Stochastic homogenization and its applications to financial mathematics

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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.