Theory of Probability and Mathematical Statistics
(Teoriya imovirnostey ta matematychna statystyka)
An application of the Malliavin calculus for calculating the precise and approximate prices of options with stochastic volatility
S. V. Kuchuk-Yatsenko, Yu. S. Mishura, Ye. Yu. Munchak
Abstract: The article is devoted to models of financial markets with stochastic volatility, which is defined by a functional of Ornstein-Uhlenbeck process or CoxIngersoll-Ross process. We study the question of exact price of European option. The form of the density function of the random variable, which expresses the average of the volatility over time to maturity is established using Malliavin calculus.The result allows calculate the price of the option with respect to minimum martingale measure when the Wiener process driving the evolution of asset price and the Wiener process, which defines volatility, are uncorrelated.
Keywords: Black-Scholes model, stochastic volatility, option pricing, Malliavin calculus.
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