Nonparametric prediction for the time-dependent volatility of the security price

In this paper we consider a continuous time model for the security price with the time-dependent volatility. It is shown that the 'non-normality' and 'non-linear dependency' of the short-term return, the major characteristics observed on many financial assets, can be incorporated into our model. In order to evaluate the option price formula on the model we propose a nonparametric predictor for the volatility function without reference to a specific functional form. We examine the so-called 'continuous record asymptotics' and show that the proposed predictor is asymptotically minimax for a wide class of the volatility functions. One of the most important results is that the application of the Black-Scholes method can be justified by plugging the proposed predictor in the standard Black-Scholes formula even if the volatility changes over time.