Stochastic homogenization and its applications to financial mathematics

随机均质化和时间尺度分离是研究者研究课题中的共同主题。在最近的一篇论文中，她和合作者开发了一种非线性偏微分方程齐次化的一般方法。该方法用于研究短期期权定价问题，其中股票价格由具有快速均值回归波动率的随机波动率模型给出。由于成熟时间短，这在概率论上是一个偏差大的问题，而波动率的平均回归时间短，这是一个平均问题。得到了期限缩时尺度和波动率均值回归条件下期权价格的渐近性。在这个项目中，研究者的目的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，研究了基于动态凸风险测度的波动率快速均值回归对期权无差异价格的影响。这种期权的无差异定价是由Sircar和Sturm在最近的一篇论文中给出的，它将有助于研究快速均值回归波动率对风险度量的影响。鉴于当前的经济危机，对财务状况风险的研究是必不可少的。研究者研究了考虑的期权价格以风险度量的形式给出的问题。这是有道理的，因为期权提供了对股票价格下跌（看跌期权）或上涨（看涨期权）风险的保护，因此它们的价格反映了期权购买者的风险厌恶。研究者分析了市场波动（一种观察到的现象）的聚类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聚类对这些重要风险度量的影响。这里考虑的主题给出了适用于所使用的均质化技术的多尺度问题的类型。虽然所考虑的具体问题源于金融数学，但自然界中存在着大量的多尺度现象，这些获得这些现象更精确近似值的方法应该得到广泛的应用。

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.