Simple Iterative Method For Pricing American Options

The aim of this paper is to price American options with stochastic volatility. Since American options can be exercised at any time before maturity, there is no analytical formula for the valuation, instead the price of American option can be presented as the sum of the price of its corresponding European option and the early exercise premium (or EEP), which captures the benefits from exercising prior to maturity. The formula of EEP is an integral with the free boundary (i.e., optimal exercise boundary), which satisfies that when the price of the underlying asset falls on the boundary, the payoff of exercising the option equals the price of the option. This equation of EEP is extremely useful in that it provides a recursive integral equation for the optimal exercise boundary. Solving the integral equation resolves the valuation problem:It identifies the exercise boundary and produces a parametric formula for the American option price. For ease of exposition, the above approach is named as EEP simple iterative method.This approach, which is based on the integral equation, is easy to implement and has significant advantages over other numerical procedures such as methods based on Partial Differential Equations (PDE), Monte Carlo simulation, or binomial lattices. The approach was first proposed to solve the problem of pricing American put options with constant volatility by Little and Pant (2000). 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. Secondly, the thesis reviews the EEP simple iterative method for pricing American options under constant volatility. Thirdly, the main part of the thesis is presented. The EEP simple iterative 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 simple iterative 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.