Convergence Of The Binomial Tree Method In Option Pricing Models

This paper is a comprehensive survey of the recent results obtained from the study of the convergence of binomial tree method in option pricing models. For vanilla options, the binomial tree method is equivalent to certain explicit difference schemes. This article summarizes the problem of convergence of the binomial tree method in option pricing models during the past few years. As tâ†'0, the convergent process of binomial tree method can to a large extent imitate the process of option pricing.This paper consists of four parts.In the second part, we mainly introduce the Black-Scholes formula. It is the classic formula in option pricing models. The Black-Scholes formula is obtained on∑: {0≤S <∞, 0≤t≤T} by solving the definite problem(1) (2):(1) is the Black-Scholes equation, definite problem(1) (2) is a backward definite problem. Let and for European call option, its pricing formula is:It is so-called Black-Scholes formula; for European put option, its pricing formula is:L. Jiang ([20]) considered that asâ–³tâ†'0, the option pricing defined by the binomial tree converges to a sufficiently smooth limit function, which is the solution of the Black-Scholes equation.In the third part, we only consider continuous monitored path-dependent options. When the price of the options is computed from the binomial tree method, we need to consider the consistency of the binomial tree method and the explicit difference schemes of this kind of problem. In this way, the problem of convergence in numerical analysis can be translated to that in the framework of partial differential equation.By no-arbitrage argument, one has for European path-dependent options:At expiration time T = Nâ–³t, we haveUsing the backward induction (3) (4), option prices can be calculated.For American path-dependent options, (3) is replaced by In order to obtain the uniform convergence of binomial tree methods, wefirstly consider the consistency of the difference function which is discrete from the partial differential equation.For the continuous model, let V(S, A,t) be the path-dependent option value. Note that S, A, t are mutually independent from the view point of PDEs. The pricing model of European path-dependent options is (see([26]或[27]))With the final value conditionIn addition, for lookback options, one has an additional boundary conditionFor American options, (6) is replaced by a variational inequality:with the final condition (7) (and boundary condition (8) for lookback options).We give the theorems and lemma below to represent the convergence of binomial tree method for path-dependent options.Theorem 1 The binomial tree methods (3)(resp.(5)) are consistent with the corresponding PDE (6)(resp.(9)).Lemma 1 The strong comparison principle holds for problem (7) (9) (and (8) for lookback options);namely, if u and v are the viscosity sub solution and supersolution of the problem, respectively, then u≤v.Theorem 2 (Convergence Theorem) Suppose that V(S, A, t) is the viscosity solution to the problem. (7) (9) (and (8 for lookback options). Then, asâ–³tâ†' 0,we have V_â–³t(S,A,t) converges uniformly to V(S, A, t) in any bounded closed subdomain of D×(0,T).Theorem 3 Binomial tree methods for European/American path-dependent options are uniformly convergent in any bounded closed domain of D×(0,T).In the fourth part, Barles, Daher and Romano ([17]) and L. Jiang,D. Min ([21]) presented a framework to prove the convergence of difference schemes for parabolic equations in diffusion models.We firstly show the binomial tree method of American option's price in jump-diffusion model.Its explicit difference scheme isTheorem 4 The binomial tree method (10) is equivalent to the explicit difference scheme (11),withσ²â–³t/â–³x² = 1 in the sense of neglecting a higher orderâ–³t^3/2.Now we show the convergence of the binomial tree method for American options. Using an argument similar to that of Pham ([38]), it can be shown that an American option's price solves the following parabolic variational inequality: Theorem 5 Asâ–³tâ†'0, we have u_â–³t(x,t) converges locally uniformly to u(x,t) in R×[0,T], where u(x,t) is the unique solution to the problem (12).Theorem 6 Suppose u and v are viscosity subsolution and supersolution of (12) andThenWe may notice from this paper that the study of the convergence of the binomial tree method is still very limited. In recent years there are many people devoting to the study of the problem of convergence, but the binomial tree method is still not convenient for pricing options (for example, the arithmetic average option). It is important to modify the binomial tree method, and necessary to improve the algorithm in order to reduce the cost of computation and storage as much as possible. Hence, we have to find some new method and perform careful analysis in studying the algorithm which needs our further efforts.