Empirical Calibration and Minimum-Variance Delta Under Log-Normal Stochastic Volatility Dynamics

Abstract

We consider calibration of log-normal stochastic volatility model and computation of option delta consistently with statistical dynamics of the asset price and its implied volatility surface. We introduce the concept of volatility skew-beta which serves as an empirical adjustment for empirical option delta. We show how to calibrate the model and make it consistent with any dynamics of implied volatility under the statistical measure and reproduce empirical option delta. The calibrated model minimizes realized volatility of delta-hedging P&L-s, especially so for non-vanilla options. We present empirical investigation using implied and realized volatilities of four major stock indices (S&P 500, FTSE 100, Nikkei 225, and STOXX 50) to validate the assumption about log-normality of both implied and realized volatilities.

Keywords: Lognormal stochastic volatility, Jumps in price and volatility, Model calibration, Implied volatility skew, Closed-form solution, Option pricing, Minimum-variance hedging

JEL Classification: C00, G00