Application of Malliavin calculus to exact and approximate option pricing under stochastic volatility

TL;DR

This paper applies Malliavin calculus to derive exact and approximate pricing formulas for European options in stochastic volatility models driven by Ornstein-Uhlenbeck or Cox-Ingersoll-Ross processes, enabling more precise valuation methods.

Contribution

It introduces a Malliavin calculus-based approach to obtain the density function of average volatility, facilitating exact option pricing under specific stochastic volatility models.

Findings

Derived the density function of average volatility using Malliavin calculus.

Provided a method to calculate option prices under the minimum martingale measure.

Addressed uncorrelated Wiener processes for asset and volatility evolution.

Abstract

The article is devoted to models of financial markets with stochastic volatility, which is defined by a functional of Ornstein-Uhlenbeck process or Cox-Ingersoll-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.