A Maximum Principle Approach to Risk Indifference Pricing with Partial Information

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Volume 2008, Article ID 821243,15pages doi:10.1155/2008/821243

Research Article

A Maximum Principle Approach to Risk

Indifference Pricing with Partial Information

Ta Thi Kieu An,¹ Bernt Øksendal,^{1, 2}and Frank Proske¹

1Centre of Mathematics for Applications (CMA), Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway

2Department of Finance and Management Science, Norwegian School of Economics and Business Administration, Helleveien 30, 5045 Bergen, Norway

Correspondence should be addressed to Ta Thi Kieu An,atkieu@math.uio.no Received 10 May 2008; Accepted 28 September 2008

Recommended by Yaozhong Hu

We consider the problem of risk indifference pricing on an incomplete market, namely on a jump diffusion market where the controller has limited access to market information. We use the maximum principle for stochastic differential games to derive a formula for the risk indifference pricep^seller_risk G,Eof a European-type claimG.

Copyrightq2008 Ta Thi Kieu An et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Suppose the value of a portfolioπt, S0tis given by

X_x^πt xπtSt S₀t, 1.1

wherexis the initial capital,Stis a semimartingle price process of a risky asset,πtis the number of risky assets held at timet, andS0tis the amount invested in the risk-free asset at timet. Then, the cumulative cost at timetis given by

Pt X^π_x t− _t

0

πu⁻dSu. 1.2

IfPt p-constant for allt, then the portfolio strategiesπt, S0tis called self-financing.

A contingent claim with expiration dateTis a nonnegativeFT-measurable random variableG

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that represents the timeT payoﬀfrom seller to buyer. Suppose that for a contingent claimG there exists a self-financing strategy such thatX^π_x T G, that is,

p _T

0

πudSu G. 1.3

Then,pis the price ofGin the complete market, that is,

pEQG, 1.4

whereQis any martingale measure equivalent toPon the probability spaceΩ,Ft, P. In an incomplete market, an exact replication of a contingent claim is not always possible. One of the approaches to solve the replicating problems in an incomplete market is the utility indifference pricing. See, for example, Grasselli and Hurd 1 for the case of stochastic volatility model, Hodges and Neuberger 2 for the financial model with constraints, and Takino3for model with incomplete information. The utility indifference price p of a claim G is the initial payment that makes the seller of the contract utility indifferent to the two following alternatives: either selling the contract with initial payment p and with the obligation to pay out G at time T or not selling the contract and hence receiving no initial payment.

Recently, several papers discuss risk measure pricing rather than utility pricing in incomplete markets. Some papers related to risk measure pricing are the following: Xu4 propose risk measure pricing and hedging in incomplete markets; Barrieu and El Karoui5 study a minimization problem for risk measures subject to dynamic hedging; Kl öppel and Schweizer6study the indifference pricing of a payoffwith a minus dynamic convex risk measure. See also the references in these papers.

In our paper, we study a pricing formula based on the risk indifference principle in a jump-diffusion market. The same problem was studied by Øksendal and Sulem 7 with the restriction to Markov controls. So the problem is solved by using the Hamilton- Jacobi-Bellman equation. In our paper, the control process is required to be adapted to a given subfiltration of the filtration generated by the underlying Lévy processes. This makes the control problem non-Markovian. Within the non-Markovian setting, the dynamic programming cannot be used. Here we use the maximum principle approach to find the solution for our problem.

The paper is organized as follows. InSection 2, we will implement the option pricing method in an incomplete market. InSection 3, we present our problem in a jump-diffusion market. In Section 4, we use a maximum principle for a stochastic differential game to find the relation between the optimal controls of the stochastic differential game and of a corresponding stochastic control problem. Using this result, we derive the relationship between the two value functions of the two problems above, and then find the formulas for the risk indifference prices for the seller and the buyer.

2. Statement of the problem

Assume that a filtered probability spaceΩ,F,{Ft}_0≤t≤T, Pis given.

