A hyperbolic diffusion model for stock prices (*)

Bo Martin Bibby and Michael SÛrensen
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Bo Martin Bibby: Department of Biometry and Informatics, Research Centre Foulum, P.O. Box 23, DK-8830 Tjele, Denmark
Michael SÛrensen: Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus, DK-8000 Aarhus C, Denmark

Finance and Stochastics, 1996, vol. 1, issue 1, 25-41

Abstract: In the present paper we consider a model for stock prices which is a generalization of the model behind the Black-Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black-Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.

Keywords: Martingale; estimating; function; ·; option; pricing; ·; quasi-likelihood; ·; simulation; ·; stochastic; differential; equation; ·; volatility. (search for similar items in EconPapers)
JEL-codes: G12 (search for similar items in EconPapers)
Date: 1996
Note: received: November 1995; final revision received: April 1996
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Citations: View citations in EconPapers (10)

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