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
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 93 (2015).
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
Abstract | References | Similar Articles | 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.
- 1. Mark Broadie, Paul Glasserman, and S. G. Kou, Connecting discrete and continuous path-dependent options, Finance Stoch. 3 (1999), no. 1, 55–82. MR 1805321, https://doi.org/10.1007/s007800050052
- 2. Lo-Bin Chang and Ken Palmer, Smooth convergence in the binomial model, Finance Stoch. 11 (2007), no. 1, 91–105. MR 2284013, https://doi.org/10.1007/s00780-006-0020-6
- 3. Hans Föllmer and Alexander Schied, Stochastic finance, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 2004. An introduction in discrete time. MR 2169807
- 4. Steve Heston and Guofu Zhou, On the rate of convergence of discrete-time contingent claims, Math. Finance 10 (2000), no. 1, 53–75. MR 1743973, https://doi.org/10.1111/1467-9965.00080
- 5. 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, https://doi.org/10.7494/OpMath.2015.35.1.99
- 6. 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, https://doi.org/10.4064/bc104-0-8
- 7. Yuliya Mishura, The rate of convergence of option prices on the asset following a geometric Ornstein-Uhlenbeck process, Lith. Math. J. 55 (2015), no. 1, 134–149. MR 3323287, https://doi.org/10.1007/s10986-015-9270-3
- 8. Yuliya Mishura, Yevheniya Munchak, and Petro 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, https://doi.org/10.15559/15-vmsta23
- 9.
Yu. S. Mishura and E. Yu. Munchak, Rate of convergence of option prices by using the method of pseudomoments, Teor. Ĭmov
r. Mat. Stat. 92 (2015) 110-124; English transl. in Theor. Probability and Math. Statist. 92 (2016), 117-133.
- 10. John B. Walsh, The rate of convergence of the binomial tree scheme, Finance Stoch. 7 (2003), no. 3, 337–361. MR 1994913, https://doi.org/10.1007/s007800200094
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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
DOI:
https://doi.org/10.1090/tpms/999
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
