Forecasting and testing a non-constant volatility

In this paper we study volatility functions. Our main assumption is that the volatility is deterministic or stochastic but driven by a Brownian motion independent of the stock. We propose a forecasting method and check the consistency with option pricing theory. To estimate the unknown volatility function we use the approach of \cite{Goldentayer Klebaner and Liptser} based on filters for estimation of an unknown function from its noisy observations. One of the main assumptions is that the volatility is a continuous function, with derivative satisfying some smoothness conditions. The two forecasting methods correspond to the the first and second order filters, the first order filter tracks the unknown function and the second order tracks the function and it derivative. Therefore the quality of forecasting depends on the type of the volatility function: if oscillations of volatility around its average are frequent, then the first order filter seems to be appropriate, otherwise the second order filter is better. Further, in deterministic volatility models the price of options is given by the Black-Scholes formula with averaged future volatility \cite{Hull White 1987}, \cite{Stein and Stein 1991}. This enables us to compare the implied volatility with the averaged estimated historical volatility. This comparison is done for five companies and shows that the implied volatility and the historical volatilities are not statistically related.