Some Mathematical Topics on Risk Mangement of Financial Markets

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This project promotes a new perspective in understanding the fundamentals of credit risk including the timing and onset of default events, introduces new mathematical tools for analyzing complex algebraic structure inherent in dependent default, and advances stochastic control theory to better analyze trading behavior and strategies in the presence of liquidity risk. Specifically, the first part of the project develops a new concept of default called the economic default. A stochastic model is used to analyze the differences between the economic default and the traditional default. The analysis is built on the classical fluctuation theory in probability. Simulation of the model is compared with recent empirical findings from defaulted bond prices. Sensitivity analysis in terms of key economic factors provides additional insights to the timing and prediction of default events. The second part of the project is to analyze default dependence using a graphical model. Techniques from algebraic geometry are explored to establish the well-posedness of the model and to provide algorithms for calibration. A more structured and dynamic version of the model is then further investigated for CDO prices, for studying loss distributions, and for comparing with other factor models. The third part of the project is devoted to a particular aspect of liquidity risk, namely, market liquidity and optimal execution. Starting from a highly stylized model, analysis of the trading behavior and strategy evolves with a further development of stochastic control theory and a better understanding of the dynamics of limit order books. Technological and financial innovation have transformed the fundamentals of the financial system over the last few decades. The ever-growing complexity of the securitization process and the vulnerability of the financial system to various market risks come to light in the current financial crisis. The crisis highlights several basic issues and poses many mathematical challenges. In particular, given the huge losses in credit derivatives and the extreme illiquidity in certain markets during the crisis, one asks the following. (i) Is the timing of default risk properly recognized? (ii) Should the pricing and hedging of complex credit derivatives be re-evaluated with an appropriate dependence structure? (iii) How to better understand and manage market liquidity for the stability of the financial system? This project explores new perspectives in understanding and modeling various market risks and develops inter-disciplinary mathematical tools to facilitate a more thorough understanding of the financial systems.