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Journal Article

Hyperbolic Distributions in Finance

Ernst Eberlein and Ulrich Keller
Bernoulli
Vol. 1, No. 3 (Sep., 1995), pp. 281-299
DOI: 10.2307/3318481
Stable URL: http://www.jstor.org/stable/3318481
Page Count: 19
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Hyperbolic Distributions in Finance
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Abstract

Distributional assumptions for the returns on the underlying assets play a key role in valuation theories for derivative securities. Based on a data set consisting of daily prices of the 30 DAX shares over a three-year period, we investigate the distributional form of compound returns. After performing a number of statistical tests, it becomes clear that some of the standard assumptions cannot be justified. Instead, we introduce the class of hyperbolic distributions which can be fitted to the empirical returns with high accuracy. Two models based on hyperbolic Lévy motion are discussed. By studying the Esscher transform of the process with hyperbolic returns, we derive a valuation formula for derivative securities. The result suggests a correction of standard Black-Scholes pricing, especially for options close to expiration.

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