Option Pricing under Stochastic Volatility and Jumps: A PIDE Framework with Empirical Evidence

Abstract

We develop a partial integro-differential equation (PIDE) framework for option pricing under joint stochastic volatility and jump dynamics, and evaluate its empirical content using the S&P 500 index option contracts across three maturities. The framework is derived from the infinitesimal generator of an affine Lévy-type process and implemented via finite-difference discretization with FFT-based treatment of the nonlocal jump operator. Calibration via GMM reveals that stochastic volatility accounts for the dominant share of pricing improvement, where relative to Black-Scholes, the Heston specification reduces implied-volatility RMSE by 39%. Jump augmentation via either Merton or CGMY specifications yields marginal improvements concentrated at short maturities and in the deep out-of-the-money region. The calibrated CGMY activity index supports a compound-Poisson structure, consistent with high-frequency evidence on S&P 500 index returns.

Keywords: Stochastic Volatility, Jump-Diffusion, Partial Integro-Differential Equations, Option Pricing, CGMY, L´Evy Measure

JEL Classification: C58, C63, G12, G14