From Spot Volatilities to Implied Volatilities

Abstract

The link between spot volatilities and implied volatilities has been actively investigated in the last two decades. Since the pioneering work of Dupire (1994), one knows how to infer the local volatility function from the implied volatility surface. Inverting this formula, i.e., computing implied volatilities from local volatilities, is not an easy task. The classical solution consists in solving numerically the forward equation for call prices. In this article, we suggest new methods for computing implied volatilities, based on a very general result which expresses the square of the implied volatility as an average over time and space of the square of the spot volatilities.

Keywords: Implied volatility, local volatility, approximation, heat kernel expansion

JEL Classification: G13