The exact Taylor formula of the implied volatility

作者：Pagliarani Stefano; Pascucci Andrea

来源：Finance and Stochastics, 2017, 21(3): 661-718.

DOI：10.1007/s00780-017-0330-x

• Implied volatility
• Local-stochastic volatility
• Local diffusions
• Feller process

In a model driven by a multidimensional local diffusion, we study the behavior of the implied volatility sigma and its derivatives with respect to log-strike k and maturity T near expiry and at the money. We recover explicit limits of the derivatives partial derivative(q)(T)partial derivative(m)(k) sigma for (T, x - k) approaching the origin within the parabolic region vertical bar x - k vertical bar <= lambda root T, with x denoting the spot log-price of the underlying asset and where. is a positive and arbitrarily large constant. Such limits yield the exact Taylor formula for the implied volatility within the parabola vertical bar x - k vertical bar <= lambda root 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.

• 出版日期2017-7