Convergence of reward functionals in a reselling model for a European option
Author:
M. S. Pupashenko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 83 (2011), 135-148
MSC (2010):
Primary 60J05, 60H10; Secondary 91Gxx, 91B70
DOI:
https://doi.org/10.1090/S0094-9000-2012-00847-8
Published electronically:
February 2, 2012
MathSciNet review:
2768854
Full-text PDF Free Access
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Additional Information
Abstract: We consider an optimal reselling problem for a European option. A modification of the Cox–Ingersoll–Ross process is used to model the implied volatility. We construct a two-dimensional binomial-trinomial exponential approximation instead of the discrete approximation proposed by Pupashenko and Kukush (2008) in Theory Stoch. Process. 14(30), no. 3–4, 114–128. We use the results concerning the convergence of reward functionals for exponential price processes with independent log-increments obtained by Lundgren et al. (2008) in J. Numer. Appl. Math. 1(96), 90–113. We proved that there is no arbitrage strategy in the proposed discrete model.
References
- M. M. Leonenko, Yu. S. Mishura, V. M. Parkhomenko, and M. I. Yadrenko, Probability-Theoretical and Statistical Methods in Economics and Finance Mathematics, Informtekhnika, Kyiv, 1995. (Ukrainian)
- A. V. Skorohod, Random processes with independent increments, Mathematics and its Applications (Soviet Series), vol. 47, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the second Russian edition by P. V. Malyshev. MR 1155400
- John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), no. 2, 385–407. MR 785475, DOI https://doi.org/10.2307/1911242
- 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
- R. Lundgren, D. Silvestrov, and A. Kukush, Reselling of options and convergence of option rewards, J. Numer. Appl. Math. 1(96) (2008), 90–113.
- Mykhailo Pupashenko and Alexander Kukush, Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process, Theory Stoch. Process. 14 (2008), no. 3-4, 114–128. MR 2498609
- S. E. Shreve, Lectures on Stochastic Calculus and Finance, Springer, New York, 1997.
References
- M. M. Leonenko, Yu. S. Mishura, V. M. Parkhomenko, and M. I. Yadrenko, Probability-Theoretical and Statistical Methods in Economics and Finance Mathematics, Informtekhnika, Kyiv, 1995. (Ukrainian)
- A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964; English transl., Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1155400 (93a:60114)
- J. Cox, J. Ingersoll, and S. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), 385–407. MR 785475
- A. G. Kukush, Yu. S. Mishura, and G. M. Shevchenko, On reselling of European option, Theory Stoch. Process. 12(28) (2006), 75–87. MR 2316567 (2008e:62171)
- R. Lundgren, D. Silvestrov, and A. Kukush, Reselling of options and convergence of option rewards, J. Numer. Appl. Math. 1(96) (2008), 90–113.
- M. Pupashenko and A. Kukush, Reselling of European option if the implied volatility varies as Cox–Ingersoll–Ross process, Theory Stoch. Process. 14(30) (2008), no. 3–4, 114–128. MR 2498609 (2010h:62318)
- S. E. Shreve, Lectures on Stochastic Calculus and Finance, Springer, New York, 1997.
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Additional Information
M. S. Pupashenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
myhailo.pupashenko@gmail.com
Keywords:
European option,
American option,
reselling problem,
reward,
convergence,
optimal stopping time,
discrete approximation,
Markov process,
binomial-trinomial approximation,
Cox–Ingersoll–Ross process
Received by editor(s):
April 22, 2010
Published electronically:
February 2, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
