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Probability Distribution in the SABR Model of Stochastic Volatility Read chapter Read first chapter

2015 | OriginalPaper | Chapter

General Asymptotics of Wiener Functionals and Application to Implied Volatilities

Author : Yasufumi Osajima

Published in: Large Deviations and Asymptotic Methods in Finance

Publisher: Springer International Publishing

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Abstract

In the present paper, we give an asymptotic expansion of probability density for a component of general diffusion models. Our approach is based on infinite dimensional analysis on the Malliavin calculus and Kusuoka-Stroock’s asymptotic expansion theory for general Wiener functionals (Kusuoka and Stroock, J. Funct. Anal. 99:1–74, 1991 [12]). The initial term of the expansion is given by the geodesic distance and we calculate it by solving Hamilton’s equation. We apply our approach to obtain asymptotic expansion formulae for implied volatilities in general diffusion models, e.g. CEV and SABR model.

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previous chapter Second Order Expansion for Implied Volatility in Two Factor Local Stochastic Volatility Models and Applications to the Dynamic -Sabr Model
next chapter Implied Volatility of Basket Options at Extreme Strikes
Appendix
Available only for authorised users
Footnotes
1
We define \(Dy( \cdot ; h)[k] = \frac{d}{d\varepsilon } y( \cdot ; h + \varepsilon k)\).
 
Literature
1.
Berestycki, H., Busca, J., Florent, I.: Computing the implied volatility in stochastic volatility models. Commun. Pure Appl. Math. 57(10), 1352–1373 (2004)MATHMathSciNetCrossRef
2.
Bismut, J.M.: Large Deviations and the Malliavin Calculus. Birkhauser, Boston (1984)MATH
3.
Deuschel, J.D., Friz, P.K., Jacquier, A., Violante, S.: Marginal density expansions for diffusions and stochastic volatility, part I: theoretical foundations. commun. Pure Appl. Math. 67(1) (2014)
4.
Deuschel, J.D., Friz, P.K., Jacquier, A., Violante, S.: Marginal density expansions for diffusions and stochastic volatility, part II: applications. Commun. Pure Appl. Math. 67(2), 321–350 (2014)MATHMathSciNetCrossRef
5.
Dunford, N., Schwartz, J.T.: Linear Operators, Part II. Wiley, New York (1988)MATH
6.
Hagan, P.S., Woodward, D.E.: Equivalent black volatilities. Appl. Math. Financ. 6, 147–157 (1999)MATHCrossRef
7.
Hagan, P.S., Kumar, D., Lesniewski, S., Woodward, D.E.: Managing smile risk. Wilmott Mag. 18(11), 84–108 (2002)
8.
Hagan, P.S., Lesniewski, S., Woodward, D.E.: Probability distribution in the SABR model of stochastic volatility. In: Friz, P., Gatheral, J., Gulisashvili, A., Jacquier, A., Teichmann, J. (eds.) Large Deviations and Asymptotic Methods in Finance. Springer Proceedings in Mathematics and Statistics, vol. 110 (2015)
9.
10.
Kunitomo, N., Takahashi, A.: The asymptotic expansion approach to the valuation of interest rate contingent claims. Math. Financ. 11, 117–151 (2001)MATHMathSciNetCrossRef
11.
Kusuoka, S., Stroock, D.W.: Applications of Malliavin Calculus, Part I. In: Ito, K. (ed.) Proceedings of the Taniguchi International Symposium on Stochastic Analysis, Kyoto and Katata, 1982, pp. 271–360. Kinokuniya, Tokyo (1984)
12.
13.
Kusuoka, S., Osajima, Y.: A remark on the asymptotic expansion of density function of Wiener functionals. J. Funct. Anal. 255, 2545–2562 (2007)MathSciNetCrossRef
14.
Osajima, Y.: The Asymptotic Expansion Formula of Implied Volatility for Dynamic SABR model and FX hybrid model, BNP Paribas, Date posted: 26 Feb 2007 SSRN working paper series
15.
Shigekawa, I.: Stochastic analysis. Am. Math. Soc. (2004)
16.
Siopacha, M., Teichmann, J.: Weak and strong Taylor methods for numerical solutions of stochastic differential equations. Quant. Financ. 11(4), 517–528 (2011)MATHMathSciNetCrossRef
17.
Watanabe, S.: Analysis of wiener functionals (Malliavin calculus) and its application to heat kernels. Ann. Probab. 15, 1–39 (1987)MATHMathSciNetCrossRef
18.
Yoshida, N.: Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe. Probab. Theory Relat. Fields 92, 275–311 (1992)MATHCrossRef
Metadata
Title
General Asymptotics of Wiener Functionals and Application to Implied Volatilities
Author
Yasufumi Osajima
Copyright Year
2015
Book
Large Deviations and Asymptotic Methods in Finance

Print ISBN: 978-3-319-11604-4

Electronic ISBN: 978-3-319-11605-1

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-11605-1_5