Continuous Mixed-Laplace Jump Diffusion Models for Stocks and Commodities

Donatien Hainaut
2017 Quantitative Finance and Economics  
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
more » ... 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 empirical study 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 year... Document type : Article de périodique (Journal article) Référence bibliographique Hainaut, Donatien. Continuous Mixed-Laplace Jump Diffusion Models for Stocks and Commodities. Abstract This paper proposes two jump diusion models with and without mean reversion, for stocks or commodities, capable to t highly leptokurtic processes. The jump component is a continuous mixture of independent point processes with Laplace jumps. As in nancial markets, jumps are caused by the arrival of information and sparse information has usually more importance than regular information, the frequencies of shocks are assumed inversely proportional to their average size. In this framework, we nd analytical expressions for the density of jumps, for characteristic functions and moments of log-returns. Simple series developments of characteristic functions are also proposed. Options prices or densities are retrieved by discrete Fourier transforms. An empirical study demonstrates the capacity of our models to t time series with a high kurtosis. The Continuous Mixed-Laplace Jump Diusion (CMLJD) is tted to four major stocks indices (MS World, FTSE, S&P and CAC 40), over a period of 10 years. The mean reverting CMLJD is tted to four time series of commodity 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 this surface to each parameters is analyzed.
doi:10.3934/ms.2017.2.145 fatcat:olmmto64a5eonkbdbqwjgezg5u