed, the function ϕ : x → xlnx is convex on [0,+∞) and admits a unique minimum in x = ¹_e: ∀x ≥ 0, xlnx≥ −¹_e.

We setτn= inf

t ∈ [0, T]| t

0 m²_s+ν_s² ds≥n

, n≥1. (τn)n≥1 is a sequence of stopping times converg- ing towards infinity. We adopt the convention inf∅= +∞.

We obtain

EMTlnMT∧τ_n = E(E(MTlnMT∧τ_n|FT∧τ_n))

= EMT∧τ_nlnMT∧τ_n.

Hence

EMT∧τ_nlnMT∧τ_n = EMT∧τ_n

T∧τ_n 0

mtdW_t⁽¹⁾+ T∧τ_n

0

νtdW_t⁽²⁾−1 2

T∧τ_n 0

m²_t+ν_t² dt

= E^Pν

T∧τ_n 0

mtd ˜Wt⁽¹⁾+ T∧τ_n

0

νtd ˜Wt⁽²⁾+1 2

T∧τ_n 0

m²t+νt²

dt

= 1

2E^Pν T∧τ_n

0

m²_t+ν²_t dt.

The sequenceτn being increasing, we get by monotone convergence theorem

n−→∞lim EMT∧τ_nlnMT∧τ_n = sup

n≥1EMT∧τ_nlnMT∧τ_n

= 1

2E^Pν T

0

m²_t+ν_t²

dt. (2.4)

But conditional Jensen’s inequality yields

MT∧τ_nlnMT∧τ_n≤E(MTlnMT|FT∧τ_n). By hypothesis,MTlnMT is integrable.

Hence the sequence (MT∧τ_nlnMT∧τ_n)_n is uniformly integrable, and we get

nlim→∞EMT∧τ_nlnMT∧τ_n = E(MTlnMT)

= 1

2E^Pν T∧τ_n

0

m²_t+ν_t²

dt <+∞. ForQ=P^ν ∈ Me, the entropy with respect toPis

h(Q|P) =E^Qln dQ

dP

=E^Q

1 2

T 0

m²_tdt+1 2

T 0

ν²_tdt

.

We will next formulate the dual problem for an investor having an initial wealthxas mentionned in Section 1.

The agent invests in the construction of a hedging portfolioV^x,θ and in the same time sells a contingent claim H at the date t = 0. We suppose that the price of the option of payoff H is included in the initial wealth.

Hence the agent’s program is:

(PH) v(x+p, H) := sup

θ∈ A(x+p)

E

−e^−γ(^V_T^x+p,θ−H)

=− inf

θ∈ A(x+p)E

e^−γ(^V_T^x+p,θ−H) . We set

(P0) v(x,0) := sup

θ∈ A(x)

E

−e^−γVTx,θ

=− inf

θ∈ A(x)E e^−γVTx,θ

.

(P0) is the program of the agent having the initial wealthx, and invests only in the construction of a portfolioθ.

In the two cases the agent is maximizing the expected utility v. Then, the indifference price (cf. [8]) is defined by the equality:

v(x+p, H) =v(x,0)

or equivalently,

1

γln [−v(x+p, H)] = 1

γln [−v(x,0)]. From now, we set

V(x, H) := 1

γln [−v(x, H)].

Henceforth, we will try to computeV(x, H) in order to get a fair characterization of the optimal hedging ˆθand the pricep.

Let us calculate now

V(x, H) = inf

θ∈ A(x)

1 γlnE

exp −γ

V_T^x,θ−H

.

Let us exhibit another set of martingale measures which will fully characterize the dual problem.

Letθbe inAb(x). We define the set Mθ:=

QP|V^x,θQ-integrable andh(Q|P)<+∞

. By using Lemma 2.1, ifθ ∈ Ab(x)

1 γlnE

exp −γ

V_T^x,θ−H

= sup

Q∈M_θ

E^Q

−VT^x,θ+H −1

γh(Q|P)

. (2.5)

Indeed, if Qis not in the setMθ,h(Q|P) = +∞. AsH is almost surely finite, and asθ ∈ Ab(x) implies E^Q

V_T^x,θ

≤x, (2.6)

hence we have 1 γ sup

Q∈M_θ

E^Q

−γ

−VT^x,θ+H

−h(Q|P)

≥ 1 γ sup

Q∈M_e

E^Q

−γ

−VT^x,θ+H

−h(Q|P)

≥ sup

Q∈M_e

E^Q

−

−VT^x,θ+H −1

γh(Q|P)

≥ −x+ sup

Q∈ Me

E^Q(H)− 1 γh(Q|P)

. (2.7)

Consequently,

V(x, H) = inf

θ∈ A(x)

1 γlnE

exp −γ

V_T^x,θ−H

= inf

θ∈ Ab(x)

1 γlnE

exp −γ

V_T^x,θ−H

.

Hence V(x, H)≥ −x+ sup

Q∈ Me

E^Q(H)−1 γh(Q|P)

. If we can find ˆθ∈ A(x) and ˆQ∈ Mesuch that

1 γlnE

exp −γ

V_T^x,ˆθ−H

=−x+E^Qˆ(H)−1

γh( ˆQ|P), the previous inequality is an equality.

We first study the problem:

sup

Q∈ Me

E^Q(H)−1 γh(Q|P)

= sup

P^ν∈ Me

E^Pν(H)− 1 2γE^Pν

T 0

m²_t+ν_t² dt

. We will solve the formulated control problem in the following section.

3. Properties of the optimal dual process

Denote byKe the set of progressively measurable processesν such thatP^ν ∈ Me. We define the process (Mt^ν)_0≤t≤T by

M_t^ν :=E^Pν

H− 1 2γ

T 0

m²_t +ν_t² dt| Ft

.

M^ν is a martingale process underP^ν.

According to the Theorem 1.1 there is a process z^ν F-adapted with values in R²such that T

0

(z_t^1,ν)² + (z²_t^,ν)²

dt <+∞ P-a.s. and:

M_t^ν :=M₀^ν+ t

0

(z_s^ν)d ˜Ws = M₀^ν+ t

0

z_s^1,νd ˜W_s⁽¹⁾+ t

0

z_s^2,νd ˜W_s⁽²⁾, 0≤t≤T.

Let

X_t^H,ν

0≤t≤T be the process defined by:

X_t^H,ν :=E^Pν

H− 1 2γ

T t

m²_s+ν²_s ds| Ft

, and

Xt^H := ess sup

ν

Xt^H,ν. It is clear that

X_t^H,ν = 1 2γ

t 0

m²_s+ν_s²

ds+M_t^ν. Moreover,M₀^ν =X₀^H,ν. We obtain using Girsanoy theorem

dXt^H,ν = 1

2γ m²t+νt²

dt+zt^1,νd ˜Wt⁽¹⁾+z²t^,νd ˜Wt⁽²⁾

= 1

2γ m²t+νt²

dt+zt^1,νdWt⁽¹⁾+z²t^,νdWt⁽²⁾−zt^1,νmtdt−zt^2,ννtdt

= 1

2γ m²_t+ν_t²

−2γ

z¹_t^,νmt+z_t^2,ννt

dt+z¹_t^,νdW_t⁽¹⁾+z_t^2,νdW_t⁽²⁾. We also remark thatX_T^H,ν =H.

Henceforth, (X^H,ν, z) verifies the backward stochastic differential equation ( BSDE in short) −dXt^H,ν = fm,t(z^ν, ν)dt−z_t^1,νdW_t⁽¹⁾−z_t^2,νdW_t⁽²⁾, 0≤t≤T

X_T^H,ν = H (3.1)