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

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We extend the model-free formula of [Fukasawa 2012] for E[Ψ(Xₜ)], where Xₜ= Sₜ/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 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 E[eᵖ ˣᵗ] on its analyticity domain, that encompasses (and extends) Matytsin's formula [Matytsin 2000] for the characteristic function E[eⁱ ηˣᵗ] and Bergomi's formula [Bergomi 2016] for E[eᵖ ˣᵗ], p ∈ [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,·) = 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.