Integration of CARMA processes and spot volatility modelling


Correspondence to: Peter Brockwell, Statistics Department, Colorado State University, Fort Collins, CO 80523, USA.

AMS 2000 Mathematics Subject Classification: Primary: 62M10, 60H10; Secondary: 62M09.


Continuous-time autoregressive moving average (CARMA) processes with a non-negative 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. In the celebrated stochastic volatility model of Barndorff-Nielsen and Shephard (2001), the spot (or instantaneous) volatility at time t, V(t), is represented by a stationary Lévy-driven Ornstein-Uhlenbeck process. This has the shortcoming that its autocorrelation function is necessarily a decreasing exponential function, limiting its ability to generate integrated volatility sequences, inline image, with autocorrelation functions resembling those of observed realized volatility sequences. (A realized volatility sequence is a sequence of estimated integrals of spot volatility over successive intervals of fixed length, typically 1 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 article, we show how to use realized volatility data to estimate parameters of a CARMA model for spot volatility and apply the analysis to a daily realized volatility sequence for the Deutsche Mark/ US dollar exchange rate.