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Convergence of reward functionals in a reselling model for a European option
Author:
M. S. Pupashenko
Translated by:
S. Kvasko
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 83 (2010).
Journal:
Theor. Probability and Math. Statist. 83 (2011), 135-148
MSC (2010):
Primary 60J05, 60H10; Secondary 91Gxx, 91B70
Posted:
February 2, 2012
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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
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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)
- 2.
A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964; English transl., Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1155400 (93a:60114)
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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, http://dx.doi.org/10.2307/1911242
- 4.
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)
- 5.
R. Lundgren, D. Silvestrov, and A. Kukush, Reselling of options and convergence of option rewards, J. Numer. Appl. Math. 1(96) (2008), 90-113.
- 6.
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)
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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
DOI:
http://dx.doi.org/10.1090/S0094-9000-2012-00847-8
PII:
S 0094-9000(2012)00847-8
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):
22/APR/2010
Posted:
February 2, 2012
Article copyright:
© Copyright 2012 American Mathematical Society
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