Minimal model of financial stylized facts

In this work we propose a statistical characterization of a linear stochastic volatility model featuring inverse-gamma stationary distribution for the instantaneous volatility. We detail the derivation of the moments of the return distribution, revealing the role of the inverse-gamma law in the emergence of fat tails and of the relevant correlation functions. We also propose a systematic methodology for estimating the parameters and we describe the empirical analysis of the Standard & Poor's 500 index daily returns, confirming the ability of the model to capture many of the established stylized facts as well as the scaling properties of empirical distributions over different time horizons.