Integration of CARMA processes and spot volatility modelling

Abstract

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, , 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.