Maximum Likelihood Approach for Stochastic Volatility Models

Abstract

English: Volatility is a measure of the amplitude of price return fluctuations. Despite it is one of the most important quantities in finance, volatility is a hidden quantity because it is not directly observable. Here we apply a known maximum likelihood process which assumes that volatility is a time-dependent diffusions coefficient of the random walk of the price return and that it is a Markov process. We use this method using the expOU, the OU and the Heston models which are previously imposed. We find an estimator of the volatility for each model and we observe that it works reasonably well for the three models. Using these estimators, we reach a way of forecasting absolute values of future returns with current volatilities. During all the process, no-correlation is introduced and at the end, we see that volatility has non-zero autocorrelation for hundreds of days and we observe a significant correlation between volatility and price return called leverage effect. We finally apply this methodology to different market indexes and we conclude that its properties are universal.