The well-established theory of large deviations describes the small probabilities that a diffusion process moves from one point to another part of the state space in a short time. This is particularly interesting in the context of financial mathematics in which one is interested in the behavior of the stock price in a short time. For instance, this will be relevant to option prices at short maturities, or probabilities to reach a default level in a short time. Multi-factor stochastic volatility models have been studied intensively in the past years, and they have been found very useful to describe the observed smile/skew of implied volatilities. At short maturities the rate functions appearing in the large deviation estimates will usually not be given in closed form, and will depend on some details of the models like volatility of volatility. On the other hand it has been shown that multiscale stochastic volatility models and their asymptotics are very useful to obtain accurate approximations of option prices and hedging strategies which can be efficiently computed with only a few group parameters easy to calibrate to data. It is then natural to study asymptotic expansions of the rate function in the regime of separation of volatility time scales. In particular, the regime in which the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor turns out to be very interesting and challenging. The research supported by this award is about deriving and fully justifying such asymptotic expansion. The problem will be formulated as logarithmic asymptotic for exponential moments of certain functionals of diffusion processes, which in turn is connected with homogenization/averaging theory for Hamilton-Jacobi-Bellman equations. The collaborative nature of the research combines expertise in the area of multiscale stochastic volatility modeling, in the areas of large deviation theory, and viscosity solution and homogenization/averaging for HJB equations.
This award will support a multidisciplinary research effort. It concerns very practical modeling issues in the area of financial mathematics as well as theoretical convergence results in the theories of large deviations for diffusion processes and homogenization of nonlinear partial differential equations. It is expected that the results obtained on small default probabilities will be very helpful in understanding default mechanisms in credit markets. This research therefore is very timely. The senior PI at UCSB is actively involved in organizing conferences, workshops and special sessions which promote interaction between academic researchers and practitioners. The research will be given full exposure in the activities of the newly created Center for Research in Financial Mathematics and Statistics (CRFMS) at UCSB. The collaborative project will also serve as a training tool for graduate students and postdoctoral fellows at both institutions, UCSB and KU.
