Numerical solutions for American options on assets with stochastic volatilities

This thesis discusses American options on assets with stochastic volatilities. First, it gives a proof of the solution uniqueness of the 2-D PDE to evaluate options for both general two-factor model and the model used in this thesis. Second, it formulates the two factor American option as a 2-D PDE free boundary problem. Third, because the solution of this 2-D PDE free boundary problem is not a very smooth function and the free boundary changes rapidly near maturity, most of the numerical methods could fail to find a reasonable solution or the numerical solution has a large truncation error. Instead of solving this 2-D PDE directly, the difference between the solution of the original 2-D PDE free boundary problem and the solution of a 1-D parabolic equation with the same final condition is calculated. The difference function is very smooth in the entire region. We can solve this new 2-D PDE free boundary problem more accurately and more efficiently. Fourth, this paper uses a coordinate transformation to map the moving boundary to a fixed boundary and applies the Singularity Separating Method (SSM) technique to separate the free boundary and find the exact location of the free boundary (the optimal exercise price). This will be very useful for arbitrage activities. Fifth, it develops numerical methods to solve the new free boundary problem and focuses on the high order implicit finite difference method. It provides several methods to solve the nonlinear system. Sixth, It discovers the put-call symmetry relation between American options in the two factor stochastic volatility model. Seventh, it uses the extrapolation technique to improve the approximation accuracy of the numerical solution. Chapter 5 gives several numerical examples.