Closed-Form Expansion of Option Prices under Stochastic Volatility Model

Under the stochastic volatility model, European option prices usually have no analytical solutions. Based on the theory of Malliavin calculus, this paper proposes a new asymptotic expansion method to approximate the price of such European options by expanding the stochastic volatility path around a constant volatility model. This method provides a closed-form expansion formula that can correct the mispricing problem and generalize the famous Black-Scholes-Merton (1973) formula. Inspired by Li¹, our asymptotic expansion can achieve correction terms of arbitrary order, resulting in accurate and effcient outcomes. As an illustrative example, this paper uses a stochastic volatility model based on the GARCH diffusion process in the numerical results section.