Polynomial adjusted Student-t densities for modeling asset returns

Abstract

We present a polynomial expansion of the standardized Student-t distribution. Our density, obtained through the polynomial adjusted method in Bagnato, Potí, and Zoia (2015. “The Role of Orthogonal Polynomials in Adjusting Hyperbolic Secant and Logistic Distributions to Analyse Financial Asset Returns.” Statistical Papers 56 (4): 1205–12340), is an extension of the Gram–Charlier density in Jondeau and Rockinger (2001. “Gram-Charlier Densities.” Journal of Economic Dynamics and Control 25 (10): 1457–1483). We derive the closed-form expressions of the moments, the distribution function and the skewness–kurtosis frontier for a well-defined density. An empirical application is also implemented for modeling heavy-tailed and skewed distributions for daily asset returns. Both in-sample and backtesting analysis show that this new density can be a good candidate for risk management.