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Title:A note on Wick products and the fractional Black-Scholes model PDF Logo
Authors:Björk, Tomas
Hult, Henrik
Issue Date:2005
Series/Report no.:SSE/EFI Working Paper Series in Economics and Finance 596
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.
Subjects:Mathematical Finance
Fractional Brownian motion
Arbitrage
option
financial derivatives
wick
JEL:G10
Document Type:Working Paper
Appears in Collections:SSE/EFI Working Paper Series in Economics and Finance, EFI - The Economic Research Institute, Stockholm School of Economics

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