Definition 2.1. A nonnegative random variableGonΩ,Ft, Pis called a European contingent claim.

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From now on, we consider a European-type option whose payoﬀat time t is some nonnegative random variable G gSt. In the rest of the paper, we will identify a contingent claim with its payoﬀfunctiong.

LetFbe the space of all equivalence classes of real-valued random variables defined onΩ.

Definition 2.2see8,9. A convex risk measureρ :F→R∪ {∞}is a mapping satisfying the following properties, forX, Y ∈F:

i convexity

ρλX 1−λY≤λρX 1−λρY, λ∈0,1; 2.1

ii monotonicityifX ≤Y, thenρX≥ρY.

If an investor sells a liability to pay out the amountgSTat timeTand receives an initial paymentpfor such a contract, then the minimal risk involved for the seller is

ΦGxp inf

π∈Pρ

X_xp^πT−gST

, 2.2

wherePis the set of self-financing strategies such thatX^π_x t≥c, for some finite constant c and for 0≤t≤T.

If the investor has not issued a claimand hence no initial payment is received, then the minimal risk for the investor is

Φ0x inf

π∈Pρ

X^π_x T

. 2.3

Definition 2.3. The seller’s risk indiﬀerence price,p p^seller_risk , of the claimGis the solutionpof the equation

ΦGxp Φ0x. 2.4

Thus,p_risk^selleris the initial paymentpthat makes an investor risk indiﬀerent between selling the contract with liability payoﬀGand not selling the contract.

In view of the general representation formula for convex risk measuressee10, we will assume that the risk measureρ, which we consider, is of the following type.

Theorem 2.4 representation theorem8, 9. A map ρ : F → R is a convex risk measure if and only if there exists a familyL of measuresQ P on FT and a convex “penalty” function ζ:L →−∞,∞with inf_Q∈LζQ 0 such that

ρX sup

Q∈L

EQ−X−ζQ

, X∈F. 2.5

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By this representation, we see that choosing a risk measureρis equivalent to choosing the familyLof measures and the penalty functionζ.

Using the representation2.5, we can write2.2and2.3as follows:

ΦGxp inf

π∈P

sup

Q∈L

EQ

−X_xp^πT gSt −ζQ

, 2.6

Φ0x inf

π∈P

sup

Q∈L

EQ

−X_x^πT −ζQ

, 2.7

for a given penalty functionζ.

Thus, the problem of finding the risk indifference price p p^seller_risk given by 2.4 has turned into two stochastic differential game problems 2.6 and 2.7. In the complete information, Markovian setting this problem was solved in 7 where the authors use Hamilton-Jacobi-Bellman-Isaacs HJBI equations and PDEs to find the solution. In our paper, the corresponding partial information problem is considered by means of a maximum principle of differential games for SDEs.

3. The setup model

Suppose in a financial market, there are two investment possibilities:

ia bond with unit priceS0t 1,t∈0, T; iia stock with price dynamics, fort∈0, T,

dStSt⁻

αtdtσtdBt

R0

γt, zNdt, dz

, S0 s >0.

3.1

HereB_tis a Brownian motion andNdt, dz Ndt, dz−νdzdtis a compensated Poisson random measure with L´evy measureν. The processesαt,σt, andγt, zareFt-predictable processes such thatγt, z>−1, for a.s.t, z, and

E _T

0

|αs|σ²s

R0

|log1γs, z|²νdz

ds

<∞a.s., 3.2

for allT ≥0.

Let Et ⊆ Ft be a given subfiltration. Denote by πt,t ≥ 0, the fraction of wealth invested in St based on the partial market information Et ⊆ Ft being available at time t.Thus, we impose on πtto beEt-predictable. Then, the total wealthX^πt with initial wealthxis given by the SDE

dX^πt πt⁻St⁻

αtdtσtdBt

R0

γt, zNdt, dz

, X^π0 x >0.

