From Implied to Spot Volatilities

This paper is concerned with the link between spot and implied volatil-ities. The main result is the derivation of the stochastic differential equa-tion driving the spot volatility based on the shape of the implied volatility surface. This equation is a consequence of no-arbitrage constraints on the implied volatility surface right before expiry. We investigate the regu-larity of this surface at maturity in the case of the Constant Elasticity of Variance and Heston models. We also show that a simple way to link spot and implied volatilities is to relate the coefficients of the implied volatility surface Taylor expansion to the coefficients of a certain chaos expansion of the spot volatility process. As a byproduct, we give expansions for the implied volatility surface for a general stochastic volatility model.