Archil Gulisashvili (Ohio University)

Distribution densities in stochastic volatility models

Stochastic volatility models were introduced in 1980s-1990s. In these models, the volatility of a stock is described by a stochastic process. For instance, in the Hull-White model, the volatility is a geometric Brownian motion, in the Stein-Stein model, the absolute value of an Ornstein-Uhlenbeck process plays the role of the volatility of a stock, and in the Heston model, the volatility is a square root process. The main object of our interest in this work is the distribution density of the stock price in a stochastic volatility model. We find explicit formulas for leading terms in asymptotic expansions of such densities and give error estimates. Using these results, we compare ``fat tails" of stock price distributions in various stochastic volatility models. We also obtain a sharp asymptotic formula for the law of a mean-square average of the volatility process. As an application of our methods, the asymptotic behavior of the implied volatility is characterized, and a sharp asymptotic formula for the price of an Asian option is obtained.
This is a joint work with E. M. Stein (Princeton University).