A Semi-Markov Model for Stock Returns with Momentum and Mean-Reversion

A vast body of empirical literature documents the existence of short-term momentum and medium-term mean reversion in various financial markets. By contrast, there is still a great shortage of theoretical models that explain the presence of these two common phenomena. We develop a semi-Markov model where the return process randomly switches between bull and bear states. In our model, the state duration times are governed by a negative binomial distribution that exhibits a positive duration dependence. We demonstrate that this model induces return momentum at short lags and reversal at subsequent lags. We calibrate our model to empirical data and show that the model-implied autocorrelation function fits reasonably well to the empirically estimated autocorrelation function.