#### Document Type

Thesis

#### Degree Name

Master of Science (MSc)

#### Department

Mathematics

#### Faculty/School

Faculty of Science

#### First Advisor

Roman Makarov

#### Advisor Role

Thesis Supervisor

#### Abstract

Many authors have used a time-changed Brownian motion as a model of log-stock returns. Using a Levy process as a stochastic time change, one obtains well known asset price models such as the variance gamma (VG) and normal inverse Gaussian (NIG) models. Following on the heels of these asset price models, it is natural to extend structural credit models by using a time-changed geometric Brownian motion and other jump-diffusion processes to model the value of a firm. To avoid the difficulties that arise in computing the associated first passage time distribution and in analogy to the time-changed Markov chain models, where the default state is an absorbing state, we propose a specific variation of the first passage time applicable to time-changed Brownian motions, but not to general jump diffusions.

This thesis deals with a time-changed bivariate Brownian motion (TCBBM) to model default in credit risk. In particular, we use a gamma process as a stochastic time change. The time of default is modelled as the first-hitting time of a default state. Analytical expressions of the probability of default for a single firm are obtained. We develop the formulas for the probability of multiple default for the case with two firms as well. The Monte Carlo method is also presented to compute the default probability under the TCBBM model in a general case.

#### Recommended Citation

Li, Hui, "First-Passage Time Models with a Stochastic Time Change in Credit Risk" (2009). *Theses and Dissertations (Comprehensive)*. 940.

http://scholars.wlu.ca/etd/940

#### Convocation Year

2009