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We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations.

A new approach to credit risk modelling is introduced that avoids the use of inaccessible stopping times. Default events are associated directly with the failure of obligors to make contractually agreed payments. Noisy information about impending cash flows is available to market participants. In this framework the market filtration is modelled explicitly, and is assumed to be generated by one or more independent market information processes. Each such information process carries partial information about the values of the market factors that determine future cash flows. For each market factor, the rate at which true information is provided to market participants concerning the eventual value of the factor is a parameter of the model. Analytical expressions that can be readily used for simulation are presented for the price processes of defaultable bonds with stochastic recovery. Similar expressions can be formulated for other debt instruments, including multi-name products. An explicit formula is derived for the value of an option on a defaultable discount bond. It is shown that the value of such an option is an increasing function of the rate at which true information is provided about the terminal payoff of the bond. One notable feature of the framework is that it satisfies an overall dynamic consistency condition that makes it suitable as a basis for practical modelling situations where frequent recalibration may be necessary.

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We propose a two-sided jump model for credit risk by extending the Leland-Toft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) The jump and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; and although in generel credit spreads and implied volatility tend to move in the same direction under exogenous default models, but this may not be true in presence of endogenous default and jumps. In terms of mathematical contribution, we give a proof of a version of the “smooth fitting ” principle for the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model. 1

An Information-Based Framework for Asset Pricing: X-Factor Theory and its Applications. This thesis presents a new framework for asset pricing based on modelling the information available to market participants. Each asset is characterised by the cash flows it generates. Each cash flow is expressed as a function of one or more independent random variables called market factors or “X-factors”. Each X-factor is associated with a “market information process”, the values of which become available to market participants. In addition to true information about the X-factor, the in-formation process contains an independent “noise ” term modelled here by a Brownian bridge. The information process thus gives partial information about the X-factor, and the value of the market factor is only revealed at the termination of the process. The market filtration is assumed to be generated by the information processes associated with the X-factors. The price of an asset is given by the risk-neutral expectation of the sum of the discounted cash flows, conditional on the information available from the filtration. The thesis develops the theory in some detail, with a variety of applica-

We discuss the valuation of credit derivatives in extreme regimes such as when the time-tomaturity is short, or when payoff is contingent upon a large number of defaults, as with senior tranches of collateralized debt obligations. In these cases, risk aversion may play an important role, especially when there is little liquidity, and utility indifference valuation may apply. Specifically, we analyze how short-term yield spreads from defaultable bonds in a structural model may be raised due to investor risk aversion. 1

for their financial backing and for giving me the opportunity to keep a toe in the credit markets. I would like to thank all those at OCIAM who have provided me with the guidance and support necessary to complete this thesis. I am particularly grateful to my supervisor, William Shaw, for allowing me the freedom to pursue my own research interests and I would especially like to thank Christoph Reisinger for more helpful discussions than I can count, not to mention the opportunity to play with his code. I would also like to mention all my friends, in Oxford and elsewhere, who have contributed in so many ways to my life over the last three years and to the ultimate form of my research. Finally, I would like to thank my family for their continued support in all my endeavours, and in particular, my sister, for giving me the impetus I needed to return to student life and Multi-asset credit derivatives trade in huge volumes, yet no models exist that are capable of properly accounting for the spread behaviour of dependent

Michaelmas 2009This thesis is dedicated to the late

The purpose of this paper is to present in a unified context the reduced form modelling approach, in which a credit event is modelled as a totally inaccessible stopping time. Once the general framework is introduced (frequently referred to as “pure intensity ” set-up), we focus on the special case where the full information at the disposal of the traders may be split in two sub-filtrations, one of them carrying the full information of the occurrence of the credit event (in general referred to as “hazard process ” approach). The general pricing rule when only one filtration is considered reveals to be non tractable in most of cases, whereas the second construction leads to much simplest formulas. Examples are given and evidence advanced that this set-up is more tractable.