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

Gérard Ben Arous; Peter Laurence

doi:10.1007/978-3-319-11605-1_4

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

Gérard Ben Arous, Peter Laurence

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference 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 language	English (US)
Title of host publication	Large Deviations and Asymptotic Methods in Finance
Editors	Peter K. Friz, Jim Gatheral, Archil Gulisashvili, Josef Teichmann, Peter K. Friz, Antoine Jacquier
Publisher	Springer New York LLC
Pages	89-136
Number of pages	48
ISBN (Print)	9783319116044
DOIs	https://doi.org/10.1007/978-3-319-11605-1_4
State	Published - 2015
Event	Workshop on Large Deviations and Asymptotic Methods in Finance, 2013 - London, United Kingdom Duration: Apr 9 2013 → Apr 11 2013

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	110
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	Workshop on Large Deviations and Asymptotic Methods in Finance, 2013
Country/Territory	United Kingdom
City	London
Period	4/9/13 → 4/11/13

Keywords

Asymptotic expansion
Heat kernels
Implied volatility
Local volatility

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-3-319-11605-1_4

Cite this

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 P. K. Friz, J. Gatheral, A. Gulisashvili, J. Teichmann, P. K. Friz, & A. Jacquier (Eds.), Large Deviations and Asymptotic Methods in Finance (pp. 89-136). (Springer Proceedings in Mathematics and Statistics; Vol. 110). Springer New York LLC. https://doi.org/10.1007/978-3-319-11605-1_4

Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-Sabr model. / Ben Arous, Gérard; Laurence, Peter.
Large Deviations and Asymptotic Methods in Finance. ed. / Peter K. Friz; Jim Gatheral; Archil Gulisashvili; Josef Teichmann; Peter K. Friz; Antoine Jacquier. Springer New York LLC, 2015. p. 89-136 (Springer Proceedings in Mathematics and Statistics; Vol. 110).

Research output: Chapter in Book/Report/Conference proceeding › Conference 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 PK Friz, J Gatheral, A Gulisashvili, J Teichmann, PK Friz & A Jacquier (eds), Large Deviations and Asymptotic Methods in Finance. Springer Proceedings in Mathematics and Statistics, 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 G, Laurence P. Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-Sabr model. In Friz PK, Gatheral J, Gulisashvili A, Teichmann J, Friz PK, Jacquier A, editors, Large Deviations and Asymptotic Methods in Finance. Springer New York LLC. 2015. p. 89-136. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-319-11605-1_4

Ben Arous, Gérard ; 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. editor / Peter K. Friz ; Jim Gatheral ; Archil Gulisashvili ; Josef Teichmann ; Peter K. Friz ; Antoine Jacquier. Springer New York LLC, 2015. pp. 89-136 (Springer Proceedings in Mathematics and Statistics).

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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. (WilmottMag. 84–108, 2003, [31]),Hagan and Lesniewski (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, [32]) and by Henry-Labord{\`e}re (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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Y1 - 2015

N2 - 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]).

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

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