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 have 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 BlackScholes 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 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 BlackScholes 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 (2003), a naive buy-and-hold strategy does not in general qualify as “selffinancing”. 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”.