Modelling the Relationships between Duration and Volatility in Asset Prices

Abstract

This paper proposes a new model that captures the interaction between duration and magnitude of changes in asset prices, and thus provides a convenient framework to test statistically the existence of such relationship. The model is flexible and contains various well known models as special cases, including, the Exponential Generalised Autoregressive Heteroskedasticity (EGARCH) model of Nelson (1991) and the Logarithmic Conditional Duration (Log-ACD) model of Bauwens and Giot (2000). Despite having the EGARCH model as a special case, the objective of the model is not trying to model conditional duration and conditional volatility jointly. As shown in Ghysels and Jasiak (1998), modelling conditional duration and volatility jointly is technically challenging. This is due to the fact that volatility is defined over a regular sampling frequency but duration is defined over irregular time intervals. Given GARCH model is not generally closed under temporal aggregation, this creates a challenging modelling problem. The aim of this paper is to avoid this challenge by not modelling the conditional volatility, but instead, model the dynamics in the magnitudes of price change. The paper argues that since volatility is a function of the magnitudes of price change, testing the relationship between duration and the magnitude of price change provides an indirect test on the relationship between duration and volatility. The paper also obtains theoretical results for the Quasi-Maximum Likelihood Estimator (QMLE) for the proposed model. Specifically, sufficient conditions for consistency and asymptotic normality are derived under mild assumptions. Monte Carlo experiments also provide further support of the theoretical results and demonstrate that the QMLE has reasonably good finite sample performance. The paper then applies the model to nine different assets from three different asset classes, namely two exchange rate, two commodities and five stocks. The two currencies are Australia/US and British Pound/US exchange rates; the two commodities are Gold and Silver and the five stocks are BHP, Rio Tinto, CBS, ANZ and Apple. The sample spans from 4 January 2010 to 30 December 2011 with an average of 100,000 observations.