Moment generating functions and normalized implied volatilities: unification and extension via Fukasawa’s pricing formula

We extend the model-free formula of Fukasawa [Math. Finance, 2012, 22, 753–762] for E[ψ(X_T)], where X_T = log S_T /F is the log-price of an asset, to functions ψ of exponential growth. The resulting integral representation is written in terms of normalized implied volatilities. Just as Fukasawa’s work provides rigorous ground for Chriss and Morokoff’s [Risk, 1999, 1, 609–641] model-free formula for the log-contract (related to the Variance swap implied variance), we prove an expression for the moment generating function E[e^pXT] on its analyticity domain, that encompasses (and extends) Matytsin’s formula [Perturbative analysis of volatility smiles, 2000] for the characteristic function E[e^iηXT] and Bergomi’s formula [Stochastic Volatility Modelling, 2016] for E[e^pXT], p ∈ [0, 1]. Besides, we (i) show that put-call duality transforms the first normalized implied volatility into the second, and (ii) analyse the invertibility of the extended transformation d(p, ·) = p d₁ + (1 - p)d₂ when p lies outside [0, 1]. As an application of (i), one can generate representations for the MGF (or other payoffs) by switching between one normalized implied volatility and the other.