Continuous Mixed-Laplace Jump Diffusion Models for Stocks and Commodities

Donatien Hainaut; Donatien Hainaut

doi:10.3934/QFE.2017.2.145

Quantitative Finance and Economics

2017, Volume 1, Issue 2: 145-173. doi: 10.3934/QFE.2017.2.145

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Continuous Mixed-Laplace Jump Diffusion Models for Stocks and Commodities

Donatien Hainaut ^,

ISBA, Université Catholique de Louvain,Voie du Roman Pays 20, 1348 Louvain-la-Neuve, Belgium.

Received: 09 June 2017 Accepted: 18 June 2017 Published: 14 July 2017

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 a continuous 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 importance than regular information, the frequencies of shocks are assumed inversely proportional to their average size. In this framework, we find 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 fit time series with a high kurtosis. The Continuous Mixed-Laplace Jump Diffusion (CMLJD) is fitted to four major stocks indices (MSWorld, FTSE, S & P and CAC 40), over a period of 10 years. The mean reverting CMLJD is fitted 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.
- jump diffusion model,
- options,
- mixed-exponential distributions,
- double exponential jump diffusion
Citation: Donatien Hainaut. Continuous Mixed-Laplace Jump Diffusion Models for Stocks and Commodities[J]. Quantitative Finance and Economics, 2017, 1(2): 145-173. doi: 10.3934/QFE.2017.2.145

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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 a continuous 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 importance than regular information, the frequencies of shocks are assumed inversely proportional to their average size. In this framework, we find 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 fit time series with a high kurtosis. The Continuous Mixed-Laplace Jump Diffusion (CMLJD) is fitted to four major stocks indices (MSWorld, FTSE, S & P and CAC 40), over a period of 10 years. The mean reverting CMLJD is fitted 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.

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