Continuous Mixed-Laplace Jump Diffusion Models for Stocks and Commodities

Abstract

This paper proposes two jump diffusion models with and without mean reversion,for stocks or commodities, capable to fit highly leptokurtic processes. The jump component is acontinuous mixture of independent point processes with Laplace jumps. As in financial markets,jumps are caused by the arrival of information and sparse information has usually more importancethan regular information, the frequencies of shocks are assumed inversely proportional to their averagesize. In this framework, we find analytical expressions for the density of jumps, for characteristicfunctions and moments of log-returns. Simple series developments of characteristic functions arealso proposed. Options prices or densities are retrieved by discrete Fourier transforms. An empiricalstudy demonstrates the capacity of our models to fit time series with a high kurtosis. The ContinuousMixed-Laplace Jump Diffusion (CMLJD) is fitted to four major stocks indices (MSWorld, FTSE, S&Pand CAC 40), over a period of 10 years. The mean reverting CMLJD is fitted to four time series ofcommodity prices (Copper, Soy Beans, Crude Oil WTI and Wheat), observed on four years. Finally,examples of implied volatility surfaces for European Call options are presented. The sensitivity of thissurface to each parameters is analyzed.

Keywords

• jump diffusion model
• options
• mixed-exponential distributions
• double exponential jump diffusion