Data Analysis and Related Applications, Volume 1

10Numerical Studies of Implied Volatility Expansions Under the Gatheral Model

The Gatheral model is a three-factor model with mean-reverting stochastic volatility that reverts to a stochastic long-run mean. This chapter reviews previous analytical results on the first- and second-order implied volatility expansions under this model. Using the Monte Carlo simulation as the benchmark method, numerical studies are conducted to investigate the accuracy and properties of these analytical expansions. Moreover, a partial calibration procedure is proposed using these expansions. This calibration procedure is implemented on real market data of daily implied volatility surfaces for an underlying market index and an underlying equity stock for periods both before and during the Covid-19 crisis.

10.1. Introduction

The classical Black–Scholes option pricing model assumes that the underlying asset follows a geometric Brownian motion with constant volatility, but there are a significant number of model extensions to ease this assumption of constant volatility. One of the recent and popular extensions is the Gatheral model, given in Gatheral (2008), where a double-mean-reverting market model is considered. The same model is later considered in Bayer et al. (2013).

Our object of interest is the asymptotic expansions of implied volatility under the Gatheral model presented in earlier research in Albuhayri et al. (2021). Albuhayri et al. (2021) obtained such asymptotic expansions by applying ...