SSE/EFI Working Paper Series in Economics and Finance
A Note on Wick Products and the Fractional Black-Scholes Model
() and Henrik Hult
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".
Keywords: Mathematical Finance; Fractional Brownian motion; Arbitrage; option; financial derivatives; wick; (follow links to similar papers)
JEL-Codes: G10; (follow links to similar papers)
13 pages, April 25, 2005
Published in: "Finance and Stochastics", Vol 9, No 2, pp 197-209, (2005).
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- This paper is published as:
Björk, Tomas and Henrik Hult, (2005), 'A Note on Wick Products and the Fractional Black-Scholes Model', Finance & Stochastics, Vol. 9, No. 2, pages 197-209
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