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
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
MathSciNet review:
3553457
Full-text PDF
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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.
References
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References
- E. Alos and Ch-O. Ewald, A note on the Malliavin differentiability of the Heston volatility, SSRN Electronic J. 09/2005 (2005).
- O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics, J. Royal Statist. Soc.: Ser. B (Statistical Methodology) 63 (2001), 167–241, MR 1841412
- J. C. Cox, J. E. Ingersoll, and S. A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), no. 2, 385–407. MR 785475
- F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage, Springer Finance, New York, 2006. MR 2200584
- S. Dereich, A. Neuenkirch, and L. Szpruch, An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process, Proc. Royal Soc. A: Mathematical, Physical and Engineering Sciences 468 (2012), 1105–1115. MR 2898556
- F. D’Ippoliti, E. Moretto, S. Pasquali, and B. Trivellato, Exact and approximated option pricing in a stochastic volatility jump-diffusion model, Mathematical and Statistical Methods for Actuarial Sciences and Finance (eds. M. Corazzo and C. Pizzi), Springer-Verlag Italia, Dordrecht–Heidelberg–London–Milan–New York, 2010, 133–142. MR 2676194
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- J. Goard, Exact and approximate solutions for options with time-dependent stochastic volatility, Appl. Math. Modell. 38 (2014), 2771–2780. MR 3201795
- S. Heston, The review of financial studies, J. Finance 6(2) (1993), 327–343.
- R. Frey, The pricing of options on assets with stochastic volatilities, J. Finance 42 (1987), 281–300.
- N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, Amsterdam, vol. 24, 1986. MR 637061
- E. Fournie, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, and T. Touzi, Applications of Malliavin calculus to Monte-Carlo methods in finance, Finance & Stochastics 3 (1999), 391–412. MR 1842285
- S. Kuchuk-Iatsenko and Yu. Mishura, Pricing the European call option in the model with stochastic volatility driven by Ornstein-Uhlenbeck process. Exact formulas, Modern Stoch. Theory Appl. 2(3) (2015), 233–249. MR 3407504
- S. Kuchuk-Iatsenko and Yu. Mishura, Pricing the European call option in the model with stochastic volatility driven by Ornstein–Uhlenbeck process. Simulation, Modern Stoch. Theory Appl. 2(4) (2015), 355–369. MR 3456143
- J. A. Leon and D. Nualart, Stochastic evolution equations with random generators, Ann. Probab. 26(1) (1998), 149–186. MR 1617045
- Yu. S. Mishura and E. Yu. Munchak, Rate of convergence of option prices by using the method of pseudomoments, Teor. Imovirnost. Mat. Stat. 92 (2015) 110–124; English transl. in Theor. Probability and Math. Statist. 92 (2016), 117–133. MR 3553430
- Yu. S. Mishura and E. Yu. Munchak, Rate of convergence of option prices for approximations of the geometric Ornstein–Uhlenbeck process by Bernoulli jumps of prices on assets, Teor. Imovirnost. Mat. Stat. 93 (2015), 127–141; English transl. in Theor. Probability and Math. Statist. 93 (2016), 137–152. MR 3553446
- Yu. Mishura, G. Rizhniak, and V. Zubchenko, European call option issued on a bond governed by a geometric or a fractional geometric Ornstein–Uhlenbeck process, Modern Stoch. Theory Appl. 1(1) (2014), 95–108. MR 3314796
- E. Nicolato and E. Venardos, Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type, Math. Finance 13 (2003), 445–466. MR 2003131
- D. Nualart, The Malliavin Calculus and Related Topics, Second edition, Probability and Its Applications, Springer-Verlag, Berlin–Heidelberg, 2006. MR 2200233
- D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands, Probab. Theory Rel. Fields 78 (1988), no. 4, 535–581. MR 950346
- G. Di Nunno, B. Øksendal, and F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance, Springer Science & Business Media, New York, 2008. MR 2460554
- Y. Ouknine, Fubini-type theorem for anticipating integrals, Random Oper. Stoch. Equ. 4 (1996), no. 4, 351–354. MR 1427723
- D. L. Ocone and I. Karatzas, A generalized Clark representation formula, with application to optimal portfolios, Stoch. Stoch. Reports 34 (1991), 187–220. MR 1124835
- J. Perelló, R. Sircar, and J. Masoliver, Option pricing under stochastic volatility: the exponential Ornstein–Uhlenbeck model, J. Stat. Mech. 1 (2008), P06010.
- M. Sanz-Sole, Malliavin Calculus with Applications to Stochastic Partial Differential Equations, EPFL Press, Lausanne, 2005. MR 2167213
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- A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific, Singapore, 1999. MR 1695318
- S. E. Shreve, Stochastic Calculus for Finance II. Continuous-Time Models, Springer-Verlag, New York, 2004. MR 2057928
- E. M. Stein and J. C. Stein, Stock price distributions with stochastic volatility: an analytic approach, Review Financial Studies, vol. 4(4), 1991, 727–752.
- J. Wiggins, Option values under stochastic volatility: Theory and empirical estimates, J. Financial Economics 19 (1987), 351–372.
- B. Wong and C. C. Heyde, On changes of measure in stochastic volatility models, J. Appl. Math. Stoch. Anal. 2006 (2006), 1–13, Article ID 18130. MR 2270326
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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
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