Monte Carlo Based Methods for Pricing American Options

In this thesis, we develop computational algorithms in a number of different regimes for pricing financial American options. As is well known, American option is a contract between two parties that gives the buyer of the option the right (but not the obligation) to buy or sell an underlying asset at a certain strike price any time before expiration date (also called maturity) of the option. Their pricing problem reduces to solving the optimal stopping problem for a Markov process. Monte Carlo is a preferred tool for solving this problem when the underlying state space is high dimensional.

We study three aspects of the pricing problem in this thesis. First, we propose a new simulation method to price American option written on a single underlying asset under the popular stochastic volatility model. We derive the optimal exercise boundary approximation and use it in conjunction with Monte Carlo to find estimator for true American option price. We also derive option price approximation to use it to form martingale control variates for proposed estimator and demonstrate considerable variance reduction.

In addition, for high-dimensional American options, under general model settings, we propose a new simulation method based on the nearest neighbor estimation technique. We derive the optimal mean square error (MSE) convergence rate for the proposed estimator and study the impact of underlying dimensionality on its error convergence rate.

Further, we perform convergence analysis for two commonly used simulation methods – stochastic mesh method and least squares method – to price American options in high dimensions. We derive respective optimal MSE convergence rates for estimators based on the two methods and compare their performance.