JAMSAJournal of Applied Mathematics and Stochastic Analysis1687-21771048-9533Hindawi Publishing Corporation82124310.1155/2008/821243821243Research ArticleA Maximum Principle Approach to Risk Indifference Pricing with Partial InformationAnTa Thi Kieu1ØksendalBernt1,2ProskeFrank1HuYaozhong1Centre of Mathematics for Applications (CMA)Department of MathematicsUniversity of Oslo P.O. Box 1053 Blindern0316 OsloNorwayuio.no2Department of Finance and Management ScienceNorwegian School of Economics and Business AdministrationHelleveien 305045 BergenNorwaynhh.no200820102008200810052008280920082008Copyright © 2008This 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.

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 price priskseller(G,) of a European-type claim G.

1. Introduction

Suppose the value of a portfolio (π(t),S0(t)) is given by Xx(π)(t)=x+π(t)S(t)+S0(t),where x is the initial capital, S(t) is a semimartingle price process of a risky asset, π(t) is the number of risky assets held at time t, and S0(t) is the amount invested in the risk-free asset at time t. Then, the cumulative cost at time t is given byP(t)=Xx(π)(t)0tπ(u)dS(u). If P(t)=p-constant for all t, then the portfolio strategies (π(t),S0(t)) is called self-financing. A contingent claim with expiration date T is a nonnegative T-measurable random variable G that represents the time T payoff from seller to buyer. Suppose that for a contingent claim G there exists a self-financing strategy such that Xx(π)(T)=G, that is,p+0Tπ(u)dS(u)=G.Then, p is the price of G in the complete market, that is,p=EQ[G],where Q is any martingale measure equivalent to P on the probability space (Ω,t,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  for the case of stochastic volatility model, Hodges and Neuberger  for the financial model with constraints, and Takino  for 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: Xu  propose risk measure pricing and hedging in incomplete markets; Barrieu and El Karoui  study a minimization problem for risk measures subject to dynamic hedging; Klöppel and Schweizer  study the indifference pricing of a payoff with 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  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. In Section 2, we will implement the option pricing method in an incomplete market. In Section 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 space (Ω,,{t}0tT,P) is given.

Definition 2.1.

A nonnegative random variable G on (Ω,t,P) is called a European contingent claim.

From now on, we consider a European-type option whose payoff at time t is some nonnegative random variable G=g(S(t)). In the rest of the paper, we will identify a contingent claim with its payoff function g.

Let 𝔽 be the space of all equivalence classes of real-valued random variables defined on Ω.

Definition 2.2 (see [<xref ref-type="bibr" rid="B6">8</xref>, <xref ref-type="bibr" rid="B7">9</xref>]).

A convex risk measureρ:𝔽{} is a mapping satisfying the following properties, for X,Y𝔽:

(convexity) ρ(λX+(1λ)Y)λρ(X)+(1λ)ρ(Y),λ(0,1);

(monotonicity) if XY, then ρ(X)ρ(Y);

If an investor sells a liability to pay out the amount g(S(T)) at time T and receives an initial payment p for such a contract, then the minimal risk involved for the seller isΦG(x+p)=infπPρ(Xx+p(π)(T)g(S(T))),where 𝒫 is the set of self-financing strategies such that Xx(π)(t)c, for some finite constant c and for 0tT.

If the investor has not issued a claim (and hence no initial payment is received), then the minimal risk for the investor isΦ0(x)=infπPρ(Xx(π)(T)).

Definition 2.3.

The seller's risk indifference price, p=priskseller, of the claim G is the solution p of the equation ΦG(x+p)=Φ0(x). Thus, priskseller is the initial payment p that makes an investor risk indifferent between selling the contract with liability payoff G and not selling the contract.

In view of the general representation formula for convex risk measures (see ), we will assume that the risk measure ρ, which we consider, is of the following type.Theorem 2.4 (representation theorem [<xref ref-type="bibr" rid="B6">8</xref>, <xref ref-type="bibr" rid="B7">9</xref>]).

