On a test for a parametric form of volatility in continuous time financial models

Abstract. A new specification test for the parametric form of the variance function in diffusion processes is proposed, which does not require specific knowledge of the functional form of the model. The corresponding test statistic has an asymptotic normal distribution under the null hypothesis and diverges at an appropriate rate under the alternative. In contrast to recent work the approach of the present paper does not require the specification of particular time points at which the hypothesis of a parametric form is checked. As a by-product we obtain a very simple test for homoscedasticity in diffusion processes. Moreover, the new test does not use nonparametric estimation techniques for estimating the variance function and is therefore independent of the specification of a particular smoothing parameter. The results are illustrated by a small simulation study and a data example is analyzed.