Moving-boundary approaches for American security valuation

This dissertation is concerned with the classical problem of pricing an American option written on a single underlying asset. Even under the simple Black-Scholes model, the pricing of the option is non-trivial. Pricing the option requires solving a free-boundary partial differential equation (PDE) problem, a problem in which along with the option price, a boundary not known a priori (known as the optimal-exercise boundary) needs to be determined. The presence of this free boundary significantly complicates the problem of pricing the option. Allowing for more-complex stochastic models to represent asset price movement, while desirable, complicates the option pricing problem even further. In this dissertation, we develop moving-boundary approaches to price American options under a variety of market models. The moving-boundary approach seeks to convert the arising free-boundary problem into a sequence of fixed-boundary problems, each of which are easy to solve, and provide the optimal-exercise boundary and associated price function on convergence. We consider market models of increasing complexity. A moving-boundary approach with approximate boundaries is used to price options under the classic Black-Scholes model. Moving-boundary approaches are also used to price options under jump-diffusion models and stochastic-volatility jump-diffusion models. The approach is general enough to encompass general jump distributions and stochastic volatility models. Convergence of the approach is proved under the various models. Extensive numerical comparisons also highlight the superior run times and accuracy of the approach compared to other methods that solve the free-boundary problems. We also show that the moving-boundary approach is general enough to use numerical methods and simulation techniques to solve the fixed-boundary problems. Finally, two theoretical results are provided with regard to price functions obtained using sub-optimal boundaries, the first proving an error bound for price functions associated with lower exercise boundaries, highlighting the efficiency of the approach, and the second proving the location of the maximum error of the price function associated with higher exercise boundaries