Discrete-Time Markov Chains for a Multivariate Stochastic Autoregressive Volatility

Abstract

Modelling and Forecasting volatility is one of the fundamental areas of research in Financial Mathematics, and thus has been the focus of many researchers; also, financial markets are known to be far from deterministic but stochastic and hence random models tend to perfectly model the markets. This study used appropriate Discrete-time Markov models to predict the multivariate stochastic Autoregressive volatility of an equity portfolio on a stock market. Therefore, the idea of modelling volatility as a stochastic process for an accurate forecast using the Markov chain on the financial data sets are based on the risks that often affect investment opportunities and the risk factors for prices changing that investors are most concerned about making decisions. The results provided more accuracy on forecasting price volatility on stock markets. We used a 3-state Discrete-Time Markov Chain (DTMC) for a portfolio of two stocks for the same sector and we compared the used model (fitted on a portfolio) to the multivariate GARCH models using real data from a stock market. The modified and generalized model provided more suitable volatility smiles compared to the Multivariate Generalized Autoregressive Conditional Heteroscedasticity (MGARCH) models and showed that working in a multivariate frame is most relevant especially when the number of state is bigger.