Hilbert Transform Methods in Financial Engineering

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该拨款为开发高性能计算方法提供资金，用于快速准确地为各种非标准收益期权合约定价。本文的研究结果将用于随机波动和跳跃模型下的期权合约定价，该模型放宽了广泛使用的Black-Scholes-Merton模型的限制性假设。基于Hilbert变换的方法将用于纯跳跃Levy过程模型中的百慕大式香草期权、回溯期权和障碍期权、平方根模型中的某些亚洲股票和利率期权以及多资产模型中的期权合约的定价。该方法还将用于有效地反演解析特征函数以获得随机变量的累积分布函数，以及模拟纯跳跃Levy过程和随机波动过程。期权合约被公司和基金经理广泛用于对冲因利率、货币汇率和股票价格波动而面临的金融风险。为了满足特定的对冲要求，许多期权合约都有复杂的收益，取决于相关金融变量的最大值、最小值或平均值。大多数期权合约可以在到期前执行。常用的Black-Scholes-Merton期权定价模型显著低估了与这些衍生产品相关的风险。如果成功，当与基础金融变量相关的风险以更合适的方式建模时，所提出的研究结果将导致许多重要类别的期权合约的准确和有效定价。研究成果将在应用概率论、工程、经济学等领域得到广泛应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)This grant provides funding for the development of a high performance computational method for the fast and accurate pricing of various option contracts with non-standard payoffs. The proposed research results will be used to price such option contracts in stochastic volatility and jump models which relax the restrictive assumptions of the widely used Black-Scholes-Merton model. A method based on the Hilbert transform will be used to price Bermudan style vanilla, lookback and barrier options in pure jump Levy process models, certain Asian equity and interest rate options in square root models, as well as option contracts in multi-asset models. The method will also be applied to efficiently invert an analytic characteristic function to obtain the cumulative distribution function of a random variable, and to simulate pure jump Levy processes and processes with stochastic volatilities. Option contracts are widely used by corporations and fund managers to hedge against financial risks they face due to the fluctuation of interest rates, currency exchange rates, and equity prices. Many option contracts have complicated payoffs, depending on the maximum, minimum, or average of the underlying financial variable, to satisfy specific hedging requirements. Most option contracts can be exercised before their maturities. The commonly used Black-Scholes-Merton option pricing model significantly underestimates risks that are associated with these derivative products. If successful, the results of the proposed research will lead to accurate and efficient pricing of many important classes of option contracts when risks associated with the underlying financial variables are modeled in more appropriate ways. The research results will also contribute to various application areas in applied probability, engineering, and economics.