Filterbased Stochastic Volatility in Continuous-Time Hidden Markov Models

TL;DR

This paper introduces a continuous-time hidden Markov model with regime-switching volatility that simplifies estimation by making the Markov chain observable through stochastic volatility, supported by theoretical and numerical analysis.

Contribution

It proposes a novel continuous-time HMM with filter-dependent volatility, providing filtering equations, approximation results, and connections to social learning for improved econometric modeling.

Findings

The model allows direct observation of the Markov chain via stochastic volatility.

Filtering equations are derived for the regime-switching model.

Numerical simulations demonstrate the model's econometric properties.

Abstract

Regime-switching models, in particular Hidden Markov Models (HMMs) where the switching is driven by an unobservable Markov chain, are widely-used in financial applications, due to their tractability and good econometric properties. In this work we consider HMMs in continuous time with both constant and switching volatility. In the continuous-time model with switching volatility the underlying Markov chain could be observed due to this stochastic volatility, and no estimation (filtering) of it is needed (in theory), while in the discretized model or the model with constant volatility one has to filter for the underlying Markov chain. The motivations for continuous-time models are explicit computations in finance. To have a realistic model with unobservable Markov chain in continuous time and good econometric properties we introduce a regime-switching model where the volatility depends on the filter for the underlying chain and state the filtering equations. We prove an approximation result for a fixed information filtration and further motivate the model by considering social learning arguments. We analyze its relation to the switching volatility model and present a convergence result for the discretized model. We then illustrate its econometric properties by considering numerical simulations.