Jump-Diffusion Models for Option Pricing versus the Black Scholes Model

Abstract

In general, the daily logarithmic returns of individual stocks are not normally distributed. This poses a challenge when trying to compute the most accurate option prices. This thesis investigates three different models for option pricing, The Black Scholes Model (1973), the Merton Jump-Diffusion Model (1975) and the Kou Double-Exponential Jump-Diffusion Model (2002). The jump-diffusion models do not make the same assumption as the Black Scholes model regarding the behavior of the underlying assets’ returns; the assumption of normally distributed logarithmic returns. This could make the models more able to produce accurate results. Both the Merton Jump-Diffusion Model and the Kou Double-Exponential Jump-Diffusion Model shows promising results, especially when looking at how they are able to reproduce the leptokurtic feature and to some extent the “volatility smile”. However, because the observed implied volatility surface is skewed and tends to flatten out for longer maturities, the two models abilities to produce accurate results are reduced. And while visual study reveals some difference between the models, the results are not significant.