Weighted-indexed semi-Markov model: calibration and application to financial modeling

We address the calibration issues of the weighted-indexed semi-Markov chain (WISMC) model applied to high-frequency financial data. Specifically, we propose to automate the discretization of the price returns and the volatility index by using four different approaches, two based on statistical quantities, namely, the quantile and sigma discretization, and two derived by the application of two popular machine learning algorithms, namely the k-means and Gaussian mixture model (GMM). Moreover, by comparing the Bayesian information criterion (BIC) scores, the GMM approach allows for the selection of the number of states of returns and index. An application to Bitcoin prices at 1-min and 1-s intervals shows the validity and usefulness of the proposed discretization approaches. In particular, GMM discretization is well suited for high-frequency returns, whereas the quantile approach works better for low-frequency intervals. Finally, by comparing the results of the Monte Carlo simulation, we show that the WISMC model, applied with the proposed discretization, can reproduce the long-range serial correlation of the squared returns, which is typical of the financial markets and, in particular, the cryptocurrency market