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 | References | Similar Articles | Additional Information

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