Viscosity solutions and the pricing of European-style options in a Markov-modulated exponential Lévy model

ABSTRACT

In this paper, we investigate the pricing problem of a European-style contingent claim under a Markov-modulated exponential Lévy model. One of the main feature of this model is the modulator factor which takes into account the empirical facts observed in asset prices dynamics such as the long-term (stochastic) variability and time inhomogeneities. Using the viscosity solutions framework, we show that the value of a European-style option is the unique viscosity solution of a system of coupled linear Partial Integro-Differential Equations when the payoff function satisfies a Lipschitz condition. Moreover, we propose a numerical scheme for approximating solution of this system and discuss its stability, consistency and convergence.

KEYWORDS:

• Viscosity solutions
• Lévy process
• regime-switching model
• partial integro-differential equations
• finite differences method
• comparison principle

Acknowledgements

We wish to thank here Manuel Morales, Mustapha Pemy and an anonymous referee for helpful comments and suggestions which have helped to improve the quality of the present paper.

Disclosure statement

No potential conflict of interest was reported by the author.