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Pricing Contingent Claims on Stocks Driven by Lévy Processes

Terence Chan
The Annals of Applied Probability
Vol. 9, No. 2 (May, 1999), pp. 504-528
Stable URL: http://www.jstor.org/stable/2667343
Page Count: 25
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Pricing Contingent Claims on Stocks Driven by Lévy Processes
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Abstract

We consider the problem of pricing contingent claims on a stock whose price process is modelled by a geometric Lévy process, in exact analogy with the ubiquitous geometric Brownian motion model. Because the noise process has jumps of random sizes, such a market is incomplete and there is not a unique equivalent martingale measure. We study several approaches to pricing options which all make use of an equivalent martingale measure that is in different respects "closest" to the underlying canonical measure, the main ones being the Föllmer-Schweizer minimal measure and the martingale measure which has minimum relative entropy with respect to the canonical measure. It is shown that the minimum relative entropy measure is that constructed via the Esscher transform, while the Follmer-Schweizer measure corresponds to another natural analogue of the classical Black-Scholes measure.

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