Applied Modeling Techniques and Data Analysis 2

2Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model

Gatheral’s (2008) double-mean-reverting model by is motivated by empirical dynamics of the variance of stock price. No closed-form solution for European option exists in the above model. In this chapter, we study the behavior of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci (2017), we explicitly calculate the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

2.1. Introduction

The history of implied volatility can be traced back at least to Latané and Rendleman (1976), where it appeared under the name “implied standard deviation”, i.e. the standard deviation of asset returns, which are implied in actual European call option prices when investors price options according to the Black-Scholes model. For a recent review of different approaches to determine implied volatility, see Orlando and Taglialatela (2017). To give exact definitions, we use Pagliarani and Pascucci (2017).

In order to briefly explain our contribution to the subject, we will introduce some notations. Let d ≥ 2 be a positive integer, let T₀ > 0 be a time horizon, let T ∈ (0, T₀], and let { Z_t : 0 ≤ t ≤ T } be a continuous ℝ^d-valued adapted Markov stochastic process on a probability space with a filtration . Assume that the first coordinate S_t of the process Z_t represents ...