For any exponential L\'evy model whose diffusion component is nonzero\E , we provide an exact series representation for the implied volatility of a European call option. Numerical examples are provided .
For any strictly positive martingale S =\E^X for which X has an analytically tractable characteristic function , we provide an expansion for the implied volatility . This expansion is explicit in the sense that it involves no integrals, but only polynomials in \log(K/S_0). We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential L\'evy model (Merton), one infinite activity exponential L\'evy model (Variance Gamma), and one stochastic volatility model (Heston). We show how this technique can be extended to compute approximate forward implied volatilities and we implement this extension in the Heston setting. Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface .
R
exponential L\'evy model whose diffusion component is nonzero
strictly positive martingale S =
A
^X for which X has an analytically tractable characteristic function
R
exact series representation
expansion
R
of a European call option. Numerical examples are provided
. This expansion is explicit in the sense that it involves no integrals, but only polynomials in \log(K/S_0). We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential L\'evy model (Merton), one infinite activity exponential L\'evy model (Variance Gamma), and one stochastic volatility model (Heston). We show how this technique can be extended to compute approximate forward implied volatilities and we implement this extension in the Heston setting. Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface