IAENG International Journal of Applied Mathematics, 41:4, IJAM_41_4_05 ______________________________________________________________________________________ Variance Reduction with Control Variate for Pricing Asian Options in a Geometric Levy´ Model Wissem Boughamoura and Faouzi Trabelsi Abstract—This paper is concerned with Monte Carlo simu- use of the path integral formulation. [19] studied a cer- lation of generalized Asian option prices where the underlying tain one-dimensional, degenerate parabolic partial differential asset is modeled by a geometric (finite-activity) Levy´ process. equation with a boundary condition which arises in pricing A control variate technique is proposed to improve standard Monte Carlo method. Some convergence results for plain Monte of Asian options. [25] considered the approximation of Carlo simulation of generalized Asian options are proven, and a the optimal stopping problem associated with ultradiffusion detailed numerical study is provided showing the performance processes in valuating Asian options. The value function is of the variance reduction compared to straightforward Monte characterized as the solution of an ultraparabolic variational Carlo method. inequality. Index Terms—Levy´ process, Asian options, Minimal-entropy martingale measure, Monte Carlo method, Variance Reduction, For discontinuous markets, [8] generalized Laplace trans- Control variate. form for continuously sampled Asian option where under- lying asset is driven by a Levy´ Process, [18] presented a binomial tree method for Asian options under a particular I. INTRODUCTION jump-diffusion model. But such Market may be incomplete. He pricing of Asian options is a delicate and interesting Therefore, it will be a big class of equivalent martingale T topic in quantitative finance, and has been a topic of measures (EMMs). For the choice of suitable one, many attention for many years now. Asian options are commonly methods have been proposed: Esscher martingale measure traded: they were introduced to avoid a problem common [13, 3], Minimal martingale measure [10], which can be for European options, where the speculators could drive viewed as time-honored Esscher transform and minimal up the gains from the option by manipulating the price of entropy martingale measure (MEMM) introduced in [11], the underlying asset near to the maturity date.They have [28] developped a simple network approach to American become very attractive for investors since they provide a exotic option valuation under Levy´ processes using the fast customized cheap way to hedge periodic cash flows [24]. Fourier transform (FFT). Finally, [22] derived bounds for the In spite of such advantages there is no closed form solution price of a discretely monitored arithmetic Asian option when for the price of an Asian option, even when the underlying the underlying asset follows an arbitrary Levy´ process. follows a Geometric Brownian model. As a remedy to this difficulty, many approaches and numerical approximations In this paper, we placed in a financial market driven by a are proposed. For instance, in continuous markets, [27] finite-activity Levy´ process and we study the pricing of gen- provided tight analytic bounds for the Asian option price, eralized Asian options under the MEMM. The choice of such [12] computed the Laplace transform of the Asian Option measure is justified by the fact that it preserves the Levy´ and price, [17] used Monte Carlo simulations, [16] investigated the statistical properties of the process. To estimate prices, a method based on binomial trees, [1] used Fast Fourier we propose a control variate variance reduction technique to Transform technique. The authors of [9] introduced change improve the Monte Carlo method and illustrate numerically of numeraire technique and those of [26] used it to reduce the its improvement for particular examples of generalized Asian PDE problem in two variables. They got a lower bound for options. The use of Monte Carlo techniques is an established the price. [15] used lattices to price fixed-strike European- approach when dealing with this kind of problems both in style Asian options that are discretely monitored. [7] derived theory and practice, and the paper makes a contribution to a closed-form solution for the price of an average strike as the literature. well as an average price geometric Asian option, by making Our paper is structured as follows: In section (II), we Manuscript received December 20, 2010; revised June 30, 2011; accepted recall the main properties of a geometric (finite activity) August 04, 2011. Levy´ process and give some particular generalized Asian Laboratory of Mathematical and Numerical Modelling in Engineering