IAENG International Journal of Applied Mathematics, 41:4, IJAM_41_4_04 ______

Pricing And Hedging of Asian Under Jumps

Wissem Boughamoura, Anand N. Pandey and Faouzi Trabelsi

Abstract—In this paper we study the pricing and hedging To price Asian options in such setting, [12] presents gener- problems of ”generalized” Asian options in a jump-diffusion alized Laplace transform for continuously sampled Asian op- model. We choose the minimal entropy martingale measure tion where underlying asset is driven by a Levy´ Process, [29] (MEMM) as equivalent martingale measure and we derive a partial-integro differential equation for their price. We discuss proposes a binomial tree method under a particular jump- the minimal variance hedging including the optimal hedging diffusion model. For the case of semimartingale models, [36] ratio and the optimal initial endowment. When the Asian payoff shows that the pricing function is satisfying a partial-integro is bounded, we show that for the exponential utility function, differential equation. For the precise numerical computation the utility indifference price goes to the Asian option price of the PIDE using difference schemes we refer interested evaluated under the MEMM as the Arrow-Pratt measure of absolute risk-aversion goes to zero. readers to explore with [11]. In [3] is derived the range of price for Asian option when underlying stock price is Index Terms—Geometric Levy´ process, Generalized Asian following geometric Brownian motion and jump (Poisson). In options, Minimal entropy martingale measure, Partial-integro differential equation, Utility indifference pricing, Exponential [30] is derived bounds for the price of a discretely monitored utility function. arithmetic Asian option when the underlying asset follows an arbitrary Levy´ process. The paper [37] develops a simple network approach to American valuation under I.INTRODUCTION Levy´ processes using the fast Fourier transform. [23] uses lattices to price fixed-strike European-style Asian options that Sian options have payoff depending on the average are discretely monitored. In [28] is studied a certain one- A price of the underlying asset during some part of their dimensional, degenerate parabolic partial differential equa- life. Therefore, their payoff must be expressed as a function tion with a boundary condition which arises in pricing of of the asset price history. But, in contrast to the European Asian options. It is proven that the generalized solution of the options, they have not a closed form solution for their price, problem is indeed a classical solution. [24] provides a semi even when the underlying follows a geometric Brownian explicit valuation formula for geometric Asian options, with model. fixed and floating strike under continuous monitoring, when In the setting of the continuous-time market models, many the underlying stock price process exhibits both stochastic approaches and numerical approximations are proposed to and jumps. price Asian options: For instance, [35] provides tight analytic To overcome the problem of replicating the option by bounds for the Asian option price, In [20] is computed the trading in the underlying asset in such models, the authors Laplace transform of the Asian Option price, [27] uses Monte of [15] where the first to introduce a mean-variance criterion Carlo simulations, [25] follows a method based on binomial for hedging contingent claims in incomplete markets. They trees. [13] introduces change of numeraire technique and [32] derive, under a martingale measure setting, a unique strategy uses it to reduce the PDE problem in two variables to obtain minimizing the variance of the total error of replication due a lower bound for the price. In a recent paper [10] is derived to hedging in a given claim. More precisely, they propose a closed-form solution for the price of an average strike as to approximate the target payoff H by optimally choosing well as an average price geometric Asian option, by making the initial capital c and a self-financing trading strategy use of the path integral formulation. (∆t)t∈[0,T ] in the risky asset S in order to minimize the But, in practice, continuous time models are unrealistic, quadratic hedging error: since the price process really presents discontinuities. In [ ∫ ]2 particular, Levy´ processes [6, 33, 2] became the most popular T Q and tractable for the market modeling and finance applica- J(c, ∆) = E c + ∆tdSt − H . (1) tions. 0 The choice of EMM Q is questionable, since the market in Manuscript received March 14, 2011; revised August 8, 2011. Laboratory of Mathematical and Numerical Modelling in Engineering presence of jumps may be incomplete. To choose an EMM Science. Higher Institute of Applied Mathematics and Computer Science many methods are proposed: Minimal martingale measure of Kairouan, Avenue Assad Ibn Fourat, 3100 Kairouan, Tunisia. Email: [16], Esscher martingale measure [21, 7] and Minimal [email protected]. Dept. of Mathematics and statistics, Indian Institute of Technology, entropy martingale measure introduced (MEMM) in [19]. Kanpur, India 208016. Email: [email protected]. The last one has the advantage that it conserves the Levy´ Laboratory of Mathematical and Numerical Modelling in Engineer- property of the process under the change of measure. In ing Science. Higher Institute of Computer Sciences and Mathematics of Monastir, Avenue de la Korniche, 5000 Monastir, Tunisia. E-mail: particular the MEMMs for geometric Levy´ processes have [email protected] and faouzi [email protected]. been discussed by [18, 19, 31].

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that the generalized Asian option price under any EMM Q In this paper, we study generalized Asian option pricing is written as: and hedging in a financial market where the underlying price Q − − ϕ(t, S ,Y ) = E [e r(T t)φ(S ,Y )/F ]. is modeled by geometric jump-diffusion Levy´ process. The t t T T t considered market is in fact incomplete. Deeply influenced III.ASIANOPTIONSINAGEOMETRICJUMP-DIFFUSION by [19], we choose the MEMM as equivalent martingale LEVYMARKETMODEL´ measure under which we show that the price process of a generalized Asian option is a solution of a two-dimensional In this section, we focus on the market where the underly- parabolic-hyperbolic partial-integro differential equation. ing price process is modeled by a geometric jump-diffusion The paper is structured as follows: In section (III) we Levy´ process. And we make an appropriate choice of EMM describe the geometric jump-diffusion market model and for such incomplete market. we provide some of their useful properties. Next, we in- troduce the concept of minimal entropy martingale measure A. Description of the market model under which we study the properties of the underlying asset Let (Ω, F, P) be a given complete probability space. We price. In section (IV), we derive a partial-integro differential consider a market which consists of a bond with price process rt equation for generalized Asian options and we make some given by Bt = e , where r denotes the (constant) interest vague study of the numerical computation. Section (V) is rate and a non-dividend paying risky asset (the stock or devoted to study minimal variance hedging including the index). The price process (St)0≤t≤T of the risky asset is optimal initial endowment and hedging strategy minimizing modeled by the following geometric jump-diffusion Levy´ the quadratic risk under the MEMM. In section (VI), we process: show that the MEMM generalized Asian price is the limit ≤ ≤ of the corresponding utility indifference price as the risk- St = S0 exp(rt + Lt), 0 t T, (4)

