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-12209363随机均质化和时间尺度的分离是研究者研究主题中的常见主题。 在最近的一篇论文中，她和合作者开发了一种通用方法，用于非线性偏微分方程的均质化。 该方法用于检查成熟度较短的定价选项的问题，在这种情况下，股票价格是由具有快速均值波动率的随机波动率模型给出的。 较小的成熟时间使这是概率理论中较大偏差的问题，而较短的均值逆转时间使这是一个平均问题。 在成熟度的缩小时间表和波动率均值下，期权价格的渐近价格得到了。 在这个项目中，研究人员旨在获得这些期权价格的高阶近似值。 这个问题自然会导致对多尺度扩散过程中大偏差的校正项的研究。 最后，研究人员研究了波动率快速均值逆转对动态凸风险措施的选择的漠不关心价格的影响。 Sircar和Sturm最近的一篇论文给出了这种漠不关心的定价，这对于研究快速均值转移波动对风险度量的影响非常有用。 鉴于当前的经济危机，对财务状况的风险进行研究至关重要。 研究人员研究了一个问题，其中考虑了正在考虑的选择价格，以风险衡量标准给出。 这是有道理的，因为期权可以保护股价下跌（PUT期权）或上升（通话期权）的风险，因此它们的价格反映了购买者的风险规避。 研究者分析了聚类在市场波动率（观察到的现象）中对通过风险度量所描述的期权价格的影响，从而间接研究了波动性聚类对这些重要风险措施的影响。 这里考虑的主题给出了与所使用的均质化技术相适应的多尺度问题类型的风味。 尽管所考虑的特定问题是由于财务数学而动机的，但自然界中的多规模现象和获得更准确的这种现象近似值的方法应找到广泛的应用。