Research in Financial Mathematics

该奖项支持的工作有三个主要组成部分，解决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近似。第一个组件在存在多尺度随机波动率的情况下分析期权价格或隐含波动率表面，结合快速和慢速波动率因素的奇异和常规扰动扩展，以及对短期行为的WKB类型分析。 第二部分研究从市场数据中推断风险度量。在许多标准的金融模型下，良好的时间一致凸风险度量的特征在于向后随机微分方程和拟线性抛物型偏微分方程（PDE）的解，校准是一个反问题。我们的目标是设计渐近和数值方法，将市场价值的工具偶然极端事件的风险措施。 第三个组成部分是开发新的模型和算法来评估和管理信用风险。自上而下的模型类是一个方便的宏观描述的违约的数量，我们的研究涉及渐近的随机系数的波动型偏微分方程。需要理解与成分自下而上（微观）模型的关系，挑战是设计有效的近似值，以通过两个层次的details.This研究项目开发数学和计算工具，用于理解和建模信用风险和波动在极端制度，并寻求构建适当的风险措施。总体目标是更好地量化评估和管理市场波动和违约风险，特别是在像目前这样的严重动荡时期。这一点尤其重要，因为对与信贷挂钩工具有关的风险认识不足，监管不力，导致投机和证券化泛滥，从而引发了当前的危机。虽然这种再保险产品可以用于良好的，其设计和使用必须通过随机分析和统计工具。这项研究将有助于这套工具。

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.