aversion parameter tends to zero. Finally, section (VII) gives where Lt is a Levy´ process given by: some concluding remarks. ∑Nt Lt = µt + σBt + Uj, (5) ENERALIZED SIANOPTIONS II.G A j=1 Let (St)0≤t≤T be the stock price process with jumps and here µ is a drift, σ is the underlying volatility supposed to be (Yt)0≤t≤T be the arithmetic average value of the underlying positive and (Nt)0≤t≤T is a Poisson process with intensity over the life time interval [0, t]: P ∫ λ > 0. The jump sizes (Uj)j≥1 are i.i.d. -integrable t 1 r.v.s with a distribution F and independent of (Nt)0≤t≤T . Yt = Sudu. (2) −rt We denote by Sˆt = e St = S0 exp(Lt) the discounted t 0 underlying price process. Consider an Asian option maturing at time T based on Y T We assume that the σ-fields generated respectively by with an FT -measurable payoff H: (Bt)0≤t≤T , (Nt)0≤t≤T and (Uj)j≥1 are independents and F ≤ ≥ ≥ H = φ(ST ,YT ), we take t := σ(Bs,Ns,Uj1{j≤Ns}; s t, j 1), t 0 as filtration. Let τ1, ··· , τj, ··· be the dates of jumps of where φ satisfies the following Lipschitz assumption: the stock price St; i.e. the jump times of (Nt)0≤t≤T and ′ ′ ∃ ∀ ∈ R4 V1, ··· ,Vj, ··· their proportions of jumps at these times, (A1) a > 0; (S, S ,Y,Y ) +, respectively; i.e. |φ(S, Y ) − φ(S′,Y ′)| ≤ a[|S − S′| + |Y − Y ′|]. − − − ≥ ∆Sτj = Sτj S = S Vj for all j 1, This kind of options will be termed as generalized Asian τj τj options. As examples we cite the payoff φ(ST ,YT ) = U where V = e j − 1. The r.v.’s (V ) ≥ are i.i.d. with a |S − Y | of a Asian option and the payoff j j j 1 T T common distribution G and takes its values in the interval φ(S ,Y ) = (ζ. (Y − K S − K ))+ , which is a fixed T T T 1 T 2 (−1, +∞). Recall that the expression (5) can be rewritten in strike Asian option, for K = 0, and a floating strike Asian 1 integral form as option for K2 = 0. The constant ζ = ±1 determines whether ∫ ∫ the option is call or put. Lt = µt + σBt + xJL(ds, dx), The price Vt of a generalized Asian option depends on t, [0,t] R St and on the path that the asset price followed up to time and L∫t is a Levy´ process with the characteristic triplet t. In particular, we can not invoke the Markov property to (µ + xν (dx), σ2, ν ) ν (dx) = λF (dx) |x|≤1 L L , where ∑L claim that Vt is a function of t and St. is the Levy´ measure, JL(dt, dx) = δ(τ ,∆L ) is the To overcome this difficulty, we increase the state variable St j τj by using the second process Y given by Equation (2) which j≥1 t random Poisson measure of L on [0, +∞[×R with intensity follow the stochastic differential equation: νL(dt, dx) := νL(dx)dt. St − Yt L dY = dt. (3) Hence has the following Levy-It´ oˆ decomposition: t t ( ∫ )

The process (St, Yt) constitutes then a two-dimensional Lt = µ + xνL(dx) t + σBt (6) |x|≤1 Markov process. Furthermore, the payoff H = φ(ST ,YT ) is ∫ ∫ ∫ ∫ a function of T and the final value (ST ,YT ) of this process. e + xJL(ds, dx) + xJL(ds, dx), This implies the existence of some function ϕ(t, S, Y ) such [0,t] |x|>1 [0,t] |x|≤1

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IAENG International Journal of Applied Mathematics, 41:4, IJAM_41_4_04 ______

