Some Studies On Pricing Asian Options

It's not only a hot topic to price the options in the financing market, but also it's a intractable job, especially to price the most popular exotic options, Asian options, it has already been a focus of attention. As the geometric average of a set of logarithm normal distribution variables still is a logarithm normal distribution variable, so it is easy to get the analytical solution for European-style geometrical average Asian options, but to arithmetic style, as we cann't make sure of the distribution of the underlying assets' arithmetic average, so up to now, we haven't got the analytical sulution for it.Based on Black-Scholes formula, we use the stochastic differential equation and the skill of portfolio in the arbitrage-free market to set up the mathematic model for all kinds of Asian options, by changing variables, we simplify the partial differential equations to common Cauchy problems. Then, a closed-form solution is given to geometric average Asian options through some mathematic methods and we work out some approximation methods to arithmetic average Asian options. Besides, by means of the model, the call-put parities for all kinds of Asian options are also found out. Then a P.D.E. is given for American-style Asian call options.