### 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) |
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Title of host publication | Large Deviations and Asymptotic Methods in Finance |

Publisher | Springer New York LLC |

Pages | 89-136 |

Number of pages | 48 |

Volume | 110 |

ISBN (Print) | 9783319116044 |

DOIs | |

State | Published - 2015 |

Event | Workshop on Large Deviations and Asymptotic Methods in Finance, 2013 - London, United Kingdom Duration: Apr 9 2013 → Apr 11 2013 |

### Other

Other | Workshop on Large Deviations and Asymptotic Methods in Finance, 2013 |
---|---|

Country | United Kingdom |

City | London |

Period | 4/9/13 → 4/11/13 |

### Fingerprint

### Keywords

- Asymptotic expansion
- Heat kernels
- Implied volatility
- Local volatility

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Large Deviations and Asymptotic Methods in Finance*(Vol. 110, pp. 89-136). 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, Gerard; Laurence, Peter.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Second order expansion for implied volatility in two factor local stochastic volatility models and applications to the dynamic λ-Sabr model

AU - Ben Arous, Gerard

AU - Laurence, Peter

PY - 2015

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]).

KW - Asymptotic expansion

KW - Heat kernels

KW - Implied volatility

KW - Local volatility

UR - http://www.scopus.com/inward/record.url?scp=84969268065&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84969268065&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-11605-1_4

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

M3 - Conference contribution

AN - SCOPUS:84969268065

SN - 9783319116044

VL - 110

SP - 89

EP - 136

BT - Large Deviations and Asymptotic Methods in Finance

PB - Springer New York LLC

ER -