Fractionally integrated COGARCH processes

Abstract

We construct fractionally integrated continuous-time GARCH models, which capture the observed long-range dependence of squared volatility in high-frequency data. Since the usual Molchan-Golosov and Mandelbrot-van-Ness fractional kernels lead to problems in the definition of the model, we resort to moderately long-memory processes by choosing a fractional parameter d ∈(-0.5,0)and remove the singularities of the kernel to obtain nonpathological sample paths. The volatility of the new fractional continuous-time GARCH process has positive features like stationarity, and its covariance function shows an algebraic decay, which makes it applicable to econometric high-frequency data. The model is fitted to exchange rate data using a simulation-based version of the generalized method of moments.