EEP High-Order Collocation Methods For Pricing American Options

The aim of this paper is to price American options with stochastic volatility. Since American options can be exercised at anytime before maturity, there is no analytical formula for the valuation. It is known that the free boundary (i.e., optimal exercise boundary) satisfies a’nonstandard’Volterra integral equation. This Volterra integral equation is numerically solved by a high-order collocation method based on graded meshes. With the computed free boundary, the Black-Scholes equation for pricing the American options is solved by a moving mesh method. For ease of exposition, the above approach is named as EEP high-order collocation method. The approach was first proposed to solve the problem of pricing American put options with constant volatility by Ma et al. (2010). The main contribution of this thesis is to solve free boundary problem arising in pricing American options with stochastic volatility and price the options.Firstly, the background and motivation of the thesis are provided. Options are widely used in the current financial market. Since American options can be exercised at any time before maturity, the value is more expensive than that of European options and thus American options are widely used. But due the uncertainty of the exercising, the price has no analytical formula. Therefore the pricing problem becomes challenging and hot in either academic or real world. Secondly, the thesis reviews the EEP high-order collocation method for pricingAmerican options under constant volatility. Thirdly, the main part of the thesis is presented. The EEP high-order collocation method is proposed and implemented to pricing American options under stochastic volatility. After pointing out the difference in the evaluation of American options with constant and stochastic volatility, the EEP high-order collocation method is constructed to solve the problem. In addition the efficiency of the approach is confirmed by numerical examples. Finally, conclusions are given and future work in this direction is pointed out.