<p>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.</p> <p>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.</p> / Doctor of Philosophy (PhD)

In this article, the authors explain the type of treatment the Call Spread options should be given. They argue that these should be treated as a unique derivative and not as one compound by two independent elements. Likewise, they outline the premium as an inherent element in the determination of any gains or losses from the financial options that it is decided to adopt. As an important point, they claim that adopting one specific side about the treatment of the Call Spread options and the premium implies having a viewpoint about their determination in the Income Tax. / En el presente artículo, los autores explican el tipo de tratamiento que se debería dar a las opciones Call Spread. Sostienen que debería ser tratado como un derivado único y no como uno compuesto por dos elementos independientes. Asimismo, señalan a la prima como un elemento inherente a la determinación de las eventuales ganancias o pérdidas definitivas generadas por las opciones financieras que se decida adoptar. Como punto importante, indican que adoptar una posición específica sobre el tratamiento de las opciones Call Spread y de las primas supone una posición sobre su determinación en el Impuesto a la Renta.