Rate of convergence of option prices by using the method of pseudomoments
Authors:
Yu. S. Mishura and E. Yu. Munchak
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 92 (2016), 117-133
MSC (2010):
Primary 60F15, 91B25, 91G20
DOI:
https://doi.org/10.1090/tpms/987
Published electronically:
August 10, 2016
MathSciNet review:
3553430
Full-text PDF Free Access
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Additional Information
Abstract: A sequence of discrete time financial markets is considered in the scheme of series. The rate of convergence for put and call option prices in the discrete-time model is studied if the prices of risky assets weakly converge to those in the Black–Scholes model. This rate of convergence is of order $O (n ^ {- 1})$, where $n$ is the number of trading periods on a fixed time interval for the prelimit model. The result follows from an author theorem concerning the rate of convergence in the central limit theorem for identically distributed random variables, and the latter is obtained by using the method of pseudomoments.
References
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References
- M. Broadie, O. Glasserman, and S. J. Kou, Connecting discrete continuous path-dependent options, Finance Stoch. 3 (1999), no. 1, 55–82. MR 1805321
- L.-B. Chang and K. Palmer, Smooth convergence in the binomial model, Finance Stoch. 11 (2007), no. 1, 91–105. MR 2284013
- H. Föllmer and A. Schied, Stochastic Finance. An Introduction in Discrete Time, Second revised and extended edition, Studies in Mathematics, vol. 27, Walter de Gruyter, Berlin, 2004. MR 2169807
- S. Heston and G. Zhou, On the rate of convergence of discrete-time contingent claims, Math. Finance 10 (2000), no. 1, 53–75. MR 1743973
- Yu. Mishura, The rate of convergence of option prices on the asset following geometric Ornstein–Uhlenbeck process, Lith. Math. J. 55 (2015), no. 1, 134–149. MR 3323287
- Yu. Mishura, The rate of convergence of option prices when general martingale discrete-time scheme approximated the Black–Scholes model, Advances in mathematics of finance, 151–165, Banach Center Publ., 104, Polish Acad. Sci. Inst. Math., Warsaw, 2015. MR 3363984
- Yu. Mishura, Diffusion approximation of recurrent schemes for financial markets, with application to the Ornstein–Uhlenbeck process, Opuscula Math. 35 (2015), no. 1, 99–116. MR 3282967
- Yu. Mishura, Ye. Munchak, and P. Slyusarchuk, The rate of convergence to the normal law in terms of pseudomoments, Modern Stoch. Theory Appl. 2 (2015), no. 2, 95–106. MR 3389584
- J.-L. Prigent, Weak Convergence of Financial Markets, Springer-Verlag, New York–Heidelberg–Berlin, 2003. MR 2036683
- J. B. Walsh, The rate of convergence of the binomial tree scheme, Finance Stoch. 7 (2003), no. 3, 337–361. MR 1994913
- J. B. Walsh and O. D. Walsh, Embedding and the convergence of the binomial and trinomial tree schemes, Numerical methods and stochastics (Toronto, ON, 1999), 101–121, Fields Inst. Commun., 34, Amer. Math. Soc., Providence, RI, 2002. MR 1944748
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Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, 01601, Kyiv, Ukraine
Email:
myus@univ.kiev.ua
E. Yu. Munchak
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, 01601, Kyiv, Ukraine
Email:
yevheniamunchak@gmail.com
Keywords:
Financial markets in discrete and continuous time,
scheme of series,
pseudomoments,
rate of convergence,
option prices,
Black–Scholes model
Received by editor(s):
May 6, 2015
Published electronically:
August 10, 2016
Article copyright:
© Copyright 2016
American Mathematical Society
