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Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model

  • Conference paper
Seminar on Stochastic Analysis, Random Fields and Applications

Part of the book series: Progress in Probability ((PRPR,volume 36))

Abstract

In mathematical finance, semimartingales are traditionally viewed as the largest class of stochastic processes which are economically reasonable models for stock price movements. This is mainly because stochastic integrals play a crucial role in the modern theory of finance, and semimartingales represent the largest class of stochastic processes for which a general theory of stochastic integration exists. However, some empirical evidence from actual stock price data suggests stochastic models that are not covered by the semimartingale setting.

In this paper, we discuss the empirical evidence that suggests that long-range dependence (also called the Joseph effect) be accounted for when modeling stock price movements. We present a fractional version of the Black—Scholes model that (i) is based on fractional Brownian motion, (ii) accounts for long-range dependence and is therefore inconsistent with the semimartingale framework (except in the case of ordinary Brownian motion) and (iii) yields the ordinary (independent) Black—Scholes model as a special case. Mathematical problems of practical importance to finance (e. g., completeness, equivalent martingale measures, arbitrage, financial gains) for this class of fractional Black—Scholes models are dealt with in an elementary fashion, namely using the (hyperfinite) fractional versions of the corresponding Cox—Ross—Rubinstein models.

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Author information

Authors and Affiliations

  1. School of Mathematics, University of Hull, Hull HU67RX, UK

    Nigel J. Cutland & P. Ekkehard Kopp

  2. Bellcore, 445 South Street, Room 2P-372, Morristown, NJ, 07960-6438, USA

    Walter Willinger

Authors
  1. Nigel J. Cutland
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  2. P. Ekkehard Kopp
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  3. Walter Willinger
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Editor information

Editors and Affiliations

  1. Institut für Mathematik, Abteilung Angewandte Mathematik, Universität Zürich-Irchel, Winterthurerstr. 190, CH-8057, Zürich, Switzerland

    Erwin Bolthausen

  2. Département de Mathématiques, Université de Nancy 1, B.P. 239, F-54506, Vandoeuvre-Les-Nancy Cedex, France

    Marco Dozzi

  3. Département de Mathématiques Institut Galilée, Université Paris-Nord, Av. Jean-Baptiste Clément, F-93430, Villetaneuse, France

    Francesco Russo

  4. BIBOS, Universität Bielefeld, D-33615, Bielefeld 1, Germany

    Francesco Russo

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Cutland, N.J., Kopp, P.E., Willinger, W. (1995). Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model. In: Bolthausen, E., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 36. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7026-9_23

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  • DOI: https://doi.org/10.1007/978-3-0348-7026-9_23

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-7028-3

  • Online ISBN: 978-3-0348-7026-9

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