The Numerical Methods Of Asian Option Pricing

In today's financial derivatives market, there are standard options and other types of exotic options. Asian option is one of the most active exotic options. The main difference between the Asian options and the Standard options lies in that when determining the returns of the options on expiration date, instead of adopting the prevailing market prices of underlying assets like the Standard options, asian options use the average value of the underlying assets some time during the contract period[1]. The value of the Standard options has nothing to do with the path, it relies only on the price of underlying assets on expiration date, so it is difficult to prevent people from manipulating the price on the expiration date to practice arbitrage[2]. However, Asian options depend on the path, they can be used to alleviate the congenial behavior, besides, they have some other advantages, for example, they are cheaper[3] and they can be used to reduce the risk in an accounting period, so Asian options developed rapidly in the marketplace. With the increasingly active development of Asian options, studies on the price of Asian options become extremely important.There are two kinds of Asian options:geometric average Asian options and arithmetic average Asian options. We can derive an analytical solution of geometric average Asian options based on the pricing model. In contrast, there does not exist such an analytical expression for arithmetic average Asian options[4]. However, it is the arithmetic average Asian options that we use most in practice, so a lot of financier and mathematicians devote their time to the numerical study of arithmetic average Asian options, hoping to find out an approximate solution to this problem. Researchers have tried to approach this problem using ternary tree method,the second order moment approximation[5] and Monte Carlo simulation method. With the development in the Black-Scholes model[1] and the discrete numerical methods adopted in solving partial differential equations, finite difference method[6] and finite volume method[7] are also widely used to solve the price of arithmetic average Asian options.Most of the present study on numerical solutions are simulated based on the pricing model of Asian options without dividend. The model thus derived can be transformed into one dimensional linear convection diffusion equation[1]. However, we cannot neglect the dividend in practice. In this paper, we will take into account of the dividend whose dividend rate is q, under this condition, we can transform the pricing model of arithmetic average Asian options into the following one dimensional nonlinear convection diffusion equation[4]:Applying the upwind difference format[6] to the above derived nonlinear equation, we get: whereε" represents the truncation error: As a consequence of the existence of the nonlinear terms, it is more complicated in the representation of the format and error terms, and it makes the analysis in the stability and convergence more difficult.We also apply the characteristic difference method[8][9] which is usually used to solve partial differential equation to find the solutions of specific pricing model of Asian options, we get the following format which combines the advantage of characteristic method and collocation method:We can get good numerical solutions using the characteristic difference method from the stability analysis and analysis of error estimate. The structure of the paper is as follows:in Chapter I, we will introduce the development background of the pricing model of Asian options briefly; In Chapter II, by the use of the Hedge Principle, we will derive the pricing models for different kinds of Asian options, The last part of this chapter is devoted to the derivation of the analytical solutions of the Euclidean geometric average Asian options with a fixed strike price; In ChapterⅢ, by the use of the upwind difference format, we will derive the numerical solutions of the arithmetic average Asian options with a floating strike price. More over, we will give a numerical experient; At last, we will apply the characteristic difference method to look for solutions of the pricing model of Asian options, besides, we can give the stability analysis estimate and the estimate of the error terms in ChapterⅣ.