We consider a one-stock market which is driven by a Geometric Levy process. We examine relations between Levy processes and infinite divisible distributions, and also discuss conditions for absolute continuity of measures corresponding to Levy processes.

As such markets are (generally) not complete there is a problem of which equivalent martingale measure to use. We consider as candidates so-called Esscher and minimal entropy martingale measures. Examples of option pricing for Compound Gauss-Poisson, LogStable, Variance-Gamma and other models are given.

The relevance of such models is discussed with the reference to the empirical density of returns of the AUD/USD exchange rate.

References

• Albanese, C., Jaimungal, S. and Rubisov, D. (2001) Jumping in line. Risk Magazine, February, pp 65-68.
• Frittelli, M. (2000) The Minimal Entropy Martingale Measures and the Valuation Problem in Incomplete Markets. Mathematical Finance, 10, No. 1, 39-52.
• Fujiwara, T. and Miyahara, Y. (2001) On the Minimal Entropy Martingale Measure for Geometric Levy Processes. (In preparation).
• Hurst, S.R., Platen, E. and Rachev, S.T. (1999) Option pricing for a logstable asset price model. Math. Comput. Modelling, 29 , no. 10-12, 105--119.
• Madan, D., Carr, P. and Chang, E. (1998) The variance gamma process and option pricing. European Finance Review, v.2, 79-105.
• Sato, K. (1999) Levy Processes and Infinitely Divisible Distributions. Cambridge University Press.

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