Estimation and Prediction of a Non-Constant Volatility

Abstract

In this paper we study volatility functions. Our main assumption is that the volatility is a function of time and is either deterministic, or stochastic but driven by a Brownian motion independent of the stock. Our approach is based on estimation of an unknown function when it is observed in the presence of additive noise. The set up is that the prices are observed over a time interval [0, t], with no observations over (t, T), however there is a value for volatility at T. This value is may be inferred from options, or provided by an expert opinion. We propose a forecasting/interpolating method for such a situation. One of the main technical assumptions is that the volatility is a continuous function, with derivative satisfying some smoothness conditions. Depending on the degree of smoothness there are two estimates, called filters, the first one tracks the unknown volatility function and the second one tracks the volatility function and its derivative. Further, in the proposed model the price of option is given by the Black–Scholes formula with the averaged future volatility. This enables us to compare the implied volatility with the averaged estimated historical volatility. This comparison is done for three companies and has shown that the two estimates of volatility have a weak statistical relation.