Open Access
September 1995 Hyperbolic distributions in finance
Ernst Eberlein, Ulrich Keller
Bernoulli 1(3): 281-299 (September 1995). DOI: 10.3150/bj/1193667819


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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Ernst Eberlein. Ulrich Keller. "Hyperbolic distributions in finance." Bernoulli 1 (3) 281 - 299, September 1995.


Published: September 1995
First available in Project Euclid: 29 October 2007

zbMATH: 0836.62107
Digital Object Identifier: 10.3150/bj/1193667819

Keywords: absolute continuous change of measure , hyperbolic distributions , hyperbolic Lévy motion , option pricing , statistical analysis of stock price data

Rights: Copyright © 1995 Bernoulli Society for Mathematical Statistics and Probability

Vol.1 • No. 3 • September 1995
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