e where JL(dt, dx) is the compensated measure of JL, i.e. B. Minimal entropy martingale measure and geometric e JL(dt,∫ dx) := (JL − νL)(dt, dx). jump-diffusion Levy´ processes | | EP | | ∞ Since R x νL(dx) = λ [ U1 ] < , it comes that the The fundamental asset pricing theorem shows that the E | | ∞ first moment of Lt (for all t) is finite ( [ Lt ] < ). choice of an arbitrage-free method is equivalent to the Consequently, the expression (6) resumes to this form: choice of an equivalent martingale measure Q ∼ P. To do ( ∫ ) ∫ ∫ t so, influenced by [19], we use minimal entropy martingale Lt = µ + xνL(dx) t+σBt + (JL −νL)(ds, dx). method for geometric Levy´ processes. R 0 R Denote by EMM(P) the set of all EMMs of S; i.e. the set of all probability measures Q on (Ω, FT ) such that the Set Xt := Lt + rt, then Xt is a Levy´ process with −rt 2 process {e St, 0 ≤ t ≤ T } is an (Ft, Q)-martingale and characteristic triplets (b, σ , νL), where Q ∼ P ∫ . Definition 1: The relative entropy of Q ∈ EMM(P), with b = µ + xνL(dx) + r. (7) respect to P, is defined by: |x|≤1  ∫ [ ]  dQ log dQ, if Q ≪ P; According to Proposition 8.22 of [8], there exists a Levy´ H(Q|P) := P ˆ  Ω d process X such that St is the Doleans-Dade exponent of ∞ otherwise. Xˆt: S = E(Xˆ). The process Xˆ is given by If an EMM P∗ satisfies the following condition: σ2 ∑ ∆Xu ∗ Xˆt = Xt + t + (1 + ∆Xu − e ) H(P |P) ≤ H(Q|P), for all Q ∈ EMM(P), (11) 2 ≤ ≤ ( 0 )u t P∗ 2 ∑ then, is called the minimal entropy martingale measure σ ∆L = L + r + t + (1 + ∆L − e u ) (MEMM) of (St)0≤t≤T . t 2 u 0≤u≤t A general feature of the MEMM is that its statistical prop- erties resemble the original process so the specification of − − − with ∆Xt = Xt Xt = Xt limu↑t Xu and satisfies the the prior is quite important. It has also the advantage that stochastic differential equation the Levy´ property is preserved for geometric Levy´ processes ∫ t and can be defined in terms of the Esscher transform and St = S0 + Su−dXˆu. Return process [14, 7]. The following theorem resumes the 0 main properties and results concerning the MEMM. Theorem 1: [19] Suppose there exists a constant β∗ such or in integral form that the following assumptions hold: ∫ ∫ t ∫(A2) x [ ] ˆ − ˜ ∗ x ∗ U1 Xt = b1t + σBt + (e 1)JL(du, dx) x β (e −1) P U1 β (e −1) | |≤ e e νL(dx) = λE 1U >1e e < ∞ ∫ ∫ 0 x 1 1 t x>1 x + (e − 1)JL(du, dx), ( ) ∫ 0 |x|>1 1 ∗ 2 x β∗(ex−1) (A3) µ+ + β σ + (e −1)e νL(dx) = 0 2 R where ∫ then, 1 2 x P∗ b1 = σ + b + (e − 1 − x)νL(dx). (8) 1) The Esscher transform define a probability measure 2 | |≤ x 1 on FT by:

∗ β∗Rˆ Note that, Xˆ is a Levy´ process under P with characteristics dP e t ∗ ˆ − t := = eβ Xt b0t, ∀t ∈ [0,T ], ˆ 2 P ∗ ˆ (b, σ , νˆ), and has the following Levy-It´ oˆ decomposition dP E β Rt Ft [e ] [19]: ˆ ∫ ∫ where (Rt)t∈[0,T ] is the simple return process given t ˆ ˆ by Equation (10), (Xt) is defined by (9), and Xˆt = bt + σBt + xJ ˆ (du, dx) (9) X ∗ ∫ 0 |x|>1 ∗ ∫ ∫ β ∗ 2 ∗ β (ex−1) t b0 = (1 + β )σ + β b + (e − 1 ˜ 2 R + xJXˆ (du, dx), ∗ 0 |x|≤1 − β x1{|x|≤1})νL(dx),

where νˆ(dy) = νoJ −1(dy) with J(x) = ex − 1, ˆb = with b defined by Equation (7). ∫ ∫ P∗ P x − − x − 2) The probability measure is actually in EMM( ). In b1 + {x<−1}(e 1)νL(dx) {log2

