Computing VAR and AVaR in Infinitely Divisible Distributions

In this paper we derive closed-form solutions for the cumulative density function and the average value-at-risk for five subclasses of the infinitely divisible distributions: classical tempered stable distribution, Kim-Rachev distribution, modified tempered stable distribution, normal tempered stable distribution, and rapidly decreasing tempered stable distribution. We present empirical evidence using the daily performance of the S&P 500 for the period January 2, 1997 through December 29, 2006.