3.3

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In the sequel, we will call a portfolioπ ∈ Padmissible ifπisEt-predictable, permits a strong solution of3.3, and satisfies

_T

0

|αt||πt|St σ²tπ²tS²t π²tS²t

R0

γ²t, zνdz

ds <∞, 3.4

as well as

πtStγt>−1 ω, t, z-a.s. 3.5

The class of admissible portfolios is denoted byΠ.

Now, we define the measuresQ_θparameterized by givenFt-predictable processesθ θ0t, θ1t, zsuch that

dQθω KθTdPω onFT, 3.6

where

dKθt Kθt⁻

θ0tdBt

R0

θ1t, zNdt, dz

, t∈0, T, Kθ0 k >0,

3.7

We assume that

θ1t, z≥ −1 for a.a.t, z, _T

0

θ²₀s

R0

log1θ₁s, z²νdz

ds <∞a.s.

3.8

Then, by the It ˆo formula, the solution of3.7is given by

Kθt kexp

− _t

0

θ0sdBs−1 2

_t

0

θ²₀sds

_t

0

R0

ln1−θ₁s, zNdt, dz

_t

0

R0

{ln1−θ₁s, z θ₁s, z}νdzds

.

3.9

We say that the control θ θ0, θ1 is admissible and writeθ ∈ Θ if θ is adapted to the subfiltrationEtand satisfies3.8and

EKθT K_θ0 k >0. 3.10

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We set

dYt

⎡

⎢⎣ dY₁t dY₂t dY3t

⎤

⎥⎦

⎡

⎢⎣ dK_θt

dSt dX^πt

⎤

⎥⎦

⎡

⎢⎣ 0 St⁻αt St⁻πtαt

⎤

⎥⎦dt

⎡

⎢⎣

Kθt⁻θ0t St⁻σt St⁻πtσt

⎤

⎥⎦dBt

R0

⎡

⎢⎣

Kθt⁻θ1t, z St⁻γt, z St⁻πtγt

⎤

⎥⎦ Ndt, dz,

Y0 y y1, y₂, y₃ k, s, x, dYt

dY1t dY₂t

dKθt dSt

, Y0 y y1, y₂ k, s.

3.11

We now define two setsL, Mof measures as follows:

L{Qθ; θ∈Θ},

M{Qθ; θ∈M}, 3.12

where

M{θ∈Θ; EMθt,y | Et 0∀t,y}, Mθt,y Mθt, k, s αt σtθ0t

R0

γt, zθ1t, zνdz. 3.13 In particular, by the Girsanov theorem, all the measuresQ_θ ∈ MwithEKθT 1 are equivalent martingale measures for theEt-conditioned marketS0t, S1t,where

dS₁t S₁t⁻

Eαt| EtdtEσt| EtdBt

R0

Eγt, z| EtNdt, dz S10 s >0

3.14

see, e.g.,11, Chapter 1.

We assume that the penalty functionζhas the form

ζQθ E

_T

0

R0

λt, θ0t,Yt, θ1t,Yt, z,Yt, zνdzdthYT

, 3.15

for some convex functionsλ∈C¹R²×R0, h∈C¹R, such that E

_T

0

R0

|λt, θ0t,Yt, θ1t,Yt, z,Yt, z|νdzdt|hYT|

<∞, 3.16

for allθ, π∈Θ×Π.

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Using theYt-notation, problem2.6can be written as follows:

Problem A. FindΦ^E_Gt, yandθ^∗, π^∗∈Θ×Πsuch that Φ^E_Gt, y: inf

π∈Π

sup

θ∈ΘJ^θ,πt, y

J^θ^∗^,π^∗t, y, 3.17

where

J^θ,πt, y Jθ, π E^y

− _T

t

Λθu,Yudu−hYT KθTgST−KθTX^πT

,

3.18

Λθ Λθt,y

R0

λt, θ0t,y, θ 1t,y, z, y, zνdz. 3.19

We will relate Problem A to the following stochastic control problem:

Ψ^E_Gsup

Q∈M{EQG−ζQ}. 3.20

Using theYt-notation, the problem gets the following form.

