UCLA

Many traditional mathematical finance models attempt to evaluate the time-varying credit risk term structure through various pricing formulae that assume certain stochastic dynamics. The Black-Scholes-Merton model has not only been recognized as one of the most significant contributions in economics with the Nobel Prize, but seen numerous extensions over the years. In naming a `fair' price of any financial instrument, a measure of risk or uncertainty needs to be carefully specified. Common mathematical models rely on partial differential equations and can conveniently express drift and volatility explicitly in a geometric Brownian motion. Such models are widely used today for their simplicity, easy interpretation and robust estimates. However, some major limitations, such as the assumption of a stationary process or market equilibrium which is unrealistic, persist despite various extensions of the basic model.

In this dissertation, we propose a statistical methodology for modeling credit risk in a financial market. Without specifying the dynamics in a partial differential equation approach, we attempt to model the time-varying implied default probability of a firm in a complete market through the use of hazard term structure. While our pricing formulae are not complicated and additive in nature, they are intuitive, theoretically sound and easy to generalize. Contrary to most other statistical approaches, normality is not directly assumed in the pricing error. We do, however, propose methods to effectively capture the correlation structure among residuals, both in cross-sectional studies and when time-series data is involved. Such implied default correlation structure could be useful in many practical applications such as, and not limited to, credit risk management, portfolio selection and stress tests. Empirical studies have been conducted using financial market data in Japan. We report findings by specifically comparing the market behavior before and after the 2007 - 2009 financial crisis. Various computational and theoretical advantages of our methodology are discussed.