options. In section (III), we introduce the minimal entropy Science, National Engineering School of Tunis (ENIT-LAMSIN), B.P. 37, 1002 Tunis-Belved´ ere,` Tunis El Manar University, Tunisia, and Higher martingale measure under which we study the properties of Institute of Applied Mathematics and Computer Science of Kairouan, the stock price. In section (IV), we study the convergence Avenue Assad Ibn Fourat, 3100 Kairouan, University of Kairouan, Tunisia. of Monte Carlo price estimator in L2 using the Reimann Email: [email protected]. Laboratory of Mathematical and Numerical Modelling in Engineering scheme and propose a particular control variate variance Science, National Engineering School of Tunis (ENIT-LAMSIN), B.P. 37, reduction technique to improve the Monte Carlo method. 1002 Tunis-Belved´ ere,` Tunis El Manar University, Tunisia, and Higher Such amelioration is illustrated numerically for particular Institute of Computer Sciences and Mathematics of Monastir, Avenue de la Korniche, B.P. 223, 5000 Monastir, University of Monastir, Tunisia. examples of generalized Asian options in section (V). Finally, E-mail: faouzi [email protected] and [email protected]. section (VI) deals with concluding remarks. (Advance online publication: 9 November 2011) IAENG International Journal of Applied Mathematics, 41:4, IJAM_41_4_05 ______________________________________________________________________________________ II. THE FRAMEWORK B. Generalized Asian options A. The market model Let Yt be the arithmetic average value of the underlying Let (Ω; F; P) be a given complete probability space. We (St)t2[0;T ] on the life time interval [0; t]: consider a market which consists of a bond with price process 1 Z t rt given by B = e , where r denotes the (constant) interest Yt = Sudu: t t rate and a non-dividend paying risky asset (the stock or 0 index). The price process (St)0≤t≤T of the risky asset is Consider an Asian option maturing at time T based on YT modeled by the following geometric Levy´ process: with an FT -measurable payoff H having the form: St = S0 exp(rt + Lt); 0 ≤ t ≤ T; (1) H = '(ST ;YT ); where Lt is a Levy´ process given by: where ' satisfies the following Lipchitz assumption: Nt 0 0 4 X (A1) 9a > 0; 8(S; S ; Y; Y ) 2 R+; Lt = µt + σBt + Uj; (2) 0 0 0 0 j=1 j'(S; Y ) − '(S ;Y )j ≤ a[jS − S j + jY − Y j]: here µ is a drift, σ is the underlying volatility supposed to be Such option is called generalized Asian option. As ex- amples we can cite the payoffs '(S ;Y ) = jS − Y j positive and (Nt)0≤t≤T is a Poisson process with intensity T T T T of a straddle Asian option and the payoff '(S ;Y ) = λ. The jump sizes (Uj)j≥1 are i.i.d. P-integrable r.v. with a T T (ζ: (Y − K S − K ))+ ; which is a fixed strike Asian distribution F and independent of (Nt)0≤t≤T . We denote by T 1 T 2 −rt option, for K = 0, and a floating strike Asian option for S^t = e St the discounted underlying price process. 1 We assume that the σ-fields generated respectively by K2 = 0. The constant ζ = ±1 determines whether the option is call or put. (Bt)0≤t≤T , (Nt)0≤t≤T and (Uj)j≥1 are independents and The price Vt of such option at time t depends on t, St we take Ft := σ(Bs;Ns;Uj1fj≤Nsg; s ≤ t; j ≥ 1); t ≥ 0 as filtration. and on the path that the asset price followed up to time t. In Let τ1; ··· ; τj; ··· be the dates of jumps of the stock price particular, we can not invoke the Markov Property to claim V t S H St; i.e. the jump times of (Nt)0≤t≤T and V1; ··· ;Vj; ··· their that t is a function of and t because is not a function proportions of jumps at these times, respectively; i.e. of T and ST ; H depends on the whole path of S. To overcome this difficulty, we increase the state St − − ∆Sτj = Sτj − S = S Vj for all j ≥ 1; τj τj by defining a second process Yt following the stochastic Uj differential equation: where Vj = e − 1. The r.v.’s (Vj)j≥1 are i.i.d. with a common distribution G and takes its values in the interval S − Y dY = t t dt: (6) (−1; +1). t t Recall that Lt is a Levy´ process with the characteris- The process (S ;Y ) is governed by the pair of equations R 2 t t tic triplet (µ + jx|≤1 xνL(dx); σ ; νL), where νL(dx) = (1) and (6) constitutes a two-dimensional Markov process. λF (dx) L . The Levy-It´ oˆ decomposition of t takes the form: Furthermore, the payoff H = '(ST ;YT ) is a function of T Z and the final value (ST ;YT ) of this process. Thus, there exists Lt = µ + xνL(dx)t + σBt jx|≤1 some function φ(t; S; Y ) such that the generalized Asian Z Z Z Z option price under a particular probability P∗ is given by xJ (ds; dx) + xJ (ds; dx); L eL (3) ∗ −r(T −t) [0;t] jxj>1 [0;t] jx|≤1 φ(t; St;Yt) = E [e '(ST ;YT )=Ft]: X where JL(dt; dx) = δ(τ ;∆L ) is the random Poisson j τj III.