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3) The Stochastic process (Lt) remains a Levy´ process On the other hand, we have: P∗ ∗ 2 ∗ ∫ ∫ under with characteristic triplet (µ , σ , νL) and t +∞ ˆ x − ∗ Levy-It´ oˆ decomposition: Ss−(e 1)νL(ds, dx) ∫ ∫ −∞ t 0∫ ∫ ∗ ∗ ∗ t − ∗ x Lt = µ t + σBt + x(JL νL)(ds, dx) = λ Sˆ ds (ex − 1)eβ (e −1)F (dx) 0 |x|≤1 s ∫ ∫ 0 (∫ R ) t t + xJ (ds, dx), (12) ∗ L = λE [V1] Sˆsds . 0 |x|>1 ∫ (0 ∫ ) ∗ ∗ 2 β∗(ex−1) t where µ = µ + β σ + xe νL(dx), ∗ 2 |x|≤1 = (µ + β σ ) Sˆsds . ∗ ∗ x− ∗ ∗ β (e 1) − 0 νL(dx) = e νL(dx) and Bt = Bt σβ t is a ∗ P -Brownian motion. where E∗ denotes the expectation under P∗. The last equality dP∗ |F An easy calculus of the density ηT := dP T of the follow from assumption (A1). It follows from the Equation ∗ MEMM P leads to the explicit expression: (13) that Sˆt satisfies: [ ∫ 1 t [ ] ∗ ∗ 2 ∗ ∗ 2 ηT = exp β σBT − (β σ) T Sˆ = S + Sˆ σdB + (µ + β σ )dt 2 t 0 s s ∫ ∫ 0 ∗ x ∑Nt + β (e − 1)JL(ds × dx) ˆ − [0,T ] R + VjSτ . ∫ ] j ( ) j=1 β∗(ex−1) −T e − 1 νL(dx) . R IV. PRICING AND PARTIAL-INTEGRO DIFFERENTIAL Remark 1: Assumption (A2) ensures that the first expo- EQUATION ∗ nential moment of Lt under P is finite, i.e. (Sˆt)0≤t≤T is ∗ U1 Recall that H = φ(ST ,YT ) is the payoff of a general- P -integrable. Consequently, the r.v. V1 = e + 1 is also P∗-integrable. ized Asian option in the exponential jump-diffusion market The decomposition (12) of L can be rewritten as: model, where φ is Borel and fulfills assumption (A1). Using   the stationary and the independence of the increments of L, N −r(T −t) ∗ ∑t the value Vt = e E [H/Ft] of the option at time t is ∗ ∗  −  Lt = µ t + σBt + Uj tλκ . a function of t, St and Yt: j=1 Vt = ϕ(t, St,Yt), P β∗(eU1 −1) where κ = E [U1e 1{|U |≤1}]. The expression of St 1 2 is finally given by: where ϕ is the following measurable function on [0,T ]×R : −r(T −t) ∗ ∏Nt ϕ(t, S, Y ) = e E [H | S = S, Y = Y ] ∗ [ ( t t St = S0 exp(µ0t + σBt ) (1 + Vj), t ∗ −r(T −t) LT −t+r(T −t) j=1 = E e φ Se , Y )] T where ∫ − ∫ S T t Lu+ru ∗ + e du . (14) ∗ 2 β (ex−1) µ0 = µ + β σ + xe νL(dx) − λκ + r T 0 |x|≤1 = µ + β∗σ2 + r. We have the following useful lemma Lemma 1: On each compact set K0 = [0,T ] × [0,S0] × Since L is a Levy´ process with characteristic triplet [0,Y0], Y0,S0 > 0, the function ϕ is Lipschitz; i.e. there exist ∗ 2 ∗ P∗ ′ ′ ′ (µ , σ , νL) under , it comes according to Proposition a constant a0 > 0 such that for all (t, S, Y ), (t ,S ,Y ) ∈ K0 8.20 of [8] and taking account of assumption (A3), that L we have: is a P∗-square integrable martingale and has the following | − ′ ′ ′ | ≤ | − ′| | − ′| | − ′| expression: ϕ(t, S, Y ) ϕ(t ,S ,Y ) a0[ t t + S S + Y Y ]. ∫ t ˆ ˆ ∗ Proof: See Appendix (A). St = S0 + σ Ss−dBs (13) ∫ ∫ 0 t +∞ ˆ x − − ∗ A. Partial-integro differential equation + Ss−(e 1)(JL νL)(ds, dx). 0 −∞ Proposition 1: Under the assumption (A1), the function The integral of the predictable random function θ(ω, s, x) = ϕ, given by (14) is continuous on [0,T ]×R+ ×R+. Further- ∗ ∗ ˆ − x − 1,2,1 ×R × R Ss (ω)(e 1), with respect to the Poisson random measure more, if ϕ is in the class C (]0,T [ + +), then it JL, is given by: fulfills the following partial-integro differential equation on ∫ ∫ ×R∗ × R∗ t +∞ ∑ [0,T [ + +: θ(s, x)JL(ds, dx) = θ(s, ∆Ls) 0 −∞ s∈[0,t] Lϕ(t, S, Y ) = 0 ∆Ls=0̸ ∑Nt ∑Nt ∑Nt with the following terminal condition Uj = θ(τj,Uj) = (e − 1)Sˆ − = VjSˆ − . τj τj j=1 j=1 j=1 ∀S > 0, ∀Y > 0, ϕ(T, S, Y ) = φ(S, Y ),

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IAENG International Journal of Applied Mathematics, 41:4, IJAM_41_4_04 ______

where (a) No obvious Dirichlet type condition can be imposed. ∂ϕ ( ) ∂ϕ (b) On any finite domain [0,Smax], the integral term appears Lϕ = + r + µ + β∗σ2 S (15) ∂t ∂S to gather information from the outside domain. According to − 2 2 [11], Smax will be taken large so that the solution can be well S Y ∂ϕ σ 2 ∂ ϕ − + + S (r + λ)ϕ approximated by a linear function of S in the region [Smax − ∫t ∂Y 2 ∂S2 +∞ δ, Smax]. δ should be chosen carefully, so that integral term β∗v + λ e ϕ(t, S(1 + v),Y )G(dv) in the PIDE in [0,Smax − δ] has sufficient data, for accurate −1 computation. In the region [S − δ, S ] × [0,Y ] L is ∂ϕ ( ) ∂ϕ S − Y ∂ϕ max max max = + r + µ + β∗σ2 S + linear in S so the PIDE is reduced to the PDE. ∂t ∂S t ∂Y We refer interested readers to [11] for the computational de- σ2 ∂2ϕ + S2 − (r + λ)ϕ tail in which Semi-Lagrangian method coupled with Implicit- 2∫ ∂S2 discretization is proposed. For the Integral term evaluation +∞ ∗ u + λ eβ (e −1)ϕ(t, Seu,Y )F (du). FFT is used and correction term δ has alleviated the wrap −∞ around effect.