A map ρ:𝔽 is a convex risk measure if and only if there exists a family of measures QP on T and a convex “penalty” function ζ:(,+) with infQζ(Q)=0 such thatρ(X)=supQ{EQ[X]ζ(Q)},XF.

By this representation, we see that choosing a risk measure ρ is equivalent to choosing the family of measures and the penalty function ζ.

Using the representation (2.5), we can write (2.2) and (2.3) as follows:ΦG(x+p)=infπP(supQ{EQ[Xx+p(π)(T)+g(S(t))]ζ(Q)}),Φ0(x)=infπP(supQ{EQ[Xx(π)(T)]ζ(Q)}),for a given penalty function ζ.

Thus, the problem of finding the risk indifference price p=priskseller 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  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:

(i) a bond with unit price S0(t)=1, t[0,T];

(ii) a stock with price dynamics, for t[0,T],dS(t)S(t)[α(t)dt+σ(t)dBt+0γ(t,z)N(dt,dz)],S(0)=s>0. Here Bt is a Brownian motion and N(dt,dz)=N(dt,dz)ν(dz)dt is a compensated Poisson random measure with Lévy measure ν. The processes α(t), σ(t), and γ(t,z) are t-predictable processes such that γ(t,z)>1, for a.s. t,z, andE[0T{|α(s)|+σ2(s)+0|log(1+γ(s,z))|2ν(dz)}ds]<a.s.,for all T0.

Let tt be a given subfiltration. Denote by π(t), t0, the fraction of wealth invested in S(t) based on the partial market information tt being available at time t. Thus, we impose on π(t) to be t-predictable. Then, the total wealth X(π)(t) with initial wealth x is given by the SDEdX(π)(t)=π(t)S(t)[α(t)dt+σ(t)dBt+0γ(t,z)N(dt,dz)],X(π)(0)=x>0.

In the sequel, we will call a portfolio π𝒫admissible if π is t-predictable, permits a strong solution of (3.3), and satisfies0T{|α(t)||π(t)|S(t)+σ2(t)π2(t)S2(t)+π2(t)S2(t)0γ2(t,z)ν(dz)}ds<,as well asπ(t)S(t)γ(t)>1(ω,t,z)-a.s.The class of admissible portfolios is denoted by Π.

Now, we define the measures Qθ parameterized by given t-predictable processes θ=(θ0(t),θ1(t,z)) such thatdQθ(ω)=Kθ(T)dP(ω)onT,wheredKθ(t)=Kθ(t)[θ0(t)dB(t)+0θ1(t,z)N(dt,dz)],t[0,T],Kθ(0)=k>0,We assume thatθ1(t,z)1fora.a.t,z,0T{θ02(s)+0(log(1+θ1(s,z)))2ν(dz)}ds<a.s.Then, by the Itô formula, the solution of (3.7) is given byKθ(t)=kexp[0tθ0(s)dB(s)120tθ02(s)ds+0t0ln(1θ1(s,z))N(dt,dz)+0t0{ln(1θ1(s,z))+θ1(s,z)}ν(dz)ds].We say that the control θ=(θ0,θ1) is admissible and write θΘ if θ is adapted to the subfiltration t and satisfies (3.8) andE[Kθ(T)]=Kθ(0)=k>0.

We setdY(t)=[dY1(t)dY2(t)dY3(t)]=[dKθ(t)dS(t)dX(π)(t)]=[0S(t)α(t)S(t)π(t)α(t)]dt+[Kθ(t)θ0(t)S(t)σ(t)S(t)π(t)σ(t)]dB(t)+0[Kθ(t)θ1(t,z)S(t)γ(t,z)S(t)π(t)γ(t)]N(dt,dz),Y(0)=y=(y1,y2,y3)=(k,s,x),dY(t)=[dY1(t)dY2(t)]=[dKθ(t)dS(t)],Y(0)=y=(y1,y2)=(k,s).

