Natural Cubic Spline Model for Estimating Volatility

Abstract

Volatility measures the dispersion of returns for a market variable since a reasonable estimation of the volatility is an appropriate starting point for assessing investment risks and monetary policymaking. These risks are usually assessed by using the GARCH (1,1) model. However, the recursive term in this model makes finding the derivatives of the likelihood function mathematically intractable. In this study, the natural cubic spline model is used to estimate the volatility by fitting it to the absolute returns of the data. In estimating the parameters, the Maximum Likelihood method was applied while a simple algebra was used to find its derivatives. The damped Newton-Raphson method was then used to maximize the likelihood function with the R programming software. The proposed method was illustrated using the absolute returns of the crude oil prices data from West Texas Intermediate, and it showed similar results with the popular GARCH (1,1) model. The natural cubic spline can be an alternative for estimating the volatility of any financial time series data.