A Jump Diffusion Model for Option Pricing

TL;DR: In this article, a double exponential jump diffusion model is proposed for option pricing, which is based on the Brownian motion and normal distribution to model the return of assets, and it is shown that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution.

Abstract: _Content: 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 abnormity called "volatility smile'' in option pricing. To incorporate both of them, this paper proposes, for the purpose of option pricing, a double exponential jump diffusion model. The main attraction of the model is its simplicity. In particular, it is simple enough to derive analytical solutions for a variety of option pricing problems, including call and put options, interest rate derivatives and path-dependent options; it seems impossible for many other alternative models to do this. Equilibrium analysis and a psychological interpretation of the model are also presented.