with G the distribution function of the r.v. V1 under P. Proof: See Appendix (B). V. MINIMAL VARIANCE HEDGING Remark( 2:) In the above PIDE it is important to note the Consider an economic agent who has sold at t = 0 the St−Yt term t . This plays an important role in the Upwinding contingent claim with terminal payoff H = φ(ST ,YT ) for technique used for numerical computation. The appearance∫ the price c and decides to hedge the associated risk by trading 1 t of this term is precisely due to assumption Yt = t 0 Sudu. in the risky asset S. Note that under the model assumptions, ∗ ∫It is interesting to note that, for example, [32] defined Yt = the payoff H is (FT , P )-square integrable. t 0 Sudu in order to compute the pricing PDE of an Asian The predictable σ-algebra is the σ-algebra generated on option when stock price follows geometric Brownian motion. [0,T ] × Ω by all adapted and left-continuous processes. Remark 3: Note that the integro differential operator (15) A stochastic process X : [0,T ] × Ω −→ R which is is not well defined at t = 0. It is easy to check that we can measurable with respect to the predictable σ-algebra is called take the following transformation predictable process. Any left-continuous process is, there- ∫ t fore, predictable (by definition). All predictable processes are At = tYt = Sudu generated from left-continuous processes. However, there are 0 predictable processes which are not left-continuous. to remove the singularity. The corresponding integro differ- 1) The admissible strategy: A hedging strategy is a R2- L K { 0 ≤ ≤ } ential operator 1 is given by: valued adapted process = (∆t , ∆t), 0 t T such 0 ( ) that ∆t (resp. ∆t ) represents the number of shares held by ∂ϕ ∗ 2 ∂ϕ ∂ϕ L1ϕ = + r + µ + β σ S + S the investor in the risky asset (resp. the riskless asst). The ∂t ∂S ∂A K 0 0 2 2 value at a time t of any strategy is given by Vt = ∆t St + σ 2 ∂ ϕ + S − (r + λ)ϕ ∆tSt. A strategy K is said to be self-financing if 2∫ ∂S2 +∞ 0 0 ∗ dV = ∆ dS + ∆ dSˆ . + λ eβ vϕ(t, S(1 + v),A)G(dv) t t t t t −1 Taking account of jumps, the components (∆0) and (∆ ) ∂ϕ ( ) ∂ϕ ∂ϕ t t = + r + µ + β∗σ2 S + S will be taken to be left-continuous with limits from the right ∂t ∂S ∂A (caglad). For any self-financing strategy K with initial value σ2 ∂2ϕ + S2 − (r + λ)ϕ V0, we have: 2 ∫ 2∫ ∂S t +∞ ∗ u ˆ K ˆ + λ eβ (e −1)ϕ(t, Seu,A)F (du). Vt( ) = V0 + ∆udSu. (16) −∞ 0 In order to make Equation (16) meaningful, we shall restrict B. Numerical computation of the PIDE ourselves to the following class Θ of admissible hedging strategies, composed by all caglad and predictable processes 0 ≤ S < ∞ Despite the original domain is , and ∆ : Ω × [0,T ] −→ R such that: 0 ≤ Y < ∞, for computational sake we need to truncate   (∫ )2 the domain with [0,Smax] × [0,Ymax], with Smax = Ymax. T ∗  ˆ  In order to make the integral defined, it is necessary to E ∆udSu < ∞. (17) approximate the solution outside the computational domain 0 S > Smax. Abide by [11], no boundary condition is required Remark 4: : Thanks to the isometry formula given by at S = 0,Y = 0,Y = Ymax, which is because, either PIDE Proposition 8.8 of Cont and Tankov [8], the condition (17) becomes ODE or the resulting degenerate parabolic PIDE is equivalent to: has a normal hyperbolic term with outgoing characteristics. [∫ ] T [11] ensured with the development of consistent, stable E∗ 2 ˆ2 ∞ ∆t St dt < . (18) and monotone discrete scheme that no data outside the 0 computational domain at S = 0 is required. For the boundary condition at S = ∞ two measure issues It ensures that the discounted value Vˆt of any admissible are of concern. strategy K is a P∗-square integrable martingale.

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The value at a time t of any admissible hedging strategy is taken to be the set of predictable processes such that the given by: corresponding wealth processes are uniformly bounded from ∫ t [ ] below. ˆ K ˆ ∗ ∗ 2 Vt( ) = V0 + ∆uSu σdBs + (µ + β σ )dt In a recent paper, [1] studies the problem of utility 0 indifference pricing in a constrained financial market, using ∑Nt a utility function defined over the positive real line. He ˆ − + ∆τj VjSτ (19) j provides a dynamic programming equation associated with j=1 the risk measure, and characterize the last as a viscosity 2) The optimal hedging ratio: Under market incomplete- solution of this equation. ness, the contingent claim H cannot be perfectly replicated Let Uα(x), α > 0 be a CARA exponential utility function by a self-financing strategy. Therefore, our objective is defined by naturally to hedge the claim H by the selection of a self- −αx financing strategy and an initial endowment that minimize Uα(x) = 1 − e the variance of the error of replication over the set of all admissible strategies and all initial endowments. The residual and c be a real constant, Γ be a suitable set of trading ∈ hedging error associated to any admissible strategy ∆ ∈ Θ strategies. The gain process of every strategy θ Γ is given is given by: by ∫ ∫ t T G(θ)t = θudSu, −rT ϵ(c, ∆) = c − Hˆ + ∆udSˆu = e (VT (K) − H), (20) 0 0 Note that exponential utility implies constant absolute risk- −rT where Hˆ = e H. Following [34], we consider the aversion, with coefficient of absolute risk-aversion: following optimization problem to price and hedge the claim −U ′′(x) H under market incompleteness: = α.   U(x) ( ∫ )2 T ∗   (P) inf E c − Hˆ + ∆udSˆu Though isoelastic utility, exhibiting constant relative risk- ∈R× (c,∆) Θ 0 aversion, is considered more plausible, exponential utility is particularly convenient for many calculations. E∗ 2 = inf(c,∆)∈R×Θ [ϵ(c, ∆) ] We denote by M a convex subset of probability measures where the expectation is taken to be under P∗. on (Ω, FT ) which are absolutely continuous with respect to Proposition 2: Let H = φ(ST ,YT ) be the payoff of a P. generalized Asian option as in the previous proposition. Let us take a bounded generalized Asian payoff H = Suppose that the function ϕ, given by Equation (14), is in φ(ST ,YT ) and consider the following assumption (A4): C1,2,1([0,T [×R × R ) and that assumptions (A ) and + + 2 1 P∗ M (A4) The MEMM belongs to i.e. (A3) are satisfied. The optimal hedging ratio of the risky asset and initial endowment that allow the minimization of the quadratic risk over the set Θ are given by: inf H(Q|P) = H(P∗|P) < ∞ (23) Q∈M c¯ = e−rT E∗[H], (21) ¯ ¯ (A2) (Duality relation) [9], [4] : Duality relation holds. ∆t = ∆(t, St− ,Yt), 4 with EP − ¯ ∫ 1 Jα(c, H) = sup [Uα(c + G(θ)T H)] ∆(t, S, Y ) = ∗ (22) ∈ 2 β (eu−1) u − 2 [θ Γ ] σ + R e (e 1) νL(du) ( ) [ ∫ Q ∂ϕ 1 ∗ u = 1 − exp − inf H(Q|P) + αc − E [αH] × σ2 (u, S, Y ) + eβ (e −1)(eu − 1) Q∈M ∂S S R Jα(c, H) × [ϕ(t, Seu,Y ) − ϕ(t, S, Y )]ν (du)] [ ( )] L 1 = 1 − e−αc exp α sup EQ[H] − H(Q|P) Proof: See Appendix (C). Q∈M α For the duality relation to hold [9] and [26] have considered VI.THE MEMM AND UTILITY INDIFFERENCE PRICING locally bounded semimartingales. But the geometric jump- The indifference price concept was first introduced by [22] diffusion Levy´ process Sˆt considered in this article does not in one dimensional Black and Scholes price model, where necessarily possess this property. [4] has shown that for un- trading in the underlying involves a transaction cost. This bounded price process infimum of equation (23) may not be concept was developed in order to define a price that could attained by an equivalent martingale measure. But [19] have not be based any more on exact replication, since transaction established existence of the MEMM for the geometric Levy´ costs where admitted, but was founded on agent preferences. process without using the assumption. So it would be natural [17] was the first to use an alternative approach to to expect that Duality theorem hold. For details see [18]. compute indifference price, applying the general theory of Now we introduce utility indifference price pα(c, H) [9]. convex analysis and duality results proven by [4]. He studied The value pα(c, H) which satisfies the following equation : an incomplete market model driven by a d-dimensional semimartingale, where the set of admissible strategies was Jα(c + pα(c, H),H) = Jα(c, 0)