Problem B. FindΨ^E_Gt,y and ˇθ∈Msuch that Ψ^E_Gt,y :sup

θ∈MJ₀^θt,y J₀^θ^ˇt,y, 3.21

where

J₀^θt,y E^y

− _T

t

Λθu,Yudu−hYT K_θTgST

. 3.22

Define the HamiltonianH:0, T×R×R×R×Θ×Π×R×R× R →Rfor Problem A by Ht, k, s, x, θ, π, p, q, r·, z

−Λt,Yt sαp₂sαπp₃kθ₀q₁sσq₂sσπq₃

R0

{kθ1r₁·, z sγt, zr2·, z sπγt, zr3·, z}νdz,

3.23

and the HamiltonianH:0, T×R×R×Θ×R×R× R →Rfor Problem B by Ht, k, s, θ, p, q, r·, z

−Λt,Yt sαp2kθ0q1sσq2

R0

{kθ1t, zr1·, z sγt, zr2·, z}νdz. 3.24

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HereRis the set of functionsr :0, T×R→ Rsuch that the integrals in3.23and3.24 converge. We assume thatHandHare diﬀerentiable with respect tok,s, andx. The adjoint equations corresponding toθ,π, andYtin the unknown adapted processes pt,qt, rt, zare the backward stochastic diﬀerential equationsBSDEs

dp₁t ∂Λ

∂kt,Yt −θ₀tq1t−

R0

θ₁t, zr1t, zνdz

dt q1tdBt

R0

r1t, zNdt, dz, p₁T −∂h

∂kYT gST−X^πT,

3.25

dp2t ∂Λ

∂st,Yt −αtp2t−σtq2t−

R0

γt, zr2t, zνdz

dt q₂tdBt

R0

r₂t, zNdt, dz, p2T −∂h

∂sYT KθTgST,

3.26

dp3t

−αtp3t−σtq3t−

R0

γt, zr3t, zνdz

dt q₃tdBt

R0

r₃t, zNdt, dz, p3T −KθT.

3.27

Similarly, the adjoint equationscorresponding to θ andYtin the unknown pro- cessespt, qt, rt, zare given by

dp1t ∂Λ

∂kt,Yt −θ0tq1t−

R0

θ1t, zr1t, zνdz

dt q₁tdBt

R0

r₁t, zNdt, dz,

p1T −∂h

∂kYT gST, dp₂t

∂Λ

∂st,Yt −αtp₂t−σtq₂t−

R0

γt, zr₂t, zνdz

dt q₂tdBt

R0

r₂t, zNdt, dz,

p2T −∂h

∂sYT KθTgST.

3.28

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Lemma 3.1. Let θ ∈ Θand suppose thatpt p1t,p2t is a solution of the corresponding adjoint equations3.28. For allπ∈R, define

p1t p1t−X^πt, 3.29

p2t p2t, 3.30

p₃t −Kθt. 3.31

Ifθ ∈M, thenpt p1t, p2t, and p3tis a solution of the adjoint equations3.25,3.26, and3.27. Then, the following relation holds:

Ht, Yt, θ, π, pt, qt, rt, z

Ht, Yt, θ,pt, qt, rt, z −StπKθt

αt 2θ0tσt 2

R0

θ1t, zγt, zνdz

. 3.32

Proof. Diﬀerentiating both sides of3.29, we get

dp1t dp1t−dX^πt

∂Λ

∂kt,Yt−θ₀tq₁t−

R0

θ₁t, zr₁t, zνdz−Stαtπt

dt

q1t−StσtπtdBt

R0

r1t, z−Stπtγt, zNdt, dz.

