Date of Award

Fall 2011

Degree Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

Supervisor

Tom Hurd

Co-Supervisor

Shui Feng and Peter Miu

Language

English

Committee Member

Shui Feng and Peter Miu

Abstract

The central theme of this thesis is to develop methods of financial mathematics to understand the dynamics of a firm's capital structure through observations of market prices of liquid securities written on the firm. Just as stock prices are a direct measure of a firm's equity, other liquidly traded products such as options and credit default swaps (CDS) should also be indicators of aspects of a firm's capital structure. We interpret the prices of these securities as the market's revelation of a firm's financial status. In order not to enter into the complexity of balance sheet anatomy, we postulate a balance sheet as simple as Asset = Equity + Debt. Using mathematical models based on the principles of arbitrage pricing theory, we demonstrate that this reduced picture is rich enough to reproduce CDS term structures and implied volatility surfaces that are consistent with market observations. Therefore, reverse engineering applied to market observations provides concise and crucial information of the capital structure.

Our investigations into capital structure modeling gives rise to an innovative pricing formula for spread options. Existing methods of pricing spread options are not entirely satisfactory beyond the log-normal model and we introduce a new formula for general spread option pricing based on Fourier analysis of the payoff function. Our development, including a flexible and general error analysis, proves the effectiveness of a fast Fourier transform implementation of the formula for the computation of spread option prices and Greeks. It is found to be easy to implement, stable, and applicable in a wide variety of asset pricing models.

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