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is called utility indifference price. Solution of above equation not robust with stopping times. Thus, an obvious extension is written as ( ) of this theory to the case of Asian-American style options is Q 1 not feasible to us. pα(c, H) = sup E [H] − H(Q|P) Q∈M α We showed that the Asian option’s price is subject to a 1 time-dependent partial-integro differential equation. To solve + inf H(Q|P) this PIDE we need to customize boundary condition viewing α Q∈M the option constraint. The obtained PIDE is not very obvious Remark that pα(c, H) does not depend on c and will be to solve numerically and simulation techniques are also denoted by pα(H); i.e. not very apparent. In subsection (IV-B) we tried to make some vague study of the numerical computation. Precise pα(H) = pα(0,H) = pα(c, H). study of the numerical computation of the PIDE regarding QH M Let α be a probability measure in such that consistency, stability and comparison with empirical data,

QH 1 H this is what we intend to do in future works. E α [H] − H(Q |P) ( α α ) We also showed that the utility indifference price for 1 bounded Asian option payoffs, converges to the risk-neutral > sup EQ[H] − H(Q|P) − α. Q∈M α valuation under the MEMM, as the risk-aversion goes to zero. For the case of unbounded generalized Asian options, Theorem 2: MEMM for bounded generalized Asian the question remains open. options Consider a generalized Asian options with bounded payoff H = e−rT φ(S ,Y ), (for example a fixed strike Asian T T APPENDIX A with payoff H = e−rT max (K − Y , 0)). Then T PROOFOF LEMMA 1 following is true. (1) lim H(QH |P) = H (P∗|P). ↓ α Proof of lemma 1: Let S0 > 0, Y0 > 0 and K0 = α 0 ′ ′ ′ H ∗ [0,T ] × [0,S0] × [0,Y0]. For (t, S, Y ), (t ,S ,Y ) ∈ K0, we (2) lim ∥Q − P ∥var = 0 where ∥.∥var denotes total ↓ α α 0 have taking into account assumption (A1): variation norm. ≥ EP∗ (3) pα(H) [H] for any α > 0. | − ′ ′ ′ | ≤ −r(T −t) × ϕ(t,[ S,( Y ) ϕ(t ,S ,Y ) e ) (4) If 0 < α < β then pα(H) ≤ pβ(H). ∫ ∗ T −t P ∗ S t S (5) lim pα(H) = E (H). E α↓0 φ ST −t, Y + Sudu S0 T TS0 The proof of the theorem will exactly follow with the same ( 0 ) ] ∫ lines of [19] for bounded contingent claim H. [5] has shown S′ t S′ T −t − ′ φ ST −t, Y + Sudu that the utility indifference price of any bounded claim lies S0 T TS0 0 within the interval of possible arbitrage-free valuations, i.e. −r(T −t) −r(T −t′) No-arbitrage consistency: For H ∈ L∞(P): + e − e [ ( ∫ ) ] Q ′ ′ T −t inf EQ[H] ≤ pα(H) ≤ sup E [H]. ∗ S t ′ S Q∈ P ×E φ S − , Y + S du EMM( ) Q∈EMM(P) T t u S0 T TS0 0 It is also clear that the utility-indifference price tends to ≤ ξ(T )[|S′ − S| + |Y ′ − Y |] + rerT |t′ − t| (|φ(0, 0)| the super-replication price, which is the supremum of the | | | | expectations of H over all equivalent martingale measures, + ξ(T )( S0 + Y0 )) ′ ′ ′ as the risk-aversion tends to infinity, price converges to the ≤ a0[|S − S| + |Y − Y | + |t − t|], super-hedging price, that is, worst price under the viability ( ) rT 1 rT − assumption. On the other hand, as the absolute risk-aversion where{ ξ(T ) := a e + r (e 1) + 1 and} a0 = rT decreases to zero, the utility indifference price tends to a max ξ(T ), re (|φ(0, 0)| + ξ(T )(|S0| + |Y0|)) . risk-neutral valuation under the minimal entropy martingale measure P∗ which is the lower bound of the interval of arbitrage-free price i.e. infimum of expectation of H over all APPENDIX B equivalent martingale measures. Above theorem for Asian PROOFOF PROPOSITION 1 put option and arguments portrays that MEMM P∗ is the ˆ −rt −rt ˆ appropriate choice for pricing generalized Asian options. Let Vt = e Vt = e ϕ(t, St,Yt) := ϕ(t, St,Yt) be the discounted value of the option at time t. The continuity of the function ϕ derives from Lemma (1). We have VII.CONCLUDING REMARKS ( ) rt ˆ ˆ ∗ In this paper, we studied generalized Asian option pricing dSt = e dSt + rStdt = St− (σdBt + ςdt) , and hedging problems in a financial market with jumps in the S − Y dY = t t dt, underlying under the minimal entropy martingale measure. t t Such measure is better than the method suggested by [16] in − E∗ ∗ 2 the presence of jumps: Their choice for semimartingale price with ς = r λ [V1] = r ∫+ µ + β σ , taking account of 1 t process in the market measure, would end up with signed (A3), and Y0 = limt→0 t 0 Sudu = S0, a.s., Then by measure. But, it is well-known that this kind of measures are application of Ito’sˆ lemma on each time interval [τj−1, τj] to

