Moment generating functions and normalized implied volatilities: unification and extension via Fukasawa’s pricing formula
扩展了Fukasawa的无模型公式到指数增长函数，用归一化隐含波动率表示矩生成函数，统一并扩展了Matytsin和Bergomi的公式，并分析了看涨看跌对偶性及其可逆性。
We extend the model-free formula of Fukasawa [Math. Finance, 2012, 22, 753–762] for , where is the log-price of an asset, to functions 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 on its analyticity domain, that encompasses (and extends) Matytsin’s formula [Perturbative analysis of volatility smiles, 2000] for the characteristic function and Bergomi’s formula [Stochastic Volatility Modelling, 2016] for , . 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 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.