Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Mark Reesor

Abstract

Valuing American options is a central problem in option pricing since the early-exercise feature is very common among financial or insurance derivatives products. For high-dimensional American options, Monte Carlo simulation is generally regarded as the only viable approach to price them, and this is the focus of our work. We propose a new regression-based Monte Carlo algorithm for pricing American options. This method typically generates an upper bound of the option value. It is computationally efficient and generates accurate price estimates.

To improve the convergence rate, we apply a bias reduction technique to the least-squares Monte Carlo estimators of American option value. It works by subtracting a bias approximation from the original option value estimators at each exercise opportunity. The bias approximation is derived using large sample properties of the least-squares regression estimators. The resulting expression is easy to evaluate, and is applicable to any payoff structures and underlying processes. Numerical results show that this technique can significantly reduce the bias. However, it introduces non-negligible computational costs, thus careful treatment is required when it is adopted in practice.

Finally, we extend the least-squares Monte Carlo algorithm to estimate the counterparty exposures of American options. The new algorithm is termed optimized least-squares Monte Carlo (OLSM), which is combined with variance reduction techniques, initial state dispersion and multiple bucketing to enhance its performance. The biggest advantage of OLSM is that it avoids nested simulations, allowing for the computation of risk measures on various time horizons under a reasonable computational budget.


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