Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-Sabr model

Gerard Ben Arous, Peter Laurence

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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. (WilmottMag. 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]).

Original languageEnglish (US)
Title of host publicationLarge Deviations and Asymptotic Methods in Finance
PublisherSpringer New York LLC
Pages89-136
Number of pages48
Volume110
ISBN (Print)9783319116044
DOIs
StatePublished - 2015
EventWorkshop on Large Deviations and Asymptotic Methods in Finance, 2013 - London, United Kingdom
Duration: Apr 9 2013Apr 11 2013

Other

OtherWorkshop on Large Deviations and Asymptotic Methods in Finance, 2013
CountryUnited Kingdom
CityLondon
Period4/9/134/11/13

Fingerprint

Implied Volatility
Stochastic Volatility Model
Dynamic Model
Volatility
Financial Mathematics
Statistics
Transition Density
Henry
Finance
Density Function
First-order
Series
Term
Modeling

Keywords

  • Asymptotic expansion
  • Heat kernels
  • Implied volatility
  • Local volatility

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-Sabr model. / Ben Arous, Gerard; Laurence, Peter.

Large Deviations and Asymptotic Methods in Finance. Vol. 110 Springer New York LLC, 2015. p. 89-136.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Ben Arous, G & Laurence, P 2015, Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-Sabr model. in Large Deviations and Asymptotic Methods in Finance. vol. 110, Springer New York LLC, pp. 89-136, Workshop on Large Deviations and Asymptotic Methods in Finance, 2013, London, United Kingdom, 4/9/13. https://doi.org/10.1007/978-3-319-11605-1_4
Ben Arous, Gerard ; Laurence, Peter. / Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-Sabr model. Large Deviations and Asymptotic Methods in Finance. Vol. 110 Springer New York LLC, 2015. pp. 89-136
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AB - 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. (WilmottMag. 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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