Implied Volatility and Forward Price Term Structures

Abstract

This paper discusses the relation between forward price models (FPM) and the so called implied volatility term structure (VTS). We start by considering the case of pure deterministic forward price volatilities and suppose both forward contracts and at-the-money (ATM) options, on a same underlying, are liquidly traded in the market. We, then, derive no arbitrage conditions between the functional form of the ATM implied VTS and the functional form of forward price volatilities. We conclude the first part by characterizing a parametric family of ATM implied VTS that is compatible with a finite dimensional realization (FDR) of forward prices. Finally, we consider the possibility of stochastic forward price volatilities and derive a no arbitrage drift condition that must hold for the dynamics of ATM implied VTS.