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Finance and Stochastics

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05-09-2019

Extreme at-the-money skew in a local volatility model

Author: Paolo Pigato

Published in: Finance and Stochastics | Issue 4/2019

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Abstract

We consider a local volatility model, with volatility taking two possible values, depending on the value of the underlying with respect to a fixed threshold. When the threshold is taken at the money, we establish exact pricing formulas for European call options and compute short-time asymptotics of the implied volatility surface. We derive an exact formula for the at-the-money implied volatility skew which explodes as \(T^{-1/2}\), reproducing the empirical steep short end of the smile. This behaviour is a consequence of the singularity of the local volatility at the money. Finally, we look at continuous, non-differentiable versions of such a model. We still find, in simulations, exploding implied skews.

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Title: Extreme at-the-money skew in a local volatility model
Author: Paolo Pigato
Publication date: 05-09-2019
Publisher: Springer Berlin Heidelberg
Published in: Finance and Stochastics / Issue 4/2019
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-019-00406-2

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