We now define two sets , of measures as follows:={Qθ;θΘ},={Qθ;θM},whereM={θΘ;E[Mθ(t,y)t]=0t,y},Mθ(t,y)=Mθ(t,k,s)=α(t)+σ(t)θ0(t)+0γ(t,z)θ1(t,z)ν(dz).

In particular, by the Girsanov theorem, all the measures Qθ with E[Kθ(T)]=1 are equivalent martingale measures for the t-conditioned market (S0(t),S1(t)), wheredS1(t)=S1(t)[E[α(t)t]dt+E[σ(t)t]dBt+0E[γ(t,z)t]N(dt,dz)]S1(0)=s>0(see, e.g., [11, Chapter 1]).

We assume that the penalty function ζ has the formζ(Qθ)=E[0T0λ(t,θ0(t,Y(t)),θ1(t,Y(t),z),Y(t),z)ν(dz)dt+h(Y(T))],for some convex functions λC1(2×0),hC1(), such thatE[0T0|λ(t,θ0(t,Y(t)),θ1(t,Y(t),z),Y(t),z)|ν(dz)dt+|h(Y(T))|]<,for all (θ,π)Θ×Π.

Using the Y(t)-notation, problem (2.6) can be written as follows:

Problem A.

Find ΦG(t,y) and (θ*,π*)Θ×Π such thatΦG(t,y):=infπΠ(supθΘJθ,π(t,y))=Jθ*,π*(t,y),whereJθ,π(t,y)=J(θ,π)=Ey[tTΛ(θ(u,Y(u)))duh(Y(T))+Kθ(T)g(S(T))Kθ(T)X(π)(T)],Λ(θ)=Λ(θ(t,y))=0λ(t,θ0(t,y),θ1(t,y,z),y,z)ν(dz).

We will relate Problem A to the following stochastic control problem:ΨG=supQ{EQ[G]ζ(Q)}.Using the Y(t)-notation, the problem gets the following form.

Problem B.

Find ΨG(t,y) and θˇ𝕄 such thatΨG(t,y):=supθMJ0θ(t,y)=J0θˇ(t,y),whereJ0θ(t,y)=Ey[tTΛ(θ(u,Y(u)))duh(Y(T))+Kθ(T)g(S(T))].

Define the Hamiltonian H:[0,T]××××Θ×Π××× for Problem A byH(t,k,s,x,θ,π,p,q,r(,z))=Λ(t,Y(t))+sαp2+sαπp3+kθ0q1+sσq2+sσπq3+0{kθ1r1(,z)+sγ(t,z)r2(,z)+sπγ(t,z)r3(,z)}ν(dz),and the Hamiltonian H:[0,T]×××Θ××× for Problem B byH(t,k,s,θ,p,q,r(,z))=Λ(t,Y(t))+sαp2+kθ0q1+sσq2+0{kθ1(t,z)r1(,z)+sγ(t,z)r2(,z)}ν(dz).Here is the set of functions r:[0,T]× such that the integrals in (3.23) and (3.24) converge. We assume that H and H are differentiable with respect to k, s, and x. The adjoint equations (corresponding to θ, π, and Y(t)) in the unknown adapted processes p(t), q(t), r(t,z) are the backward stochastic differential equations (BSDEs)dp1(t)=(Λk(t,Y(t))θ0(t)q1(t)0θ1(t,z)r1(t,z)ν(dz))dt+q1(t)dB(t)+0r1(t,z)N(dt,dz),p1(T)=hk(Y(T))+g(S(T))X(π)(T),dp2(t)=(Λs(t,Y(t))α(t)p2(t)σ(t)q2(t)0γ(t,z)r2(t,z)ν(dz))dt+q2(t)dB(t)+0r2(t,z)N(dt,dz),p2(T)=hs(Y(T))+Kθ(T)g(S(T)),dp3(t)=(α(t)p3(t)σ(t)q3(t)0γ(t,z)r3(t,z)ν(dz))dt+q3(t)dB(t)+0r3(t,z)N(dt,dz),p3(T)=Kθ(T).

