Mathematical Finance

自从1973年费希尔·布莱克、罗伯特·默顿和迈伦·斯科尔斯的开创性工作（预示着将复杂的数学思想引入定价期权问题，更广泛地说，引入金融行业的风险分析）以来，先进的数学技术在管理保险和金融行业的风险方面发挥了越来越重要的作用。为了表彰他们的工作，默顿和斯科尔斯被授予1997年诺贝尔经济学奖，布莱克已于1995年去世。金融领域的这些发展得益于许多不同领域的思想，包括传统的数学和统计学，但也来自电气工程和凝聚态物理等领域。然而，著名的布莱克-斯科尔斯-默顿模型不再被认为是合适的，试图用改进的模型取代它的尝试，构成了当前数学金融领域激烈研究活动的基础。我计划利用我在纯数学和应用数学领域的研究经验——数学物理、偏微分方程和控制理论——来研究以下相关的研究课题：第一，组合保险和相关结构性投资产品的风险分析；第二，随机控制理论方法在最优投资组合管理和交易模型中的应用；第三，开发新的风险资产随机过程期权定价、交易和风险分析模型，包括随机波动率和跳跃扩散模型，以替代black - scholes - merton模型；第四，这些模型的数值实现。我将从罗格斯大学（Rutgers University）休假期间在纽约彭博社（Bloomberg LP）工作，与帕特里克·哈根（Patrick Hagan）和彭博社定量研究主管彼得·卡尔（Peter Carr）合作。在我2005年秋季回到罗格斯大学后，我计划继续我的工作，在数学金融领域开发一个新的应用数学硕士学位课程，以及我在该领域的研究。我们的硕士学位课程将为我们的学生在金融、保险和风险分析行业带来更好的就业机会。我的提案中描述的研究活动可以通过产生改进的金融建模和风险分析算法来造福社会，从而更好地保护养老基金，降低购房者和行业的借贷成本。

Since the pioneering work of Fischer Black, Robert Merton, and Myron Scholes in 1973, which heralded the introduction of sophisticated mathematical ideas to the problem of pricing options and, more broadly, to risk analysis in the financial industry, advanced mathematical techniques have played an increasingly important role in managing risk across a wide range of insurance and financial industries. In recognition of their work, Merton and Scholes were awarded the 1997 Nobel Prize in Economics, Black having died in 1995. These developments in finance have benefited from ideas in many different fields, including the traditional ones of mathematics and statistics, but also from fields such as electrical engineering and condensed matter physics. The celebrated Black-Scholes-Merton model is no longer viewed as adequate, however, and attempts to replace it with improved models form the basis of intense current research activity in mathematical finance. I plan to bring my research experience in both pure and applied areas of mathematics -- mathematical physics, partial differential equations, and control theory -- to investigate the following interrelated research topics: First, risk analysis for portfolio insurance and related structured investment products;second, the application of methods of stochastic control theory to optimal portfolio management and trading models; third, the development of new option pricing, trading and risk analysis models for risky-asset stochastic processes, including stochastic volatility and jump diffusion models, as replacements for theBlack-Scholes-Merton model; fourth, the numerical implementation of these models.I will work on this project while at Bloomberg LP, New York, on leave from Rutgers University, working with Patrick Hagan and Peter Carr, Head of Quantitative Research at Bloomberg. Upon my return to Rutgersin Autumn 2005, I plan to continue my work on the development of a new applied mathematics master's degree program in mathematical finance, as well as my research in that field. Our master's degree program will lead to better career opportunities for our students in the financial, insurance, and risk analysis professions. The research activities described in my proposal can benefit society by generating improvedalgorithms for financial modeling and risk analysis, leading to greater protection of pension funds and to lower borrowing costs for home purchasers and for industry.