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

Abigail Anokyewaa Mensah, Ayush Jha, Hongwei Mei, Rui Wang, Svetlozar T. Rachev and Frank J. Fabozzi

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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&P500 index option contracts across three maturities. The framework is derived from the infinitesimal generator of an affine L\'evy-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&P500 index returns.

Date: 2026-05
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