Stochastic homogenization and its applications to financial mathematics

随机均质化及其在金融数学中的应用

• 批准号: 1209363
• 负责人: Rohini Kumar
• 金额: $ 9.71万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1209363&HistoricalAwards=false
• 关键词: Stochastic; homogenization; its; applications; financial

KumarDMS-1209363 Stochastic homogenization and a separation of time scales are the common themes in the topics the investigator studies. In a recent paper, she and collaborators developed a general method for homogenization of nonlinear partial differential equations. This method was used to examine the problem of pricing options with short maturity, where the stock price was given by a stochastic volatility model with fast mean-reverting volatility. The small maturity time made this a problem of large deviations in probability theory, while the shorter mean-reversion time of volatility made this an averaging problem. Asymptotics of option price under shrinking time-scales of maturity and mean-reversion of volatility were obtained. In this project the investigator aims to obtain higher order approximations for these option prices. This problem naturally leads to the investigation of correction terms for large deviations in multi-scale diffusion processes. Lastly, the investigator studies the effect of fast mean-reversion of volatility on the indifference price of options given in terms of dynamic convex risk measures. This indifference pricing of options was given in a recent paper by Sircar and Sturm and would be useful for studying the effect of fast mean-reverting volatility on risk measures. In view of the current economic crisis, the study of risk of financial positions is essential. The investigator studies a problem in which the option prices under consideration are given in terms of risk measures. This makes sense as options offer protection against the risk of stock prices falling (put options) or rising (call options) and thus their price reflects the option buyer's risk aversion. The investigator analyzes the effect of clustering in market volatility (an observed phenomenon) on the option price described via risk measures, thus indirectly studying the effect of volatility clustering on these important risk measures. The topics considered here give a flavor of the types of multi-scale problems that are amenable to the homogenization techniques used. While the specific problems considered are motivated from financial mathematics, multi-scale phenomena abound in nature and these methods for obtaining more accurate approximations of such phenomena should find wide application.

KumarDMS-1209363随机均质化和时间尺度分离是调查者研究的主题中的共同主题。在最近的一篇论文中，她和合作者发展了一种非线性偏微分方程齐化的一般方法。该方法被用来研究短期期权的定价问题，其中股票价格是由具有快速均值回复波动率的随机波动率模型给出的。较小的到期日使其成为概率论中偏差较大的问题，而波动率较短的均值回归时间使其成为平均化问题。得到了期权价格在到期日缩水时间尺度和波动率均值回归时间尺度下的渐近性。在这个项目中，调查者的目标是获得这些期权价格的高阶近似。这个问题自然导致了对多尺度扩散过程中大偏差的修正项的研究。最后，研究了波动率快速均值回归对基于动态凸性风险度量的期权无差别价格的影响。Sircar和Sturm在最近的一篇论文中给出了期权的这种无差别定价，这将有助于研究快速均值回归波动对风险衡量的影响。鉴于当前的经济危机，对金融头寸风险的研究是必不可少的。调查者研究了一个问题，在这个问题中，所考虑的期权价格是根据风险度量给出的。这是有道理的，因为期权提供了对股价下跌(看跌期权)或上涨(看涨期权)的风险的保护，因此它们的价格反映了期权买家的风险厌恶。研究人员分析了市场波动率集聚(一种观察到的现象)对通过风险度量描述的期权价格的影响，从而间接研究了波动率集聚对这些重要风险度量的影响。这里讨论的主题给出了一种多尺度问题的类型，这些类型的问题服从于所使用的齐化技术。虽然所考虑的具体问题源于金融数学，但多尺度现象在自然界中比比皆是，这些获得此类现象更准确近似的方法应该得到广泛的应用。