### Talk by Rohini Kumar (UCSB)

Title: Small time asymptotics for fast mean-reverting stochastic volatility models

Abstract: An understanding of small-time behavior of stock price, S_t, given by a stochastic volatility model, is useful for computing prices of options with short maturity. In our model, S_t satisfies a stochastic differential equation with volatility (diffusion coefficient) given by a fast mean-reverting process. We take the time scale to be of order \epsilon << 1 and the mean reversion time of the volatility process to be of order \epsilon^{\alpha}; we consider two regimes: \alpha = 2 and \alpha = 4. This separation of time scales leads to an averaging/homogenization type of problem as \epsilon -> 0. Viscosity solution techniques are used to obtain a large deviation principle (LDP) for stock price - an LDP is a statement about the rate of exponential decay in the probabilities of rare events. Asymptotics of option prices and implied volatility are obtained as a consequence of the LDP. This is joint work with Prof. Fouque at UCSB and Prof. Jin Feng at Kansas University.