The structure of the stopping region in a Lévy model
Authors:
A. G. Moroz and G. M. Shevchenko
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 84 (2012), 107-115
MSC (2010):
Primary 60G40, 60G51
DOI:
https://doi.org/10.1090/S0094-9000-2012-00864-8
Published electronically:
July 31, 2012
MathSciNet review:
2857421
Full-text PDF Free Access
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Additional Information
Abstract: The optimal stopping problem in a Lévy model is investigated. We show that the stopping region is nonempty for a wide class of models and payoff functions. In the general case, we establish sufficient conditions on the payoff function that provide nonemptiness of the stopping region. For a zero discounting rate we also give conditions for the stopping region to have a threshold structure.
References
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- A. G. Kukush, Yu. S. Mishura, and G. M. Shevchenko, On reselling of European option, Theory Stoch. Process. 12 (2006), no. 3-4, 75–87. MR 2316567
- A. Moroz and G. Shevchenko, Asymptotic behavior of the American type option prices in the Lévy model if the time interval is extending unboundedly, Visn. Kyiv Univ. Mat. Mech. 24 (2010), 39–43. (Ukrainian)
- H. Jönsson, A. G. Kukush, and D. S. Silvestrov, Threshold structure of optimal stopping strategies for American type option. I, Teor. Ĭmovīr. Mat. Stat. 71 (2004), 82–92; English transl., Theory Probab. Math. Statist. 71 (2005), 93–103. MR 2144323, DOI https://doi.org/10.1090/S0094-9000-06-00650-8
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- A. Papapantoleon, An Introduction to Lévy Processes with Applications in Finance, Lecture notes, 2008; arXiv/0804.0482.
- Philip E. Protter, Stochastic integration and differential equations, 2nd ed., Applications of Mathematics (New York), vol. 21, Springer-Verlag, Berlin, 2004. Stochastic Modelling and Applied Probability. MR 2020294
References
- S. Villeneuve, Exercise regions of American options on several assets, Finance Stoch. 3 (1999), no. 3, 295–322.
- D. Lamberton and M. Mikou, The critical price for the American put in an exponential Lévy model, Finance Stoch. 12 (2008), no. 4, 561–581. MR 2447412 (2009j:91100)
- A. Kukush, Yu. Mishura, and G. Shevchenko, On reselling of European option, Theory Stoch. Process. 12(28) (2006), no. 1–2, 75–87. MR 2316567 (2008e:62171)
- A. Moroz and G. Shevchenko, Asymptotic behavior of the American type option prices in the Lévy model if the time interval is extending unboundedly, Visn. Kyiv Univ. Mat. Mech. 24 (2010), 39–43. (Ukrainian)
- H. Jönsson, A. G. Kukush, and D. S. Silvestrov, Threshold structure of optimal stopping strategies for American type option. I, Teor. Imovir. Matem. Statist. 71 (2004), 82–92; English transl. in Theor. Probability and Math. Statist. 71 (2005), 93–103. MR 2144323 (2006h:91075)
- H. Jönsson, A. G. Kukush, and D. S. Silvestrov, Threshold structure of optimal stopping strategies for American type option. II, Teor. Imovir. Matem. Statist. 72 (2005), 42–53; English transl. in Theor. Probability and Math. Statist. 72 (2006), 47–58. MR 2168135 (2006i:62070)
- A. Papapantoleon, An Introduction to Lévy Processes with Applications in Finance, Lecture notes, 2008; arXiv/0804.0482.
- P. E. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, Berlin–Heidelberg, 2004. MR 2020294 (2005k:60008)
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Additional Information
A. G. Moroz
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine
Email:
mag-87@inbox.ru
G. M. Shevchenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Lévy processes,
American option,
payoff function,
stopping region,
threshold structure
Received by editor(s):
April 11, 2011
Published electronically:
July 31, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
