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Theory of Probability and Mathematical Statistics

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Convergence of option rewards for Markov type price processes modulated by stochastic indices. I

Authors: D. S. Silvestrov, H. Jönsson and F. Stenberg
Translated by: the authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 79 (2008).
Journal: Theor. Probability and Math. Statist. 79 (2009), 153-170
MSC (2000): Primary 60J05, 60H10; Secondary 91B28, 91B70
Published electronically: December 28, 2009
MathSciNet review: 2494545
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Abstract: A general price process represented by a two-component Markov process is considered. Its first component is interpreted as a price process and the second component as an index process modulating the price component. American type options with pay-off functions admitting power type upper bounds are studied. Both the transition characteristics of the price processes and the pay-off functions are assumed to depend on a perturbation parameter $ \delta \geq 0$ and to converge to the corresponding limit characteristics as $ \delta \rightarrow 0$. In the first part of the paper, asymptotically uniform skeleton approximations connecting reward functionals for continuous and discrete time models are given. In the second part of the paper, these skeleton approximations are used for obtaining results about the convergence of reward functionals of American type options with perturbed price processes for both cases of discrete and continuous time. Examples related to modulated exponential price processes with independent increments are given.

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Additional Information

D. S. Silvestrov
Affiliation: Mälardalen University, Västerås, Sweden

H. Jönsson
Affiliation: Eurandom, Eindhoven University of Technology, The Netherlands

F. Stenberg
Affiliation: Nordea bank, Stockholm, Sweden

Keywords: Reward, convergence, optimal stopping, American option, skeleton approximation, Markov process, price process, modulation, stochastic index
Received by editor(s): August 25, 2008
Published electronically: December 28, 2009
Article copyright: © Copyright 2009 American Mathematical Society