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Cox, Ross and Rubinstein (1979) developed a simple discrete time lattice model to deal with the complexity of option valuation, as they considered the methods of Black and Scholes (1973) 'quite advanced and have tended to obscure the underlying economics' Cos, Ross and Rubinstein (1979). The use of lattice models such as the one by Cox, Ross and Rubinstein (1979) is the simplicity of its application.

The most significant drawback of the Cox, Ross and Rubinstein (1979) model, is to increase its accuracy the number of time intervals must increase, in order to approach a continuous time model, which will significantly increase the computational time, needed for processing the entire tree in order to derive the option value.

Others such as Hull and White (1988), (1993) and Trigeorgis (1991) have extended the model of Cox, Ross and Rubinstein (1979).

Hull and White (1988) present a study of the use of lattice models for underlying assets with known dividends instead of known divided yields. They also consider the use of a control variate to price a option numerically, by a the lattice model, using the price of a similar option calculated analytically. While Trigeorgis (1991) proposes 'a log transformed variation of binomial option pricing designed to overcome problems of consistency, stability and efficiency encountered in the Cox, Ross and Rubinstein (1979)' focusing on the pricing of exotic options. Hull and White (1993) also present an application of binomial and trinomial procedures for exotic path dependent options, where they developed a model faster than Monte Carlo simulation and faster than other numerical methods.

Usually the analytical procedures are applicable to simple payoffs of the American Options, but in the cases where this is not possible numerical solutions must be developed. Geske and Shastri (1985) give a detailed comparison of the lattice methods to the different numerical methods, finite difference methods and other simulation methods.

The model proposed by Brennan and Schwartz (1978) for valuing options was the first approach that used the finite difference method. This approach was used due to the fact that most of the times an analytical solution for the option pricing problem does not exist. The finite difference method uses the heat equation derived from the Black and Sholes PDE to obtain an approximation of the option price. Courtadon (1998) goes further to reduce the approximation error of the Brennan and Schwartz (1978) model but only applies his findings only to simple option pay offs.

Geske and Shastri (1985) give a good description of the finite difference method: 'The finite difference technique analyze the partial differential equation (...) by using discrete estimates of the changes in the options value for small changes in time or the underlying stock price to form equations as approximations to the continuous partial derivatives.' Usually the approximations is done using forward, backward or central difference theorem, which respectively result in the explicit, implicit and Crank Nicolson schemes, the procedure used in this study 本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。