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Theory of Probability and Mathematical Statistics

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An application of the Malliavin calculus for calculating the precise and approximate prices of options with stochastic volatility


Authors: S. V. Kuchuk-Yatsenko, Yu. S. Mishura and Ye. Yu. Munchak
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 94 (2016).
Journal: Theor. Probability and Math. Statist. 94 (2017), 97-120
MSC (2010): Primary 91B25, 91G20; Secondary 60H07
DOI: https://doi.org/10.1090/tpms/1012
Published electronically: August 25, 2017
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Abstract: This paper is devoted to mathematical models of financial markets with stochastic volatility defined as a functional of either the Ornstein-Uhlenbeck process or Cox-Ingersoll-Ross process. We study the question on the exact price of a European type option. Using Malliavin calculus, we establish the probability density of the average value of the volatility in the time interval until the maturity. This result allows us to express the price of an option in terms of the minimum martingale measure for the case where the Wiener process driving the evolution of asset prices is uncorrelated with the Wiener process that defines the volatility.


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Additional Information

S. V. Kuchuk-Yatsenko
Affiliation: Department of Integral and Differential Equations, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email: kuchuk.iatsenko@gmail.com

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/1012
Keywords: Black--Scholes model, stochastic volatility, pricing the options, Malliavin calculus
Received by editor(s): April 6, 2016
Published electronically: August 25, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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