A Jump-Diffusion Model for Option Pricing

TLDR

Black–Scholes relies on Brownian motion and normal returns, yet empirical studies reveal leptokurtic distributions and volatility smiles. This paper introduces a double‑exponential jump‑diffusion model to address these empirical anomalies in option pricing. The model remains analytically tractable, yielding closed‑form solutions for calls, puts, interest‑rate derivatives, and path‑dependent options. Equilibrium analysis and a psychological interpretation of the model are also provided.

Abstract

Brownian motion and normal distribution have been widely used in the Black–Scholes option-pricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called “volatility smile” in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jump-diffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of option-pricing problems, including call and put options, interest rate derivatives, and path-dependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.