WEAK APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS AND APPLICATION TO DERIVATIVE PRICING SYOITI NINOMIYA1 AND NICOLAS VICTOIR2 Abstract. The authors present a new simple algorithm to approximate weakly stochastic dierential equations in the spirit of [9][13]. They apply it to the problem of pricingAsian options under theHeston stochastic volatilitymodel, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast. 1. Introduction 1.1. The Problem and its Motivation. We consider a stochastic dierential equa- tion written in the Stratonovich form Y(t; x) = x + Z t 0 V0 (Y(s; x)) ds + dX i=1 Z t 0 Vi (Y(s; x)) dBis; V j 2 C1b RN;RN ; (1) where B = B1; ;Bd is a standard Brownian motion, and C1b RN;RN denotes the set of RN-valued smooth functions dened over RN whose derivatives of any order are bounded. In particular, we will use the classical notation V f (x) =PN i=1 V i (x) @ f=@xi (x) for V 2 C1b (RN;RN) and f a dierentiable function from Rn into R: This stochastic dierential equation can be written in It o form: Y(t; x) = x + Z t 0 V0 (Y(s; x)) ds + dX i=1 Z t 0 Vi (Y(s; x)) dBis; where Vi0 y = Vi0 y + 1 2 dX j=1 V jVij y : Now, given a function f with some regularity, how can one approximate e- ciently E f (Y(1; x)) It is equivalent to the following deterministic problem: if L is the dierential operatorV0+ (1=2) Pd i=1 V 2 i and u is the solution of the heat equation @u @t (t; x) = Lu; u (0; x) = f (x); 2000Mathematics Subject Classication. 65C30, 65C05. Key words and phrases. Hestonmodel, numericalmethods for stoc
hastic dierential equations, math- ematical nance, quasi-Monte Carlo method. This research was partially supported by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientic Research (C), 15540110, 2003. 1 2 S. NINOMIYA AND N. VICTOIR howdoes one approximate u (1; x) (which is equal toE f (Y(1; x)) by Feynman-Kac theorem [7]). This problem has had a lot of attention because of its practical importance: it gives the evolution of the temperature in some media, and also represents price of nancial derivatives under stochastic nancial models such as Black-Scholes [1]. Non-probabilistic methods to solve the PDE (such as nite dierence methods) seem to only work well when L is elliptic and in low dimension. We refer to [11] for a more detailed discussion on the subject. We will focus in this paper on probabilistic methods. 1.2. Notation. If V is a smooth vector eld, i.e. an element of C1b RN;RN , exp (V) (x) denotes the solution at time 1 of the ordinary dierential equation dzt dt = V (zt) ; z0 = x: For x 2 R, bxc denotes the integer part of x. 1.3. Probabilistic Methods. 1.3.1. Order 1. The most popular probabilistic method to approximate E h f Yx1 i is called the Euler-Maruyama method [8]. We rst x n independent d-dimensional random variables Z1; ;Zn such that, if X denotes a standard normal random variables, E p (Zk) = E p (X) for all polynomial of degree less than or equal to 3. 1 Then one denes recursively the following random variables: X(EM);n0 = x; X(EM);n(k+1)=n = X (EM);n k=n + 1 n V X(EM);nk=n + 1p n dX i=1 Vi X(EM);nk=n Zik+1: Then, one can show [8][17] that for all nice enough function f (2)