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A note on Wick products and the fractional Black-Scholes model

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  • Tomas Björk
  • Henrik Hult

Abstract

In some recent papers (Elliott and van der Hoek 2003; Hu and Øksendal 2003) a fractional Black-Scholes model has been proposed as an improvement of the classical Black-Scholes model (see also Benth 2003; Biagini et al. 2002; Biagini and Øksendal 2004). Common to these fractional Black-Scholes models is that the driving Brownian motion is replaced by a fractional Brownian motion and that the Itô integral is replaced by the Wick integral, and proofs have 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 objective 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 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 2003, a naive buy-and-hold strategy does not in general qualify as “self-financing”. We also show that in Hu and Øksendal 2003, a portfolio consisting of a positive number of shares of a stock with a positive price may, with positive probability, have a negative “value”. Copyright Springer-Verlag Berlin/Heidelberg 2005

Suggested Citation

  • Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
    • Handle: RePEc:spr:finsto:v:9:y:2005:i:2:p:197-209
    DOI: 10.1007/s00780-004-0144-5
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    References listed on IDEAS

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    1. Fred Espen Benth, 2003. "On arbitrage-free pricing of weather derivatives based on fractional Brownian motion," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(4), pages 303-324.
    2. Biagini, Francesca & Hu, Yaozhong & Øksendal, Bernt & Sulem, Agnès, 0. "A stochastic maximum principle for processes driven by fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 233-253, July.
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    More about this item

    Keywords

    Mathematical finance; fractional Brownian motion; arbitrage;
    All these keywords.

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)

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