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Pricing contingent claims on stocks driven by Lévy processes. (English) Zbl 1054.91033

This paper studies martingale measures in a model where the asset price \(S\) is given as the stochastic exponential of a Lévy process \(Y\) satisfying an exponential moment condition and having jumps bounded from below; more precisely, \(S = {\mathcal E}\left( \int \sigma(s)\,dY_s + \int b(s)\,ds \right)\) for deterministic continuous functions \(\sigma,b\). The author determines the minimal equivalent martingale measure, a multiplicative variant of this, and shows that the minimal entropy martingale measure is given by a generalized Esscher transform. (This result is different from the one in H. U. Gerber and E. S. W. Shiu [Trans. Soc. Actuar. 69, 99–191 (1994)] because the latter paper considers a model with \(S = \exp( \sigma Y + b t)\).) Numerical examples show that prices computed under these measures differ very substantially.

MSC:

91B28 Finance etc. (MSC2000)
60G35 Signal detection and filtering (aspects of stochastic processes)
60J27 Continuous-time Markov processes on discrete state spaces
60J75 Jump processes (MSC2010)
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