10. Parameter Estimation in a Weak Hidden Markov Model with Independent Drift and Volatility

We develop a multivariate higher-order Markov model, also known as weak hidden Markov model (WHMM), for the evolution of asset prices. The means and volatilities of asset’s log-returns are governed by a second-order Markov chain in discrete time. WHMM enriches the usual HMM by incorporating more information from the past thereby capturing presence of memory in the underlying market state. A filtering technique in conjunction with the Expectation-Maximisation algorithm is adopted to develop the optimal estimates of model parameters. To ensure that the errors between the “true” parameters and estimated parameters are due only to the estimation method and not from model uncertainty, recursive filtering algorithms are implemented to a simulated dataset. Using goodness-of-fit metrics, we show that the WHMM-based filtering techniques are able to recover the “true” underlying parameters.