This paper introduces alternative measurements that use additional information of prices during the day: opening, minimum, maximum, and closing prices. Using the binomial model as the distribution of the stock price we prove that these alternative measurements are more efficient than the...

This paper provides several estimates of the GARCH models' parameters for the S&P500 index, based on returns and CBOE VIX. Using a daily sample collected from 2007 to 2022, we can conclude that adding the VIX information improves the estimates of the long-term volatility. By providing an...