Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

Details Asymptotic Implied Volatility at the Second Order with Application to the SABR Model Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

2009-06-03

arxiv.org/abs/0906.0658v2

Economy

Quantitative Finance

Pricing of Securities

27 pages; v2: typos fixed and a few notation changes

Scientific paper

We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kernel expansion. Furthermore, the first correction in the heat kernel expansion gives the second order correction for implied volatility, which we also give exactly at all strikes. As an application, we compute this asymptotic expansion at order 2 for the SABR model.

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