Moment generating functions and Normalized implied volatilities: unification and extension via Fukasawa's pricing formula

We extend the model-free formula of [Fukasawa 2012] for $\mathbb E[\Psi(X_T)]$, where $X_T=\log S_T/F$ is the log-price of an asset, to functions $\Psi$ of exponential growth. The resulting integral representation is written in terms of normalized implied volatilities. Just as Fukasawa's work provides rigourous ground for Chriss and Morokoff's (1999) model-free formula for the log-contract (related to the Variance swap implied variance), we prove an expression for the moment generating function $\mathbb E[e^{p X_T}]$ on its analyticity domain, that encompasses (and extends) Matytsin's formula [Matytsin 2000] for the characteristic function $\mathbb E[e^{i \eta X_T}]$ and Bergomi's formula [Bergomi 2016] for $\mathbb E[e^{p X_T}]$, $p \in [0,1]$. Besides, we (i) show that put-call duality transforms the first normalized implied volatility into the second, and (ii) analyze the invertibility of the extended transformation $d(p,\cdot) = p \, d_1 + (1-p)d_2$ 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.