Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Faculty/School

Faculty of Science

First Advisor

Joe Campolieti

Advisor Role

Co-supervisor

Second Advisor

Roman Makarov

Advisor Role

Co-supervisor

Abstract

This thesis introduces and studies two new credit risk models: structural occupation time and occupation time hazard rate models. The defaults within the models are characterized in the form of the U.S. Bankruptcy Code Chapter 7 (a liquidation process) and Chapter 11 (a reorganization process). Structural occupation time models assume credit events are triggered as soon as the occupation time, which measures the total amount of time the firm's asset value is below a certain threshold, exceeds the grace period. Occupation time hazard rate models extend the structural occupation time models and may presume that other random factors may lead to credit events. Both models take in a grace period to ensure the firm could fulfill its obligations during the grace period.

We arrive at new closed-form pricing formulae for credit derivatives, containing the risk-neutral probability of defaults and credit default swap (CDS) spreads as special cases, which are derived analytically via a spectral expansion methodology. This methodology works for any solvable diffusion, such as a geometric Brownian motion (GBM) and several non-linear diffusion processes, and allows us to write the pricing formulae explicitly as rapidly converging infinite series. In particular, we derive the explicit pricing formulae where the firm's asset value is governed by the GBM process, and then extend them to some nonlinear solvable processes. We also study other numerical Laplace inverse transform methods, such as the Talbot and Gavor--Stehfest methods. The models are calibrated to typical market credit default swap (CDS) spreads, resulting in a near-perfect fit.

Convocation Year

2025

Convocation Season

Fall

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