### 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 |

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 | |

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 |
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Volume | 110 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | Workshop on Large Deviations and Asymptotic Methods in Finance, 2013 |
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Country | United Kingdom |

City | London |

Period | 4/9/13 → 4/11/13 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*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