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

2015 | OriginalPaper | Chapter

A Remark on Gatheral’s ‘Most-Likely Path Approximation’ of Implied Volatility

Authors : Martin Keller-Ressel, Josef Teichmann

Published in: Large Deviations and Asymptotic Methods in Finance

Publisher: Springer International Publishing

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Abstract

We give a new proof of the representation of implied volatility as a time-average of weighted expectations of local or stochastic volatility. With this proof we clarify the question of existence of ‘forward implied variance’ in the original derivation of Gatheral, who introduced this representation in his book ‘The Volatility Surface’.

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previous chapter Small-Time Asymptotics for the At-the-Money Implied Volatility in a Multi-dimensional Local Volatility Model
next chapter Implied Volatility from Local Volatility: A Path Integral Approach
Footnotes
1
See Gatheral [1, p. 29ff] for details.
 
2
See also Lee [3, Sect. 2.3], who remarks that the proof in Gatheral [1] hinges upon the assumption of the existence of \(v_{K,T}(t)\).
 
Literature
1.
Gatheral, J.: The Volatility Surface. Wiley Finance (2006)
2.
Guyon, J., Henry-Labordère, P.: Nonlinear Option Pricing. CRC Press, Boca Raton (2013)
3.
Lee, R.: Implied volatility: statics, dynamics, and probabilistic interpretation. Recent Advances in Applied Probability. Springer, New York (2004)
Metadata
Title
A Remark on Gatheral’s ‘Most-Likely Path Approximation’ of Implied Volatility
Authors
Martin Keller-Ressel
Josef Teichmann
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_8