Rate of convergence of option prices for approximations of the geometric Ornstein–Uhlenbeck process by Bernoulli jumps of prices on assets
Authors:
Yu. S. Mishura and Ye. Yu. Munchak
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 93 (2016), 137-152
MSC (2010):
Primary 91B24, 91B25, 91G20
DOI:
https://doi.org/10.1090/tpms/999
Published electronically:
February 7, 2017
MathSciNet review:
3553446
Full-text PDF Free Access
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Additional Information
Abstract: We consider the discrete approximation scheme for the price of an asset that is modeled by the geometric Ornstein–Uhlenbeck process. The approximation scheme corresponds to Euler type discrete-time approximations where the increments of the Wiener process are changed by independent identically distributed Bernoulli random variables. The rate of convergence of both objective and fair option prices is estimated by using the classical results on the rate of convergence to the normal law of the distribution functions of sums of identically distributed random variables. We analyze option prices and specific changes in a model where the martingale measure is used instead of the objective measure.
References
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- Yuliya 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, DOI 10.7494/OpMath.2015.35.1.99
- Yuliya Mishura, The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black-Scholes model, Advances in mathematics of finance, Banach Center Publ., vol. 104, Polish Acad. Sci. Inst. Math., Warsaw, 2015, pp. 151–165. MR 3363984, DOI 10.4064/bc104-0-8
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- John B. Walsh, The rate of convergence of the binomial tree scheme, Finance Stoch. 7 (2003), no. 3, 337–361. MR 1994913, DOI 10.1007/s007800200094
References
- M. Broadie, O. Glasserman, and S. J. Kou, Connecting discrete continuous path-dependent options, Finance Stochast. 3 (1999), no. 1, 55–82. MR 1805321
- L.-B. Chang and K. Palmer, Smooth convergence in the binomial model, Finance Stochast. 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, 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, 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, The rate of convergence of option prices when general martingale discrete-time scheme approximates the Black–Scholes model, Banach Center Publications. Advances in Mathematics of Finance. 104 (2015), 151–165. MR 3363984
- 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, Ye. Munchak, and P. Slyusarchuk, The rate of convergence to the normal law in terms of pseudomoments, Mod. Stoch. Theory Appl. 2 (2015), no. 2, 95–106. MR 3389584
- Yu. S. Mishura and E. Yu. Munchak, Rate of convergence of option prices by using the method of pseudomoments, Teor. Ĭmov$\bar {\text {\i }}$r. Mat. Stat. 92 (2015) 110–124; English transl. in Theor. Probability and Math. Statist. 92 (2016), 117–133.
- J. B. Walsh, The rate of convergence of the binomial tree scheme, Finance Stochast. 7 (2003), no. 3, 337–361. MR 1994913
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Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
myus@univ.kiev.ua
Ye. Yu. Munchak
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
yevheniamunchak@gmail.com
Keywords:
Financial market,
rate of convergence,
Ornstein–Uhlenbeck process,
option prices
Received by editor(s):
June 11, 2015
Published electronically:
February 7, 2017
Additional Notes:
This paper was prepared following the talk at the International conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015
Article copyright:
© Copyright 2017
American Mathematical Society
