Integration of CARMA processes and spot volatility modelling. (18th December 2012)

• Record Type: Journal Article
• Authors: Brockwell, Peter; Lindner, Alexander
• Is Part Of: Journal of time series analysis. Volume 34:Number 2(2013:Mar.)
• Page Start: 156
• Page End: 167
• Publication Date: 2012-12-18
• Subjects: Time-series analysis -- Periodicals; 519.232
• DOI: 10.1111/jtsa.12011
• Languages: English
• ISSNs: 0143-9782

Abstract

<abstract abstract-type="main" xml:lang="en"> <title> <x xml:space="preserve">Abstract</x> </title> <p>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 <xref ref-type="link" rid="b4">Barndorff‐Nielsen and Shephard (2001)</xref>, the spot (or instantaneous) volatility at time <italic>t</italic>, <italic>V</italic>(<italic>t</italic>), 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-graphic xlink:href="ark:/27927/pgg1xm355nv" mimetype="image" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, 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.</p> </abstract>