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), 135148
MSC (2010):
Primary 60J05, 60H10; Secondary 91Gxx, 91B70
Published electronically:
February 2, 2012
MathSciNet review:
2768854
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Additional Information
Abstract: We consider an optimal reselling problem for a European option. A modification of the CoxIngersollRoss process is used to model the implied volatility. We construct a twodimensional binomialtrinomial exponential approximation instead of the discrete approximation proposed by Pupashenko and Kukush (2008) in Theory Stoch. Process. 14(30), no. 34, 114128. We use the results concerning the convergence of reward functionals for exponential price processes with independent logincrements obtained by Lundgren et al.(2008) in J. Numer. Appl. Math. 1(96), 90113. We proved that there is no arbitrage strategy in the proposed discrete model.
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G. Kukush, Yu.
S. Mishura, and G.
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Process. 12 (2006), no. 34, 75–87. MR 2316567
(2008e:62171)
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R. Lundgren, D. Silvestrov, and A. Kukush, Reselling of options and convergence of option rewards, J. Numer. Appl. Math. 1(96) (2008), 90113.
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Mykhailo
Pupashenko and Alexander
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S. E. Shreve, Lectures on Stochastic Calculus and Finance, Springer, New York, 1997.
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 M. M. Leonenko, Yu. S. Mishura, V. M. Parkhomenko, and M. I. Yadrenko, ProbabilityTheoretical and Statistical Methods in Economics and Finance Mathematics, Informtekhnika, Kyiv, 1995. (Ukrainian)
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 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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 J. Cox, J. Ingersoll, and S. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), 385407. MR 785475
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 A. G. Kukush, Yu. S. Mishura, and G. M. Shevchenko, On reselling of European option, Theory Stoch. Process. 12(28) (2006), 7587. MR 2316567 (2008e:62171)
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 R. Lundgren, D. Silvestrov, and A. Kukush, Reselling of options and convergence of option rewards, J. Numer. Appl. Math. 1(96) (2008), 90113.
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 M. Pupashenko and A. Kukush, Reselling of European option if the implied volatility varies as CoxIngersollRoss process, Theory Stoch. Process. 14(30) (2008), no. 34, 114128. 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/S009490002012008478
Keywords:
European option,
American option,
reselling problem,
reward,
convergence,
optimal stopping time,
discrete approximation,
Markov process,
binomialtrinomial approximation,
Cox–Ingersoll–Ross process
Received by editor(s):
April 22, 2010
Published electronically:
February 2, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
