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We extend the model-free formula of [Fukasawa 2012] for $\mathbbE[\Psi(X_T)]$, where $X_T=\log S_T/F$ is the log-price of an asset, tofunctions $\Psi$ of exponential growth. The resulting integral representationis written in terms of normalized implied volatilities. Just as Fukasawa's workprovides rigourous ground for Chriss and Morokoff's (1999) model-free formulafor the log-contract (related to the Variance swap implied variance), we provean expression for the moment generating function $\mathbb E[e^{p X_T}]$ on itsanalyticity domain, that encompasses (and extends) Matytsin's formula [Matytsin2000] for the characteristic function $\mathbb E[e^{i \eta X_T}]$ and Bergomi'sformula [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 impliedvolatility into the second, and (ii) analyze the invertibility of the extendedtransformation $d(p,\cdot) = p \, d_1 + (1-p)d_2$ 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 andthe other.

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