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Computing VaR and AVaR in infinitely divisible distributions

Kim Y. S., Rachev S. T., Bianchi M. L., Fabozzi F. J.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 2

223--245

In this paper we derive closed-form solutions for the cumulative distribution 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.

The modified tempered stable distribution, GARCH models and option pricing

Kim Y. S., Rachev S. T., Chung D. M., Bianchi M. L.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 1

91--117

We introduce a new variant of the tempered stable distribution, named the modified tempered stable (MTS) distribution and we develop a GARCH option pricing model with MTS innovations. This model allows the description of some stylized empirical facts observed in financial markets, such as volatility clustering, skewness, and heavy tails of stock returns. To demonstrate the advantages of the MTS-GARCH model, we present the results of the parameter estimation.