A multivariate Lévy process model with linear correlation

Abstract

In this paper, we develop a multivariate risk-neutral Lévy process model and discuss its applicability in the context of the volatility smile of multiple assets. Our formulation is based upon a linear combination of independent univariate Lévy processes and can easily be calibrated to a set of one-dimensional marginal distributions and a given linear correlation matrix. We derive conditions for our formulation and the associated calibration procedure to be well-defined and provide some examples associated with particular Lévy processes permitting a closed-form characteristic function. Numerical results of the option premiums on three currencies are presented to illustrate the effectiveness of our formulation with different linear correlation structures.

Keywords:

• Volatility modelling
• Stochastic jumps
• Non-Gaussian option pricing
• Non-Gaussian distributions
• Multivariate volatility
• Model calibration
• Mathematical finance
• Implementation of pricing

Acknowledgements

Part of this work was carried out at Daiwa Securities SMBC Co. Ltd. The author would like to thank Tatsuya Toshida and the Financial Engineering team for their support and encouragement.