A jump diffusion model for option pricing with three properties: leptokurtic feature, volatility smile, and analytical tractability

Abstract

Brownian motion and normal distribution have been widely used to study option pricing and the return of assets. Despite the successes of the Black-Scholes-Merton model based on Brownian motion and normal distribution, two puzzles which emerged from many empirical investigations, have had much attention recently: 1) the leptokurtic and asymmetric features; 2) the volatility smile. Much research has been conducted on modifying the Black-Scholes models to explain the two puzzles. To incorporate the leptokurtic and asymmetric features, a variety of models have been proposed. The article proposes a novel model which has three properties: 1) it has leptokurtic and asymmetric features, under which the return distribution of the assets has a higher peak and two heavier tails than the normal distribution, especially the left tail; 2) it leads to analytical solutions to many option pricing problems, including: call and put options, and options on futures; interest rate derivatives such as caplets, caps, and bond options; exotic options, such as perpetual American options, barrier and lookback options; 3) it can reproduce the "volatility smile".