Finite Difference Methods For Pricing Asian Options

An option is a contract with the conditions that, at a prescribed time in the future, the holder of the option may purchase or sell the underlying asset for a prescribed amount. Since 1973 when options on stocks were first traded on an organized exchange, there has been a dramatic growth in options markets. Now, as an important financial derivative product, options are traded on many exchanges throughout the world.Asian options are some sort of exotic options derived from standard options. At present, they are traded very actively in the over-the-counter market as well as on exchanges. Asian option price depends on the average value of the underlying assets over a certain time interval. It consists of average strike options and average rate options. In this paper, we discuss average strike options.Asian options are path-dependent options. Their mathematical model is a final value parabolic problem with three independent variables. Though most Asian options permit a reduction in the dimensionality of the problem by use of a similarity variable, we can only obtain explicit solution for the price of geometric average Asian options by an appropriate Black-Scholes model. It is complicated to obtain an explicit solution for the price of arithmetic average Asian options using the Black-Scholes model. So, numerical techniques are preferable. At present, the most commonly used methods include: Monte Carlo methods, binomial tree, finite element and finite difference methods. In this work, we use finite difference methods for pricing the Asian options.In chapter one, we introduce the basic option theory and the Black-Scoles model. In chapter two, we introduce Asian options and deduce the basic model for their valuation, including the continuously and discretely sampled averages. We reduce the option problem from three to two dimensions by use of a similarity variable, in chapter three. Finally, in chapter four, we demonstrate the finite difference methods for pricing Asian options by discretising the models for the European and American average strikes.