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Zbl 0836.62107
Eberlein, Ernst; Keller, Ulrich
Hyperbolic distributions in finance. (English)
[J] Bernoulli 1, No.3, 281-299 (1995). ISSN 1350-7265

Summary: 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.\par Two models based on hyperbolic Levy 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.

MSC 2000:: *62P20 Appl. of statistics to economics
91B28 Finance etc.
60J65 Brownian motion

Keywords: absolute continuous change of measure; option pricing; stock price data; distributional form of compound returns; hyperbolic distributions; hyperbolic Levy motion; Esscher transform; valuation formula; derivative securities; Black-Scholes pricing

Cited in: Zbl 1239.65008 Zbl 1053.91059 Zbl 0996.91067

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Zentralblatt MATH Berlin [Germany]