A note on Wick products and the fractional Black-Scholes model - Tomas Björk; Henrik Hult (Stockholm School of Economics, The Economic Research Institute (EFI), 2005)
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: G10; 330; Mathematical Finance; Fractional Brownian motion; Arbitrage; option; financial derivatives; wick; Finanzmathematik; Black-Scholes-Modell; Arbitrage Pricing; Stochastischer Prozess; Theorie