A new approach to volatility modeling: the High-Dimensional Markov model

A new model -the high-dimensional Markov (HDM) model - is proposed for financial returns and their latent variances. It is also applicable to model directly realized variances. Volatility is modeled as a product of three components: a Markov chain driving volatility persistence, an independent discrete process capable of generating jumps in the volatility, and a predictable (data-driven) process capturing the leverage effect. The Markov chain and jump components allow volatility to switch abruptly between thousands of states. The transition probability matrix of the Markov chain is structured in such a way that the multiplicity of the second largest eigenvalue can be greater than one. This distinctive feature generates a high degree of volatility persistence. The statistical properties of the HDM model are derived and an economic interpretation is attached to each component. In-sample results on six financial time series highlight that the HDM model compares favorably to the main existing volatility processes. A forecasting experiment shows that the HDM model significantly outperforms its competitors when predicting volatility over time horizons longer than five days.