An application of the Malliavin calculus for calculating the precise and approximate prices of options with stochastic volatility

This paper is devoted to mathematical models of financial markets with stochastic volatility defined as a functional of either the Ornstein-Uhlenbeck process or Cox-Ingersoll-Ross process. We study the question on the exact price of a European type option. Using Malliavin calculus, we establish the probability density of the average value of the volatility in the time interval until the maturity. This result allows us to express the price of an option in terms of the minimum martingale measure for the case where the Wiener process driving the evolution of asset prices is uncorrelated with the Wiener process that defines the volatility.