Research in Financial Mathematics

该奖项支持的工作有三个主要组成部分，涉及金融问题的数学分析中的一些核心问题，特别是信用风险，风险度量的构建和校准，以及衍生品估值问题的渐近逼近。第一部分分析期权价格，或隐含波动率面，在多尺度随机波动率存在的情况下，结合快速和缓慢波动率因素的奇异和规则扰动扩展，以及对短期行为的wkb型分析。第二部分研究从市场数据推断风险测度。在许多标准的金融模型中，好的时间一致凸风险度量是由倒向随机微分方程和拟线性抛物型偏微分方程（PDEs）的解来表征的，其校准是一个逆问题。目标是设计渐近和数值方法，将工具的市场价值取决于极端事件转化为风险度量。第三部分是开发评估和管理信用风险的新模型和算法。这类自顶向下模型是对默认值数量的一种方便的宏观描述，我们的研究涉及到带随机系数的波浪型偏微分方程的渐近性。需要理解与组成的自下而上（微观）模型的关系，并且挑战在于设计有效的近似以在两个细节级别之间传递。本研究项目开发数学和计算工具，以理解和模拟极端制度下的信用风险和波动，并寻求构建适当的风险措施。总体目标是更好地对市场波动和违约风险进行量化评估和管理，尤其是在当前这种严重动荡的时期。这一点尤其重要，因为对信贷相关工具相关风险的不了解和监管不力，导致了肆无忌惮的投机和证券化，从而引发了当前危机的爆发。虽然这类再保险产品可以用来造福人类，但它们的设计和使用必须借助随机分析和统计工具。这项研究将有助于这个工具集。

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 thatspurred 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.