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A hyperbolic diffusion model for stock prices (*)

Author

Listed:
  • 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)

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.

Suggested Citation

  • Bo Martin Bibby & Michael SÛrensen, 1996. "A hyperbolic diffusion model for stock prices (*)," Finance and Stochastics, Springer, vol. 1(1), pages 25-41.
    • Handle: RePEc:spr:finsto:v:1:y:1996:i:1:p:25-41
    Note: received: November 1995; final revision received: April 1996
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    Citations

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    Cited by:

    1. Jing Zhao & Hoi Ying Wong, 2012. "A closed-form solution to American options under general diffusion processes," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 725-737, July.
    2. Oriol Zamora Font, 2024. "Pricing VIX options under the Heston-Hawkes stochastic volatility model," Papers 2406.13508, arXiv.org.
    3. Gangadhar Nayak & Subhranshu Sekhar Tripathy & Agbotiname Lucky Imoize & Chun-Ta Li, 2024. "Application of Extended Normal Distribution in Option Price Sensitivities," Mathematics, MDPI, vol. 12(15), pages 1-18, July.
    4. Robert Brooks & Xibin Zhang & Emawtee Bissoondoyal Bheenick, 2007. "Country risk and the estimation of asset return distributions," Quantitative Finance, Taylor & Francis Journals, vol. 7(3), pages 261-265.
    5. Danijel Grahovac & Nenad Suvak, 2015. "Heavy-tailed modeling of CROBEX," Financial Theory and Practice, Institute of Public Finance, vol. 39(4), pages 411-430.
    6. Nenghui Kuang & Huantian Xie, 2015. "Sequential Maximum Likelihood Estimation for the Hyperbolic Diffusion Process," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 373-381, June.
    7. Y. K. Tse & Xibin Zhang & Jun Yu, 2004. "Estimation of hyperbolic diffusion using the Markov chain Monte Carlo method," Quantitative Finance, Taylor & Francis Journals, vol. 4(2), pages 158-169.
    8. David R. Ba~nos & Salvador Ortiz-Latorre & Oriol Zamora Font, 2022. "Change of measure in a Heston-Hawkes stochastic volatility model," Papers 2210.15343, arXiv.org.
    9. Ahmed Nafidi & Ilyasse Makroz & Ramón Gutiérrez Sánchez, 2021. "A Stochastic Lomax Diffusion Process: Statistical Inference and Application," Mathematics, MDPI, vol. 9(1), pages 1-9, January.
    10. Ahmed Nafidi & Abdenbi El Azri & Ramón Gutiérrez-Sánchez, 2023. "A Stochastic Schumacher Diffusion Process: Probability Characteristics Computation and Statistical Analysis," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-15, June.

    More about this item

    Keywords

    Martingale estimating function · option pricing · quasi-likelihood · simulation · stochastic differential equation · volatility.;

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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