Similarly, the adjoint equations (corresponding to θ and Y(t)) in the unknown processes p(t), q(t), r(t,z) are given bydp1(t)=(Λk(t,Y(t))θ0(t)q1(t)0θ1(t,z)r1(t,z)ν(dz))dt+q1(t)dB(t)+0r1(t,z)N(dt,dz),p1(T)=hk(Y(T))+g(S(T)),dp2(t)=(Λs(t,Y(t))α(t)p2(t)σ(t)q2(t)0γ(t,z)r2(t,z)ν(dz))dt+q2(t)dB(t)+0r2(t,z)N(dt,dz),p2(T)=hs(Y(T)Lemma 3.1.

Let θΘ and suppose that p(t)=(p1(t),p2(t)) is a solution of the corresponding adjoint equations (3.28). For all π, definep1(t)=p1(t)X(π)(t),p2(t)=p2(t),p3(t)=Kθ(t). If θ𝕄, then p(t)=(p1(t),p2(t),andp3(t)) is a solution of the adjoint equations (3.25), (3.26), and (3.27). Then, the following relation holds: H(t,Y(t),θ,π,p(t),q(t),r(t,z))=H(t,Y(t),θ,p(t),q(t),r(t,z))S(t)πKθ(t)(α(t)+2θ0(t)σ(t)+20θ1(t,z)γ(t,z)ν(dz)).

Proof.

Differentiating both sides of (3.29), we getdp1(t)=dp1(t)dX(π)(t)=(Λk(t,Y(t))θ0(t)q1(t)0θ1(t,z)r1(t,z)ν(dz)S(t)α(t)π(t))dt+(q1(t)S(t)σ(t)π(t))dB(t)+0(r1(t,z)S(t)π(t)γ(t,z))N(dt,dz).Comparing this with (3.25) by equating the dt, dB(t), N(dt,dz) coefficients, respectively, we getΛk(t,Y(t))θ0(t)q1(t)0θ1(t,z)r1(t,z)ν(dz)=Λk(t,Y(t))θ0(t)q1(t)0θ1(t,z)r1(t,z)ν(dz)S(t)α(t)π(t),q1(t)=q1(t)S(t)σ(t)π(t),r1(t,z)=r1(t,z)S(t)γ(t,z)π(t).Substituting (3.35) and (3.36) into (3.34), we getS(t)π(t)(α(t)+θ0(t)σ(t)+0θ1(t,z)γ(t,z)ν(dz))=0.Since θ𝕄, (3.37) is satisfied, and hence p1(t) is a solution of (3.25).

Proceeding as above with the processes p2(t) and p3(t), we getq2(t)=q2(t),r2(t)=r2,α(t)p3(t)σ(t)q3(t)0γ(t,z)r3(t,z)ν(dz)=0,q3(t)=Kθ(t)θ0(t),r3(t,z)=Kθ(t)θ1(t,z).With the values p3(t), q3(t), and r3(t,z) defined as above, relation (3.39) is satisfied if θ𝕄. Hence, p1(t), p2(t), and p3(t) are solutions of (3.29), (3.30), and (3.31), respectively.

Equations (3.23) and (3.24) give the following relation between H and H:H(t,y,θ,π,p,q,r(,z))=H(t,y,θ,p,q,r(,z))+sπ(αp3+σq3+0γ(t,z)r3(,z)ν(dz)).Hence,H(t,Y(t),θ,π,p(t),q(t),r(t,z))=H(t,Y(t),θ,p1(t),p2(t),q1(t),q2(t),r1(t,z),r2(t,z))S(t)π(t)(α(t)p3(t)+σq3(t)+0γ(t,z)r3(t,z)ν(dz))=H(t,Y(t),θ,p1(t),p2(t),q1(t),q2(t),r1(t,z),r2(t,z))S(t)σ(t)π(t)Kθ(t)θ0(t)0S(t)γ(t,z)π(t)Kθ(t)θ1(t,z)ν(dz)S(t)π(t)Kθ(t)(α(t)+σ(t)θ0(t)+0γ(t,z)θ1(t)ν(dz))=H(t,Y(t),θ,p(t),q(t),r(t,z))sπKθ(t)(α(t)+2σ(t)θ0(t)+20γ(t,z)θ1(t,z)ν(dz)).

