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(XT)]$ \mathbb E [\Psi (X_T)] $, where XT=logST/F$ X_T=\log S_T/F $ is the log-price of an asset, to functions Ψ$ \Psi $ 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[epXT]$ \mathbb E [e^{p X_T}] $ on its analyticity domain, that encompasses (and extends) Matytsin’s formula [Perturbative analysis of volatility smiles, 2000] for the characteristic function E[eiηXT]$ \mathbb E [e^{i \eta X_T}] $ and Bergomi’s formula [Stochastic Volatility Modelling, 2016] for E[epXT]$ \mathbb E [e^{p X_T}] $, p∈[0,1]$ p \in [0,1] $. Besides, we (i) show that put-call duality transforms the first normalized implied volatility into the second, and (ii) analyse the invertibility of the extended transformation d(p,·)=pd1+(1-p)d2$ d(p,\cdot ) = p \, d_1 + (1-p)d_2 $ 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.