Enhancing Option Pricing with Jump-Diffusion and Tail-Sensitive Monte Carlo Methods

This paper enhances classical option pricing methods by integrating jump diffusion dynamics and tail-sensitive Monte Carlo simulations. We begin with a comparison of the Geometric Brownian Motion (GBM) and Merton Jump-Diffusion (JD) models, highlighting the JD models superior ability to capture extreme market events. By incorporating Poisson-distributed jumps and log-normal amplitudes, our JD framework reflects discontinuous price behaviors commonly observed during financial shocks. Using Its Lemma, we derive the stochastic differential form of asset evolution and implement numerical estimations of Delta, Gamma, and Vega. To better reflect tail risk, we introduce a tail reweighting function that amplifies sensitivity metrics in extreme price regions. Additionally, we explore variance reduction techniquesAntithetic Variables, Sobol Sequences, and Control Variatesand compare their performance in terms of RMSE, convergence, and runtime. Our findings show that Antithetic and Sobol methods significantly improve stability, while tail-weighted Greeks offer better alignment with real-world market sensitivities. Finally, model calibration using historical return data demonstrates that the JD model yields more accurate option prices and risk assessments under volatile conditions. This work offers a reproducible Python-based framework for advanced pricing and risk management in quantitative finance.