The work supported by this award has three main components addressing some central concerns in the mathematical analysis of financial problems, specifically credit risk, construction and calibration of measures of risk, and asymptotic approximations for derivatives valuation problems. The first component analyzes option prices, or implied volatility surfaces, in the presence of multiscale stochastic volatility, combining singular and regular perturbation expansions from fast and slow volatility factors, and a WKB-type analysis for the short-time behaviour. The second component studies the inference of risk measures from market data. Under many standard financial models, good time-consistent convex risk measures are characterized by solutions of backward stochastic differential equations and quasilinear parabolic partial differential equations (PDEs), for which calibration is an inverse problem. The goal is design of asymptotic and numerical methods to translate market values of instruments contingent on extreme events into risk measures. The third component is to develop new models and algorithms for valuing and managing credit risk. The class of top-down models is a convenient macroscopic description of the number of defaults, and our study involves asymptotics for wave-type PDEs with random coefficients. The relationship with constituent bottom-up (microscopic) models needs to be understood, and the challenge is to design effective approximations to pass between the two levels of detail.
This research project develops mathematical and computational tools for understanding and modeling credit risk and volatilities in extreme regimes, and seeks to construct appropriate risk measures. The broad goal is better quantitative assessment and management of market volatility and default risk, especially in times of heavy turmoil like the present. This is particularly important given that poor understanding and weak regulation of risks related to credit-linked instruments allowed untamed speculation and securitization that spurred the onset of the current crisis. While such re-insurance products can be used for the good, their design and use has to be informed by tools of stochastic analysis and statistics. This research will contribute to this tool set.
