Abstracts

This paper proposes a jump diffusion model with and without mean reversion, for stocks and commodities prices in which jump sizes are continuous mixtures of Laplace random variables. The jump component is like an infinite sum of jumps whith Laplace distributed amplitudes. The jump times and the jump amplitudes are respectively interpreted as dates of information disclosures and as their impact on asset prices. Given that low frequency information has usually a bigger impact on asset prices than frequent information, the frequencies of jumps decrease and are inversely proportional to the average size of jumps. In this framework, we infer analytical expressions of distribution of jumps size, characteristic functions and moments. Simple series developments of characteristic functions are also proposed and options prices and densities are retrieved by discrete Fourier transforms. To motivate this research from an econometric point of view, we provide some empirical evidence is presented about the ability of the Continuous Mixed-Laplace Jump Diffusion (CMLJD) to represent daily returns of four major stocks indices (MS World, FTSE, S&P and CAC 40), over a period of 10 years. So as to illustrate its utility, the mean reverting CMLJD is fitted to four time series of commodity prices that exhibit this feature (Copper, Soy Beans, Crude Oil WTI and Wheat, observed on four years). Finally, examples of implied volatility surfaces for European Call options are presented. And the sensitivity of this surface to each parameters is analyzed.