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

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”.

Authors and Affiliations

1. Department of Finance, Stockholm School of Economics, PO Box 6501, 113 83, Stockholm, Sweden

   Tomas Björk

2. Department of Applied Mathematics and Statistics, Universitetsparken 5, 2100, Copenhagen, Denmark

   Henrik Hult

Corresponding author

Correspondence to Tomas Björk.

Received: August 2004,

Mathematics Subject Classification (2000):

91B28, 60H05

JEL Classification:

G10

Support of the first author from the Jan Wallander and Tom Hedelius foundation is gratefully acknowledged. The research of the second author is supported by the Swedish Research Council.

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