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ˆ ϕ(t, St, Yt), t < T , we get after addition Ct, the process k vanishes almost surely and we consequently ∫ t obtain: Vˆ = ϕ(0,S ,Y ) − r e−ruϕ(u, S ,Y )du t 0 0 u u Vˆ − C = ϕ(0,S ,Y ) (24) ∫ 0 t t ∫ 0 0 t t −ru ∂ϕ −rt ∂ϕ ∗ + e (u, S ,Y )du + σ e (u, Su,Yu)SudB . ∂t u u ∂S u ∫0 0 t −ru ∂ϕ ∗ + e (u, Su,Yu)Su− (σdBu + ςdu) APPENDIX C 0 ∂S ∫ PROOFOF PROPOSITION 2 t − −ru ∂ϕ Su Yu + e (u, Su,Yu) du Since[ for] all admissible strategy ∆ ∈ Θ, ∂Y u ∫ 0∫ E∗ T ˆ t 2 0 ∆udSu = 0, one has : 1 −ru ∂ ϕ 2 2 + e (u, Su,Yu) σ S − du + Σt, 2 ∂S2 u E∗ 2 − E∗ ˆ 2 0 [ϵ(c, ∆) ] = (c  [H])  ( ∫ )2 where T E∗  E∗ ˆ − ˆ ˆ  ∑Nt [ ] + [H] H + ∆udSu . ˆ − ˆ − 0 Σt = ϕ(τj,Sτj ,Yτj ) ϕ(τj,S ,Yτj ) . τj j=1 The last equation shows that the optimal initial capital for the ∗ U +X − ∆X +X − ˆ j τ τj τ problem (P) is given by c¯ = E [H]. After replacing c by c¯ − j j Since Sτj = (1 + Vj)Sτ = S0e = S0e j in the expression of the quadratic criterion to be minimized, with Xt = Lt + rt, we obtain: we obtain:   ( ∫ )2 Σt = T [ ] ∗ 2 ∗ ∗ ∑Nt E E  E ˆ − ˆ ˆ  ∆X +X − X − [ϵ(¯c, ∆) ] = [H] H + ∆udSu . τj τ τ ˆ j − ˆ j 0 ϕ(τj,S0e ,Yτj ) ϕ(τj,S0e ,Yτj ) j=1 ∫ ∫ −rt −rt t [ ] Let Vˆt = e Vt = e ϕ(t, St,Yt) be the discounted value ˆ u+Xs− ˆ Xs− = ϕ(s, S0e ,Ys) − ϕ(s, S0e ,Ys) of the option at time t. Equation (24) is written as: 0 R ∫ ×J (ds, du) t ∂ϕ ∫ L∫ ˆ ˆ ∗ t [ ] ϕ(t, St,Yt) = ϕ(0,S0,Y0) + σ (u, Su,Yu)SudBu u 0 ∂S ˆ − − ˆ = ϕ(s, Ss e ,Ys) ϕ(s, Ss−,Ys) + C , 0 R t × JL(ds, du) = Ct + Dt, where ∫ ∫ t [ ] where u ˆ − − ˆ ∫ ∫ [ ] Ct = ϕ(s, Ss e ,Ys) ϕ(s, Ss−,Ys) t R ˆ u ˆ 0 Ct = ϕ(s, S − e ,Ys) − ϕ(s, Ss−,Ys) ∗ s × (JL − ν )(ds, du). 0 R L ∗ × (JL − ν )(ds, du) ∫ ∫ [L ] We have t P∗ ˆ u ˆ Lemma 2: The process Ct is a -square integrable mar- Dt = ϕ(s, Ss− e ,Ys) − ϕ(s, Ss−,Ys) 0 R tingale. × ν∗ (du)ds Proof: See Appendix (D). L∫ ∫ t [ ∗ Then it comes from Proposition 8.8 of [8], that the β (eu−1) ˆ u ∗ = λ e ϕ(s, Ss− e ,Ys) compensated Poisson integral Ct is a P -square integrable R 0 ] martingale. ˆ − ϕ(s, Ss−,Ys) F (du)ds Now, from Equation (19) we obtain: ∫ ∫ t +∞ [ ∗ ˆ ˆ ˆ ˆ β v ˆ H − VT (K) = ϕ(T,ST ,AT ) − VT (K) = αT + γT , = λ e ϕ(s, Ss− (1 + v),Ys) − 0 1 ] where ˆ ∫ ( ) − ϕ(s, Ss−,Ys) G(dv)ds, t ∂ϕ − ˆ ∗ αt = (u, Su,Yu) ∆u σSudBu Hence, 0 ∂S ∫ ∫ t ∑Nt t − ∂ϕ ∗ ∗ ˆ ru − ˆ − E ˆ Vt = ϕ(0,S0,Y0) + σ e (u, Su,Yu)SudBu γt = Ct ∆τj VjSτ + λ [V1] du∆uSu. ∂S j ∫ 0 j=1 0 t ∫ −ru ∑Nt t + e kudu + Ct ∗ 2 − ˆ − − ˆ 0 = Ct ∆τj VjS (µ + β σ ) du∆uSu. τj j=1 0 where k. is the process defined by: since λE∗[V ] = −(µ + β∗σ2) by assumption (A ). We can k = Lϕ(u, S ,Y ), 1 3 u u u easily verify that: L ∫ ∫ being the parabolic-hyperbolic integro differential operator ∑Nt t u defined by Equation (15). ˆ − ˆ − − ∆τj VjSτ = ∆sSs (e 1)JL(ds, du). ˆ j R Since the process Vt is a martingale, as well as the process j=1 0

