Numerical Methods for American Option Pricing with Nonlinear Volatility
Options are a fundamental and important type of financial derivatives with stocks as the underlying asset. Investors frequently trade options, making option pricing an important research area in both finance and applied mathematics. There exist mathematical models describing option pricing, amongst which the best known model is the Black-Scholes Equation. The Black-Scholes Equation is a parabolic PDE on the option price as a function of stock price and time. The volatility is one of the crucial parameters in the Black-Scholes Equation with a great impact on the behavior of the option price. Most existing works in this area focus on the cases where the volatility is either a constant or a deterministic function of stock price and time. However, in real markets, such presumptions rarely hold. In this dissertation, we concentrate on the theoretical modeling and the numerical computations for the Black-Scholes Equation with uncertain and nonlinear volatility. First, we introduce a theoretical model for uncertain volatility. We derive several properties of the solutions to the Black-Scholes Equation with uncertain volatility based on the maximum principle of parabolic PDE initial boundary value problems. For American options, the global spread for the option price is proved when the volatility depends on the underlying security and time. This result is confirmed by the observed real financial data in option markets. The Black-Scholes Equation for American options is not readily solvable for exact solutions. It is imperative to study the numerical solutions to the Black-Scholes Equation for American options. Next, we introduce two mixed-type finite difference schemes for solving such equations with nonlinear volatility. We compare these schemes with the well-known explicit and implicit finite difference schemes by implementing the algorithms and running experiments. This dissertation is organized as follows: Chapter 1 is an introduction to option pricing theory; Chapter 2 focuses on theoretical model of uncertain volatility; Chapter 3 introduces the numerical methods; Chapter 4 shows the experiment results; Chapter 5 summarizes the work and points out some future research directions.