The exact Taylor formula of the implied volatility

Stefano Pagliarani () and Andrea Pascucci
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Stefano Pagliarani: Università di Trieste

Finance and Stochastics, 2017, vol. 21, issue 3, No 3, 718 pages

Abstract: Abstract In a model driven by a multidimensional local diffusion, we study the behavior of the implied volatility σ ${\sigma}$ and its derivatives with respect to log-strike k $k$ and maturity T $T$ near expiry and at the money. We recover explicit limits of the derivatives ∂ T q ∂ k m σ ${\partial_{T}^{q}} \partial_{k}^{m} \sigma$ for ( T , x − k ) $(T,x-k)$ approaching the origin within the parabolic region | x − k | ≤ λ T $|x-k|\leq\lambda\sqrt{T}$ , with x $x$ denoting the spot log-price of the underlying asset and where λ $\lambda$ is a positive and arbitrarily large constant. Such limits yield the exact Taylor formula for the implied volatility within the parabola | x − k | ≤ λ T $|x-k|\leq\lambda\sqrt{T}$ . In order to include important models of interest in mathematical finance, e.g. Heston, CEV, SABR, the analysis is carried out under the weak assumption that the infinitesimal generator of the diffusion is only locally elliptic.

Keywords: Implied volatility; Local-stochastic volatility; Local diffusions; Feller process; 60J60; 60J70; 91G20 (search for similar items in EconPapers)
JEL-codes: C02 C60 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (9)

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DOI: 10.1007/s00780-017-0330-x

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