3.33

Comparing this with3.25by equating thedt,dBt,Ndt, dz coeﬃcients, respectively, we get

∂Λ

∂kt,Yt −θ₀tq1t−

R0

θ₁t, zr1t, zνdz ∂Λ

∂kt,Yt−θ₀tq₁t−

R0

θ₁t, zr₁t, zνdz−Stαtπt,

3.34

q₁t q₁t−Stσtπt, 3.35

r₁t, z r₁t, z−Stγt, zπt. 3.36

Substituting3.35and3.36into3.34, we get

Stπt

αt θ0tσt

R0

θ1t, zγt, zνdz

0. 3.37

Sinceθ∈M,3.37is satisfied, and hencep₁tis a solution of3.25.

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Proceeding as above with the processesp2tandp3t, we get

q2t q2t, r2t r2, 3.38

−αtp3t−σtq3t−

R0

γt, zr3t, zνdz 0, 3.39

q3t −Kθtθ0t, r3t, z −Kθtθ1t, z. 3.40 With the valuesp3t,q3t, andr3t, zdefined as above, relation3.39is satisfied ifθ∈M.

Hence,p₁t,p₂t, andp₃tare solutions of3.29,3.30, and3.31, respectively.

Equations3.23and3.24give the following relation betweenHandH:

Ht, y, θ, π, p, q, r·, z Ht, y, θ, p, q, r·, z sπ

αp3σq3

R0

γt, zr3·, zνdz

. 3.41 Hence,

Ht, Yt, θ, π, pt, qt, rt, z

Ht, Yt, θ, p1t, p2t, q1t, q2t, r1t, z, r2t, z

−Stπt

αtp3t σq₃t

R0

γt, zr3t, zνdz

Ht, Yt, θ,p1t,p2t,q1t,q2t,r1t, z,r2t, z

−StσtπtKθtθ0t−

R0

Stγt, zπtKθtθ1t, zνdz

−StπtKθt

αt σtθ0t

R0

γt, zθ1tνdz

Ht, Yt, θ,pt, qt, rt, z

−sπKθt

αt 2σtθ0t 2

R0

γt, zθ1t, zνdz

.

3.42

Lemma 3.2. Letp1t,p2t, andp3tbe as inLemma 3.1. Suppose that, for allπ ∈R, the function θ−→EHt, Yt, θ, πt, pt, qt, rt, z| Et, θ∈Θ, 3.43

has a maximum point atθθπ. Moreover, suppose that the function

π −→EHt, Yt,θπ, π, pt, qt, r t, z| Et, π ∈R, 3.44 has a minimum point atπ ∈R. Then,

Mθ π 0. 3.45

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Proof. The first-order conditions for a maximum point θ θπ of the function EHt, Yt, θ, πt, pt, qt, rt, z| Etis

E

∇θHt, Yt, θ, πt, pt, qt, rt, z_θ_θπ | Et 0, 3.46

where∇θ ∂/∂θ0, ∂/∂θ₁is the gradient operator. The first-order condition for a minimum pointπof the functionEHt, Yt,θπ, π, pt, qt, rt, z | Etis

E

∇πHt, Yt,θπ , πt, pt, qt, rt, z_π_π| Et 0, 3.47

that is,

E

∇θHt, Yt, θ,π, pt, qt, rt, z _θ_θ_π

dθπ dπ

ππ

∇πHt, Yt, θ, π, pt, qt, rt, z_π_{π, θ} θπ | Et

0.

3.48

Chooseππ. Then, by3.46and3.48, we have E

∇πHt, Yt, θ, π, pt, qt, rt, z_π_{π, θ} θπ | Et 0, 3.49

that is,

E

Stαtp3t Stσtq3t

R0

Stγt, zr3t, zνdz| Et

0. 3.50

Substituting the valuesp₃t,q₃t, andr₃t, zas inLemma 3.1into3.50, we get

E

StKθt

αt σtθ0t

R0

γt, zθ1t, zνdz

| Et

0. 3.51

This gives,

Mθ π 0. 3.52

4. Maximum principle for stochastic differential games

Problem A is related to what is known as stochastic games studied in12. Applying in12, Theorem 2.1 to our setting we get the following jump-diﬀusion version of the maximum principleof Ferris and Mangasarian type13.

