On polynomial extension of t-distribution and its financial applications

Abstract

As the fat-tail property is a well-known observational result in financial return distributions, the Student's t-distribution became popular in financial return distribution modeling. We introduce an extension of the t-distribution model in this paper, called polynomial-t-distribution model, in which we use the product of the t-distribution density and a polynomial to fit the density function of financial returns. This extended model allows arbitrary moment parameters of the return distribution, in addition of the fat-tail adjustment. The formulae of the European option prices, VaR and CVaR are given. Numerical illustrations are made based on S&P 500 index returns.