Integration of CARMA Processes and Spot Volatility Modelling

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a non-decreasing Lévy process constitute a useful and very general class of stationary, non-negative continuous-time processes which have been used, in particular, for the modelling of stochastic volatility. Brockwell, Davis and Yang (2011) considered the fitting of CARMA models to closely and uniformly spaced data, illustrating their results by fitting a CARMA(2,1) model to daily realized volatility of the Deutsche Mark/US dollar (DM/US$) exchange rate from December 1986 through June, 1999. A more fundamental quantity in financial modelling is the (unobserved) spot, or instantaneous, volatility process. In the celebrated stochastic volatility model of Barndorff-Nielsen and Shephard (2001), the spot volatility is represented by a stationary Lévy-driven Ornstein-Uhlenbeck process. This has the shortcoming that its autocorrelation function is necessarily a decreasing exponential function, which limits its ability to generate integrated volatility series with autocorrelation functions of the forms encountered in practice. (A realized volatility series is a sequence of estimated integrals of spot volatility over successive intervals of fixed length, typically one day.) If instead of the stationary Ornstein-Uhlenbeck process, we use a CARMA process to represent spot volatility, we can overcome the restriction to exponentially decaying autocorrelation function and obtain a more realistic model for the dependence observed in realized volatility. In this paper we show how to use realized volatility data to estimate parameters of a CARMA model for spot volatility and apply the analysis to the DM/US$ exchange rate.