Fractionally Integrated COGARCH Processes

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.* We thank Aleksey Min from the Chair of Mathematical Finance at the Technical University of Munich for access to the Chair’s Thomas Reuters database. Furthermore, we would like to thank Thiago do Rêgo Sousa for interesting discussions and useful comments on the simulation-based version of the generalized method of moments.