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

2015 | OriginalPaper | Chapter

Second Order Expansion for Implied Volatility in Two Factor Local Stochastic Volatility Models and Applications to the Dynamic \(\lambda \)-Sabr Model

Authors : Gérard Ben Arous, Peter Laurence

Published in: Large Deviations and Asymptotic Methods in Finance

Publisher: Springer International Publishing

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Abstract

Using an expansion of the transition density function of a two dimensional time inhomogeneous diffusion, we obtain the first and second order terms in the short time asymptotics of the local volatility function in a family of time inhomogeneous local-stochastic volatility models. With the local volatility function at our disposal, we show how recent results (Gatheral et al., Math. Financ. 22:591–620, 2012, [28]) for one dimensional diffusions can be applied to also determine expansions for call prices as well as for the implied volatility. The results are worked out in detail in the case of the dynamic Sabr model, thus generalizing earlier work by Hagan et al. (Wilmott Mag. 84–108, 2003, [31]), Hagan and Lesniewski (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, [32]) and by Henry-Labordère (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, Geometry, and Modeling in Finance. Chapman & Hall/CRC Financial Mathematics Series, 2008, [39, 40]).

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previous chapter Unifying the BGM and SABR Models: A Short Ride in Hyperbolic Geometry
next chapter General Asymptotics of Wiener Functionals and Application to Implied Volatilities
Appendix
Available only for authorised users
Footnotes
1
Depending on whether forward or log of forward is taken as state variable.
 
2
This expression was coined by Marco Avellaneda, who in 1999, without prior knowledge of Gyöngy or Dupire’s work, independently discovered the technique.
 
3
The expansion up to the first order (4.12) appears in several sources including in textbook form, as in Bender and Orszag [9], p. 273. On the other hand we have not been able to locate a source for the second order expansion given here as (4.15).
 
4
In (4.17) we once again suppressed on the right hand side the dependence of variables other than t, T and f. In the next section, when we combine the asymptotics for local volatility with the above asymptotics for implied volatility, given local volatility, it will be important that the local volatility depends on the initial time t and on the final time T in the particular way indicated.
 
5
Reference [28] explains how to adjust the results to allow for a non zero but constant interest (or other constant yield) rate.
 
6
Recall that a time change (6.21) gets rid of the extra factor \(\nu ^2\).
 
7
Note this angle is not the same angle as that used above in geodesic polar coordinates.
 
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Metadata
Title
Second Order Expansion for Implied Volatility in Two Factor Local Stochastic Volatility Models and Applications to the Dynamic -Sabr Model
Authors
Gérard Ben Arous
Peter Laurence
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_4