Dynamic Probabilistic Forecasting with Uncertainty

We introduce a dynamical model for the time evolution of probability density functions incorporating uncertainty in the parameters. The uncertainty follows stochastic processes, thereby defining a new class of stochastic processes with values in the space of probability densities. The purpose is to quantify uncertainty that can be used for probabilistic forecasting. Starting from a set of traded prices of equity indices such as FTSEMIB, FTSE100 and S&P500 we do some empirical studies. We apply our dynamic probabilistic forecasting to option pricing, where our proposed notion of model uncertainty reduces to uncertainty on future volatility. A distribution of option prices follows, reflecting the uncertainty on the distribution of the underlying prices. We associate measures of model uncertainty of prices in the context of Cont (2006). As a further application we look at the Sharpe ratio and the VaR measure of market risk as well, proposing some decision rules for investors, regulators and risk managers