Credit Risk Modeling with Affine Processes - Stanford University

2002 Cattedra Galileana Lectures Credit Risk Modeling with Affine Processes Darrell Duffie1 Stanford University and Scuola Normale Superiore, Pisa April, 2002. Revised, June, 2004 1This is the written version of the Cattedra Galileana lectures, Scuola Normale Superiore, in Pisa, 2002, made possible through the wonderful organizational work of Maurizio Pratelli, to whom I am most grateful. I am also grateful for support for the course offered by the Associazione Amici della Scuola Normale Superiore, who were generously represented by Mr. Carlo Gulminelli. 1 1 Introduction This is a written version of the Cattedra Galileana lectures, presented in 2002 at the Scuola Normale in Pisa. The objective is to combine an orientation to credit-risk modeling (emphasizing the valuation of

corporate debt and credit derivatives) with an introduction to the analytical tractability and richness of affine state processes. This is not a general survey of either topic, but rather is designed to introduce researchers with some background in mathematics to a useful set of modeling techniques and an interesting set of applications. Appendix A contains a brief overview of structural credit risk models, based on default caused by an insufficiency of assets relative to liabilities, including the classic Black-Scholes-Merton model of corporate debt pricing as well as a standard structural model, proposed by Fisher, Heinkel, and Zechner [1989] and solved by Leland [1994], for which default occurs when the issuer’s assets reach a level so small that the issuer finds it optimal to declare bankruptcy. The alternative, and our main objective, is to treat default by a “reduced-form” approach, that is, at an exogenously specified intensity process. As a special tractable case, we often suppose that the default intensity and interest rate processes are linear with respect to an “affine” Markov state process. Section 2 begins with the notion of default intensity, and the related calculation of survival probabilities in doubly-stochastic settings. The underlying mathematical foundations are found in Appendix E. Section 3 introduces the notion of affine processes, the main source of example calculations for the remainder. Technical foundations for affine processes are found in Appendix C. Section 4 explains the notion of risk-neutral probabilities, and provides the change of probability measure associated with a given change of default intensity (a version of Girsanov’s Theorem). Technical details for this are found in Appendix E. By Section 5, we see the basic model for pricing default able debt in a setting with stochastic interest rates and stochastic risk-neutral default intensities, but assuming no recovery at default. The following section extends the pricing models to handle default recovery under alternative parameterizations. Section 7 introduces multi-entity default modeling with correlation. Section 8 turns to applications such as default swaps, credit guarantees, irrevocable lines of credit, and ratings-based step-up bonds. Appendix F provides some directions for further reading. 2 2 Intensity-Based Modeling of Default This section introduces a model for a default time as a stopping time τ with a given intensity process, as defined below. From the joint behavior of the default time, interest-rates, the promised payment of the security, and the model of recovery at default, as...

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