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

Paolo Pigato

Finance and Stochastics, 2019, vol. 23, issue 4, No 2, 827-859

Abstract: 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 $T^{-1/2}$ , 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.

Keywords: Local volatility; Implied volatility; Implied skew; Asymptotic pricing; Discontinuous coefficients; 91G20; 60H30; 60F05; 91G80 (search for similar items in EconPapers)
JEL-codes: C13 C22 G12 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (12)

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DOI: 10.1007/s00780-019-00406-2

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