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In this paper, the authors 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. They present empirical evidence using the daily performance of the S&P 500 for the period January 2, 1997 through December 29, 2006. In finance, numerous studies of return and price distributions of different asset classes and national financial markets reject the notion that the distributions are normal.
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