Dr. Rohini Kumar (UCSB Statistics and Applied Probability)

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 $\epsilon\ll 1$ 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 $\epsilon\to 0$. 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. %A closed form formula for the rate function is given for the regime $\alpha=4$ and a variational representation of the rate function is obtained in the other regime. 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. [abstract in PDF]