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

Published electronically:
February 2, 2012

MathSciNet review:
2768854

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Abstract | References | Similar Articles | 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.

**1.**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. 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****3.**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**, 10.2307/1911242**4.**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****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.**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****7.**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:
https://doi.org/10.1090/S0094-9000-2012-00847-8

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):
April 22, 2010

Published electronically:
February 2, 2012

Article copyright:
© Copyright 2012
American Mathematical Society