Lemma 3.2.

Let p1(t), p2(t), and p3(t) be as in Lemma 3.1. Suppose that, for all π, the functionθE[H(t,Y(t),θ,π(t),p(t),q(t),r(t,z))t],θΘ,has a maximum point at θ^=θ^(π). Moreover, suppose that the functionπE[H(t,Y(t),θ^(π),π,p(t),q(t),r(t,z))t],π,has a minimum point at π^. Then,Mθ^(π^)=0.

Proof.

The first-order conditions for a maximum point θ^=θ^(π) of the function E[H(t,Y(t),θ,π(t),p(t),q(t),r(t,z))t] isE[θ(H(t,Y(t),θ,π(t),p(t),q(t),r(t,z)))θ=θ^(π)t]=0,where θ=(/θ0,/θ1) is the gradient operator. The first-order condition for a minimum point π^ of the function E[H(t,Y(t),θ^(π),π,p(t),q(t),r(t,z))t] isE[π(H(t,Y(t),θ^(π),π(t),p(t),q(t),r(t,z)))π=π^t]=0,that is,E[θ(H(t,Y(t),θ,π^,p(t),q(t),r(t,z)))θ=θ^(π^)(dθ^(π)dπ)π=π^+π(H(t,Y(t),θ,π,p(t),q(t),r(t,z)))π=π^,θ=θ^(π^)t]=0.Choose π=π^. Then, by (3.46) and (3.48), we haveE[π(H(t,Y(t),θ,π,p(t),q(t),r(t,z)))π=π^,θ=θ^(π^)t]=0,that is,E[S(t)α(t)p3(t)+S(t)σ(t)q3(t)+0S(t)γ(t,z)r3(t,z)ν(dz)t]=0.Substituting the values p3(t), q3(t), and r3(t,z) as in Lemma 3.1 into (3.50), we getE[S(t)Kθ(t){α(t)+σ(t)θ0(t)+0γ(t,z)θ1(t,z)ν(dz)}t]=0.This gives,Mθ^(π^)=0.

4. Maximum Principle for Stochastic Differential Games

Problem A is related to what is known as stochastic games studied in . Applying Theorem 2.1 in  to our setting we get the following jump-diffusion version of the maximum principle (of Ferris and Mangasarian type ).Theorem 4.1 (maximum principle for stochastic differential games [<xref ref-type="bibr" rid="B1">12</xref>]).

Let (θ^,π^)Θ×Π and suppose that the adjoint equations (3.25), (3.26), and (3.27) admit solutions (p^1(t),q^1(t),r^1(t,z)), (p^2(t),q^2(t),r^2(t,z)), and (p^3(t),q^3(t),r^3(t,z)), respectively. Moreover, suppose that, for all t[0,T], the following partial information maximum principle holds:supθΘE[H(t,Y(t),θ,π^(t),p^(t),q^(t),r^(t,z))t]=E[H(t,Y(t),θ^(t),π^(t),p^(t),q^(t),r^(t,z))t]=infπΠE[H(t,Y(t),θ^(t),π,p^(t),q^(t),r^(t,z))t].SupposeθJ(θ,π^)isconcave,πJ(θ^,π)is convex. Then (θ*,π*):=(θ^,π^) is an optimal control andΦG(x)=infπΠ(supθΘJ(θ,π))=supθΘ(infπΠJ(θ,π))=supθΘJ(θ,π^)=infπΠJ(θ^,π)=J(θ^,π^).

Theorem 4.2.

