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A Note on Wick Products and the Fractional Black-Scholes Model

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  • Björk, Tomas

    () (Dept. of Finance, Stockholm School of Economics)

  • Hult, Henrik

    () (Department of Applied Mathematics and Statistics)

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    Abstract

    In some recent papers, such as Elliott & van der Hoek, Hu & Öksendal, a fractional Black-Scholes model have been proposed as an improvement of the classical Black-Scholes model. Common to these fractional Black-Scholes models, is that the driving Brownian motion is replaced by a fractional Brownian motion and that the Ito integral is replaced by the Wick integral, and proofs has been presented that these fractional Black-Scholes models are free of arbitrage. These results on absence of arbitrage complelety contradict a number of earlier results in the literature which prove that the fractional Black-Scholes model (and related models) will in fact admit arbitrage. The object of the present paper is to resolve this contradiction by pointing out that the definition of the self-financing trading strategies and/or the definition of the value of a portfolio used in the above cited papers does not have a reasonable economic interpretation, and thus that the results in these papers are not economically meaningful. In particular we show that in the framework of Elliott and van der Hoek, a naive buy-and-hold strategy does not in general qualify as "self-financing". We also show that in Hu and Öksendal, a portfolio consisting of a positive number of shares of a stock with a positive price may, with positive probability, have a negative "value".

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    File URL: http://swopec.hhs.se/hastef/papers/hastef0596.pdf
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    Bibliographic Info

    Paper provided by Stockholm School of Economics in its series Working Paper Series in Economics and Finance with number 596.

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    Length: 13 pages
    Date of creation: 25 Apr 2005
    Date of revision:
    Publication status: Published in Finance & Stochastics, 2005, pages 197-209.
    Handle: RePEc:hhs:hastef:0596

    Note: Published in: "Finance and Stochastics", Vol 9, No 2, pp 197-209, (2005).
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    Postal: The Economic Research Institute, Stockholm School of Economics, P.O. Box 6501, 113 83 Stockholm, Sweden
    Phone: +46-(0)8-736 90 00
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    Web page: http://www.hhs.se/
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    Related research

    Keywords: Mathematical Finance; Fractional Brownian motion; Arbitrage; option; financial derivatives; wick;

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    Cited by:
    1. Høg, Espen P. & Frederiksen, Per H., 2006. "The Fractional Ornstein-Uhlenbeck Process: Term Structure Theory and Application," Finance Research Group Working Papers F-2006-01, University of Aarhus, Aarhus School of Business, Department of Business Studies.
    2. Alberto Ohashi, 2008. "Fractional term structure models: No-arbitrage and consistency," Papers 0802.1288, arXiv.org, revised Sep 2009.

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