Data Analysis and Related Applications 3

8Numerical Studies of Implied Volatility Expansions Under the Gatheral Model

The Gatheral double stochastic volatility model is a three-factor model with mean-reverting stochastic volatility that reverts to a stochastic long-run mean. Our previous paper investigated the performance of the first- and second-order implied volatility expansions under this model. Moreover, a simple partial calibration method has been proposed. This chapter reviews and extends previous results to the third-order implied volatility expansions under the same model. Using the Monte Carlo simulation as the benchmark method, extensive numerical studies are conducted to investigate the accuracy and properties of the third-order expansion.

8.1. Introduction

Since the advent of the Black–Scholes model (1973), option pricing has been under the spotlight and received significant attention. The model was the first nontrivial model that was completely solved in the sense that it gives mathematical rigor to option pricing. It calculates the theoretical price of European-style options that is unique regardless of the underlying asset’s expected returns. Although the Black–Scholes model has been incredibly successful in the financial world, with their finding winning a Nobel prize in 1997, it was not without its drawbacks. The main issue of the model is the assumption that the underlying asset follows a geometric Brownian motion with constant volatility. This was strongly conflicting with empirical findings. ...