Abstract
In this article, we propose a new framework for addressing multivariate time-varying volatilities. By employing methods of differential geometry, our model respects the geometric structure of the covariance space, i.e., symmetry and positive definiteness, in a way that is independent of any local coordinate parametrization. Its parsimonious specification makes it particularly suitable for large dimensional systems. Simulation studies suggest that our model embraces much of the nonlinear behaviour of the covariance dynamics. Applied to the US and the UK stock markets, the model performs well, especially when applied to risk measurement. In a broad context, our framework presents a new approach treating nonlinear properties observed in the financial market, and numerous areas of application can be further considered.
Highlights
Since the introduction of the ARCH model by Engle (1982), time-varying volatility models have played an important role in finance and have been successfully applied to various financial problems
By employing methods of differential geometry, our model respects the geometric structure of the covariance space, i.e., symmetry and positive definiteness, in a way that is independent of any local coordinate parametrization
The areas of application of time-varying volatility models are extensive; potentially all the areas where covariance dynamics comes into play, such as asset pricing and portfolio optimization, can be considered
Summary
Since the introduction of the ARCH model by Engle (1982), time-varying volatility models have played an important role in finance and have been successfully applied to various financial problems. Beyond the aesthetic appeal of describing the dynamics on a curved space in an intrinsic fashion, the most critical reason, that has a profound effect on computational results, is that ignoring the geometric structure more often than not will lead to arbitrary results without physical meaning To illustrate this with an example pointed out by Fletcher and Joshi (2004), one can define a linear average of a collection of positive definite symmetric matrices by taking their sum and dividing by the number of elements. XiX>i ; ð8Þ where Xi is the tangent vector at P of the geodesic connecting P and Pi, i.e., Xi 1⁄4 Log PðPiÞ Based on these mean and covariance formulas, a generalized normal distribution on P(n) can be constructed by taking the curvature into account; see Lenglet et al (2006) for details.
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