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Theorem 4.1 maximum principle for stochastic diﬀerential games 12. Let θ, π ∈ Θ × Π and suppose that the adjoint equations 3.25, 3.26, and 3.27 admit solutions p₁t,q₁t,r₁t, z, p₂t,q₂t,r₂t, z, and p₃t,q₃t,r₃t, z, respectively. Moreover, suppose that, for allt∈0, T, the following partial information maximum principle holds:

sup

θ∈ΘEHt, Yt, θ,πt, pt, qt, rt, z| Et EHt, Yt,θt, πt, pt, qt, rt, z | Et inf

π∈ΠEHt, Yt,θt, π, pt, qt, rt, z| Et.

4.1

Suppose

θ−→Jθ,π is concave,

π −→Jθ, π is convex. 4.2

Thenθ^∗, π^∗: θ,πis an optimal control and Φ^E_Gx inf

π∈Π

sup

θ∈ΘJθ, π

sup

θ∈Θ

π∈ΠinfJθ, π sup

θ∈ΘJθ,π inf

π∈ΠJθ, π Jθ, π.

4.3

Theorem 4.2. Letp1t,p2tbe, respectively, solutions of adjoint equations3.28, and letp1t, p₂t,p₃tbe defined as inLemma 3.1. Supposeθ →Ht, Yt, θ,pt; qt, rt,·is concave. Let θπ,π be an optimal pair for Problem A, as given inLemma 3.2. Then,

θˇ:θπ 4.4

is optimal for Problem B.

Proof. ByTheorem 4.1for Problem B, ˇθsolves Problem B under partial informationEtif sup

θ∈MEHt, Yt, θ,pt, qt, rt, z| Et EHt, Yt,θ,ˇ pt, qt, rt, z| Et, 4.5 that is, if there existsCCtsuch that

E

∇θHt, Yt, θ,pt, qt, rt, z −CtMθ_θθˇ | Et 0, 4.6

EMθtˇ | Et 0. 4.7

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Letπ, θ π be as inLemma 3.2. Then, E

∇θHt, Yt, θ,πt, pt, qt, rt, z _θθπt | Et 0, 4.8

EMθ πt | Et 0. 4.9

Hence, byLemma 3.1,

0E

∇θ

Ht, Yt, θ,pt, qt, rt, z

−StπtK θt

αt 2σtθ02

R0

γt, zθ1zνdz

θθπt

| Et

E

∇θHt, Yt, θ,pt, qt, rt, z −2StπtK θtMθ_θ_θ_πt | Et .

4.10

Therefore, if we choose

Ct 2StπtKθt, 4.11

we see that4.6holds with ˇθθπ, as claimed.

5. Risk indifference pricing

Letθ^∗, π^∗ θ,ˇ π be as inTheorem 4.2with the corresponding state processY^∗ Y^θ^∗^,π^∗. Suppose thatY Y^θ^π,π is the state process corresponding to an optimal controlθπ, π.

Then, the value functionΦ^E_G, which is defined by3.17and3.18, becomes Φ^E_Gt, y

inf

π∈Π

sup

θ∈ΘJ^θ,πt, y

inf

π∈Π

sup

θ∈Θ

E^y

− _T

t

Λθu,Yudu −hKθT, ST K_θTgST−K_θTX^πT

inf

π∈Π

E^y

− _T

t

Λθ^∗u,Y^∗udu−hKθ^∗T, ST K_θ^∗TgST−K_θ^∗TX^πT

. 5.1

We have that, for allπ∈Π, E^y

K_θ^∗TX^πT E^y

K_θTX^πT kE^k,s,x_1/kQ

θˇ

X^πT kx, 5.2

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since1/kQθˇ is an equivalent martingale measure forEt-conditioned market. On the other hand, the first part of5.1does not depend on the parameterπ. Hence,5.1becomes

Φ^E_Gt, y E^y

− _T

t

Λθu,ˇ Yudu−hKθˇT, ST KθˇTgST

−kx sup

θ∈MJ₀^θt,y −kx Ψ^E_Gt,y −kx.