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∗ ∗ 2 It follows that: Since λE [V1] = −(µ + β σ ) by assumption (A3) and ∫ ∫ [ ∗ 2 t µ0 = µ + β σ + r it comes that u γ = ϕˆ(s, S − e ,Y ) − ϕˆ(s, S −,Y ) ( ) t s s s s E∗ 2 2 2 E∗ 2 0 R ] [St ] = S0 exp 2(σ + r)t + λt [V1 ] . (26) u ∗ − ˆ − − − ∆sSs (e 1) (JL νL)(ds, du). Denote: Since the processes α and γ are two P∗-zero-mean square u t t ρ(t, u) := ϕ(t, St− e ,Yt) − ϕ(t, St−,Yt). integrable martingales, it comes that αtγt is a martingale null at time t = 0. Hence, according to the isometry formula Taking account of assumptions (A1) and (A2), we have for ′ 3 given by Proposition 8.8 of [8] we get: all (t, S, S ,Y ) ∈ [0,T ] × R : [( ) ] ′ 2 |ϕ(t, S, Y ) − ϕ(t, S ,Y )| E∗ Hˆ − Vˆ (K) = E∗[α2 ] + E∗[γ2 ] [ ∫ ] T T T T −t [ ] a ′ ∗ 1 ∫ ( ) ≤ |S − S |E S − + S du T 2 S T t T u E∗ ∂ϕ − 2 ˆ2 E∗ 2 0 0 = (u, Su,Yu) ∆u σ Sudu + [γT ]. ( ∫ ) 0 ∂S T −t a ′ ∗ 1 ∗ ≤ |S − S | E [ST −t] + E [Su]du We have S0 T 0 [∫ ∫ T ≤ | ′ − | ∗ 2 ∗ u b S S , E E ˆ − [γT ] = [ϕ(t, St e ,Yt) [ ] 0 R rT 1 rT − ∞ ] where b = a e + rT (e 1 < . Hence u 2 ∗ [ ] − ˆ − ˆ − − ∫ ∫ ϕ(t, St−,Yt) ∆tSt (e 1)] νL(du)dt . T E∗ 2 ∗ ρ (t, u)νL(du)dt The quadratic risk is finally written as: 0 R [∫ ( ) (∫ ) ∫ T 2 T ∗ u K E∗ ∂ϕ − 2 ˆ2 ≤ 2E∗ 2 u − 2 β (e −1) RT ( ) = (t, St,Yt) ∆t σ St dt b St dt (e 1) e ν(du) 0 ∂S 0 R ∫ ∫ (∫ ) T T u ∗ ∗ ˆ − − ˆ 2 E 2 E 2 + [ϕ(t, St e ,Yt) ϕ(t, St−,Yt) = b [St ]dt [V1 ] R 0 ] 0 ( ) u 2 ∗ ∗ ∗ − ˆ − − ≤ 2 2 2 E 2 E 2 ∞ ∆tSt (e 1)] νL(du)dt . b TS0 exp 2(σ + r)T + λT [V1 ] [V1 ] < , The optimal strategy K¯ = (∆¯0, ∆)¯ is obtained by differenti- where the last inequality is obtained thanks to (26). ating the last quadratic expression, and should satisfy P∗-a.s. ( ) ACKNOWLEDGMENT ∂ϕ − ¯ 2 ˆ2 (t, St,Yt) ∆t σ St The author gratefully thanks the anonymous reviewers for ∂S∫ their valuable comments, which have led to the improvement ˆ u ˆ u + St− (e − 1)[ϕ(t, St− e ,Yt) of this paper. R u ∗ −ϕˆ(t, S −,Y ) − ∆¯ Sˆ − (e − 1)]ν (du) = 0, t t t t L REFERENCES which is rewritten again as: [1] T. Adachi, “On dynamic programming equations for ( ) ∂ϕ utility indifference pricing under delta constraints,” Jour- (t, S ,Y ) − ∆¯ σ2S 2 ∂S t t t t nal of Mathematical Analysis and Applications, vol. 380, ∫ no. 1, pp. 264-288, 1 August 2011. β(eu−1) u u + St− e (e − 1)[ϕ(t, St− e ,Yt) [2] D. Applebaum, Levy´ Processes and Stochastic Calculus. R u Cambridge University Press, Cambridge, 2004. − − ¯ − − ϕ(t, St−,Yt) ∆tSt (e 1)]νL(du) = 0. [3] N. Bellamy and M. Jeanblanc, “Incomplete markets with

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