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

2026-05-28

AI Summary

This study addresses the limitations of traditional option pricing models in simultaneously capturing stochastic volatility and jump risk, which often leads to inadequate fits of the implied volatility surface. The authors develop a unified partial integro-differential equation (PIDE) framework by deriving the pricing equation from the infinitesimal generator of an affine Lévy process. They efficiently handle the non-local jump component through a combination of finite difference methods and the fast Fourier transform (FFT), and calibrate the model to S&P 500 option data using generalized method of moments (GMM). Within this unified PIDE setting, they systematically quantify the marginal contributions of stochastic volatility and two distinct jump specifications—Merton and CGMY—to option pricing accuracy. Their results show that jump effects are concentrated in short maturities and deep out-of-the-money options; relative to Black-Scholes, the Heston stochastic volatility reduces implied volatility RMSE by 39%, while the CGMY jump structure yields further modest improvements, supporting a compound Poisson jump mechanism consistent with high-frequency return characteristics.

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 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&P500 index returns.

Problem

• Option Pricing
• Stochastic Volatility
• Jumps
• PIDE

Innovation

• PIDE
• stochastic volatility
• jumps
• FFT-based discretization
• affine Lévy process