5.3

We have proved the following result for the relation between the value function for Problem A and the value function for Problem B in the partial information case that is the same as in Øksendal and Sulem7for the full information case.

Lemma 5.1. The relationship between the value function Ψ^E_Gt,y for Problem B and the value functionΦ^E_Gt, yfor Problem A is

Φ^E_Gt, y Ψ^E_Gt,y −kx. 5.4

We now apply Lemma 5.1to find the risk indiﬀerence price p p^seller_risk , given as a solution of the equation

Φ^E_Gt, k, s, xp Φ^E₀t, k, s, x. 5.5

ByLemma 5.1, this becomes

Ψ^E_Gt, k, s−kxp Ψ^E₀t, k, s−kx, 5.6

which has the solution

pp^seller_risk k⁻¹

Ψ^E_Gt, k, s−Ψ^E₀t, k, s

. 5.7

In particular, choosingk 1i.e., all measuresQ ∈ Lare probability measures, we get the following.

Theorem 5.2. Suppose that the conditions ofTheorem 4.2hold. Then, the risk indiﬀerence price for the seller of claimG,p^seller_risk G,E, is given by

p^seller_risk G,E sup

Q∈M{EQG−ζQ} −sup

Q∈M{−ζQ}. 5.8

References

1 M. R. Grasselli and T. R. Hurd, “Indiﬀerence pricing and hedging in stochastic volatility models,”

Tech. Rep., McMaster University, Hamilton, Canada, 2005.

2 S. D. Hodges and A. Neuberger, “Optimal replication of contingent claims under transaction costs,”

Review of Futures Markets, vol. 8, no. 2, pp. 222–239, 1989.

(15)

3 K. Takino, “Utility indiﬀerence pricing in an incomplete market model with incomplete information Discussion Papers in Economics and Business,” Tech. Rep., Osaka University, Graduate School of Economics and Osaka School of International Public PolicyOSIPP, Osaka, Japan, 2007.

4 M. Xu, “Risk measure pricing and hedging in incomplete markets,” Annals of Finance, vol. 2, no. 1, pp.

51–71, 2006.

5 P. Barrieu and N. El Karoui, “Optimal derivatives design under dynamic risk measures,” in Mathematics of Finance, vol. 351 of Contemporary Mathematics, pp. 13–25, American Mathematical Society, Providence, RI, USA, 2004.

6 S. Kl ¨oppel and M. Schweizer, “Dynamic indiﬀerence valuation via convex risk measures,”

Mathematical Finance, vol. 17, no. 4, pp. 599–627, 2007.

7 B. Øksendal and A. Sulem, “Risk indiﬀerence pricing in jump diﬀusion markets,” to appear in Mathematical Finance.

8 H. F ¨ollmer and A. Schied, “Convex measures of risk and trading constraints,” Finance and Stochastics, vol. 6, no. 4, pp. 429–447, 2002.

9 M. Frittelli and E. R. Gianin, “Putting order in risk measures,” Journal of Banking & Finance, vol. 26, no. 7, pp. 1473–1486, 2002.

10 H. F ¨ollmer and A. Schied, “Robust preferences and convex measures of risk,” in Advances in Finance and Stochastics: Essays in Honor of Dieter Sondermann, K. Sandmann and J. Schonbucher, Eds., pp. 39–56, Springer, Berlin, Germany, 2002.

11 B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diﬀusions, Universitext, Springer, Berlin, Germany, 2nd edition, 2007.

12 T. T. K. An and B. Øksendal, “A maximum principle for stochastic diﬀerential games with partial information,” to appear in Journal of Optimization Theory and Applications.

13 M. C. Ferris and O. L. Mangasarian, “Minimum principle suﬃciency,” Mathematical Programming, vol.

57, no. 1–3, pp. 1–14, 1992.

(16)

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