Efficient 'Black-Scholes' Approximations of the Heston and Multi-Factor Heston Option Prices for a Correct Spot Price

Abstract

This paper proposes Black-Scholes closed-form approximations for a correct spot price to estimate prices of European call and put options in the Heston and multi-factor Heston stochastic volatility models. These approximations are inspired by the Romano and Touzi (1997) formula and are obtained by a perturbation solution to the Riccati equations which define the probability density function of the log-price process as the volatility of volatility (vol of vol for short) goes to zero. This approach, which can be extended to affine jump models, shows that the accuracy of these Black-Scholes formulas depends on how fast the Airy function for a specific argument depending on vol of vol converges to the Dirac delta function when the vol of vol goes to zero. Simulations and empirical studies show that performance of these formulas is twofold. They outperform the Heston and multi-factor Heston option pricing formula in terms of computational time while satisfactorily approximating the model option prices so they can be used to develop fast and efficient procedures to estimate the model parameters. Finally, this perturbation approach provides an alternative way to approximate the implied volatility for small values of the vol of vol for models in the larger affine class.