This project develops stochastic models of credit risk. Credit risk is the risk that a financial counterparty will default on its financial obligation. The research objective of the project is to develop models describing multivariate stochastic dynamics of credit risk and analytical and computational tools to help implement the models in industrial practice and public policy. The mathematical modeling framework for arrivals of defaults of multiple counterparties (obligors) will be based on multivariate random time changes of Markov processes. The key features of the proposed modeling framework important in applications are that they can achieve any desired level of correlation among default (failure) times of multiple obligors and that the simultaneous defaults (failures) are possible (have positive probability), thus enabling the researcher to model default clustering phenomena. The project will develop an analytical methodology for this class of models based on the spectral theory. This will allow explicit analytical calculations for multivariate default (failure) time distributions by explicitly computing spectral expansions in problems of moderate size. The project will also develop a simulation methodology to deal with large problems with many obligors that cannot be efficiently solved by the analytical spectral method.
If successful, stochastic models and analytical and computational tools developed in this project will help better understand and model credit risk in the financial industry and will aid in the development of public policy to regulate financial institutions. In addition to applications in finance, the mathematical modeling framework and analytical and computational tools developed in the project will advance general stochastic modeling methodology that is applicable to a wide range of reliability and failure applications, including computer and communications networks, electric power grid, manufacturing, and biological systems.
Summary. The project developed mathematical and computational tools to evaluate the risk of default on a debt instrument, such as a corporate bond. The focus of the project was on modeling probabilities of default via stochastic arrival rates of default events, as well as on modeling dependence among defaults of multiple borrowers. This topic is of importance to financial institutions managing credit portfolios and to financial regulators assessing the risk in the financial system. Intellectual merit of project outcomes. 1. The project developed a novel jump-diffusion extension of the classical Cox-Ingersoll-Ross (CIR) diffusion default intensity model. The CIR model assumes that the default arrival rate follows a diffusion process. It is in wide use in the financial industry, but suffers from some significant limitations. In particular, its multivariate version does not allow for the possibility of simultaneous defaults of several borrowers (default clustering). Our Subordinate CIR (SubCIR) default intensity model is fully analytically tractable and yields explicit closed-form pricing of credit-sensitive securities, such as corporate bonds and credit default swaps. The paper was published in Annals of Applied Probability. 2. Developed a novel mathematical representation for the family of Marshall-Olkin multivariate exponential distributions and realized it via an efficient Monte Carlo simulation algorithm to allow simulation of large credit portfolios with up to thousands of borrowers in reasonable computation times on a laptop computer. Results published in proceedings of the Winter Simulation Conference. 3. Developed a novel multi-firm unified credit-equity modeling framework, where the observed stock price of each firm in the portfolio serves as the state variable driving the default arrival rate of the corporate bonds of that firm, and the correlation among the stock returns of multiple firms drives the default correlation for firms' corporate bonds. The paper is accepted for publication in Mathematical Finance. Broader impact. The project has provided opportunities for training and professional development to PhD students in the financial engineering major within the Department of Industrial Enginering and Management Sciences at Northwestern. Two PhD students attained faculty positions at US universities upon graduation. Two PhD students attained positions as researchers in the financial industry upon graduation.
