From volatility smiles to the volatility of volatility

The paper reviews models of the option surface and reduced-form models for stochastic volatility in continuous time, under the risk-neutral measure. It defines “forward volatilities,” analogous to forward interest rates in the theory of the term structure, and provides a proof that the forward volatility is a conditional expected value, under the risk-neutral measure, of the future spot volatility. The theory developed here is the analog of Heath–Jarrow–Morton bond-pricing theory. The link is established between forward volatilities and so-called “model-free” volatility measures such as the VIX.