The correlation dimension of returns with stochastic volatility

Signs of low-dimensional scaling of correlation integrals of financial returns have often been reported in the literature. In this paper conditions are considered under which correlation integrals of returns generated by stochastic volatility models show scaling. Subject to certain conditions on the marginal volatility distribution, scaling can indeed occur for short time horizon returns, with a scaling exponent that is directly related to the marginal volatility distribution. More generally, correlation integrals of short horizon returns are shown to be similar to those of deterministic processes with i.i.d. normal observational noise. The parameters describing the correlation integral can be estimated using the Gaussian kernel algorithm. Empirical applications indicate that the model describes the correlation integrals of index returns well, even over timescales as long as one trading day.