Let p1(t), p2(t) be, respectively, solutions of adjoint equations (3.28), and let p1(t), p2(t), p3(t) be defined as in Lemma 3.1. Suppose θH(t,Y(t),θ,p(t);q(t),r(t,)) is concave. Let (θ^(π^),π^) be an optimal pair for Problem A, as given in Lemma 3.2. Then,θˇ:=θ^(π^)is optimal for Problem B.

Proof.

By Theorem 4.1 for Problem B, θˇ solves Problem B under partial information t ifsupθME[H(t,Y(t),θ,p(t),q(t),r(t,z))t]=E[H(t,Y(t),θˇ,p(t),q(t),r(t,z))t],that is, if there exists C=C(t) such thatE[θ(H(t,Y(t),θ,p(t),q(t),r(t,z)))C(t)M(θ))θ=θˇt]=0,E[Mθˇ(t)t]=0.Let π^, θ^(π^) be as in Lemma 3.2. Then,E[θ(H(t,Y(t),θ,π^(t),p(t),q(t),r(t,z))θ=θ^(π^(t))t]=0,E[Mθ^(π^)(t)t]=0.Hence, by Lemma 3.1, 0=E[θ{H(t,Y(t),θ,p(t),q(t),r(t,z))S(t)π^(t)Kθ(t)(α(t)+2σ(t)θ0+20γ(t,z)θ1(z)ν(dz))}θ=θ^(π^(t))t]=E[θ(H(t,Y(t),θ,p(t),q(t),r(t,z))2S(t)π^(t)Kθ(t)Mθ)θ=θ^(π^(t))t].Therefore, if we chooseC(t)=2S(t)π^(t)Kθ(t),we see that (4.6) holds with θˇ=θ^(π^), as claimed.

5. Risk Indifference Pricing

Let (θ*,π*)=(θˇ,π^) be as in Theorem 4.2 with the corresponding state process Y*=Yθ*,π*. Suppose that Y=Yθ^(π^),π is the state process corresponding to an optimal control (θ^(π^),π). Then, the value function ΦG, which is defined by (3.17) and (3.18), becomesΦG(t,y)=infπΠ(supθΘJθ,π(t,y))=infπΠ(supθΘEy[tTΛ(θ(u,Y(u)))duh(Kθ(T),S(T))+Kθ(T)g(S(T))Kθ(T)X(π)(T)])=infπΠ(Ey[tTΛ(θ*(u,Y*(u)))duh(Kθ*(T),S(T))+Kθ*(T)g(S(T))Kθ*(T)X(π)(T)]).We have that, for all πΠ,Ey[Kθ*(T)X(π)(T)]=Ey[Kθ(T)X(π)(T)]=kE(1/k)Qθˇk,s,x[X(π)(T)]=kx,since (1/k)Qθˇ is an equivalent martingale measure for t-conditioned market. On the other hand, the first part of (5.1) does not depend on the parameter π. Hence, (5.1) becomesΦG(t,y)=Ey[tTΛ(θˇ(u,Y(u)))duh(Kθˇ(T),S(T))+Kθˇ(T)g(S(T))]kx=supθMJ0θ(t,y)kx=ΨG(t,y)kx.

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 Sulem  for the full information case.Lemma 5.1.

The relationship between the value function ΨG(t,y) for Problem B and the value function ΦG(t,y) for Problem A isΦG(t,y)=ΨG(t,y)kx.

We now apply Lemma 5.1 to find the risk indifference price p=priskseller, given as a solution of the equationΦG(t,k,s,x+p)=Φ0(t,k,s,x).By Lemma 5.1, this becomesΨG(t,k,s)k(x+p)=Ψ0(t,k,s)kx,which has the solutionp=priskseller=k1(ΨG(t,k,s)Ψ0(t,k,s)).In particular, choosing k=1 (i.e., all measures Q are probability measures), we get the following.Theorem 5.2.

Suppose that the conditions of Theorem 4.2 hold. Then, the risk indifference price for the seller of claim G, priskseller(G,), is given bypriskseller(G,)=supQ{EQ[G]ζ(Q)}supQ{ζ(Q)}.

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