A note on Wick products and the fractional Black-Scholes model. (English) Zbl 1092.91021

The purpose of the paper is to discuss precisely whether new concepts connected with financial market driven by a fractional Brownian motion are reasonable from the economic point of view. The authors compare concepts of the Wick product and self-financing introduced by R. J. Elliott and J. van der Hoek [Math. Finance 13, No. 2, 301–330 (2003; Zbl 1069.91047)] and Y. Hu and B. Øksendal [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, No. 1, 1–32 (2003; Zbl 1045.60072)]. The conclusion is unfavourable: the definition of Wick-financing portfolios used by Elliott-van der Hoek has no economic interpretation as a self-financing condition. Also, if one insists to use their pricing theory for trading purposes, then it will in some cases lead to easily implementable naive arbitrage opportunities for the counterpart. The same, replacing the standard definition of portfolio value by the Wick one as in Hu-Øksendal papers cannot be motivated from the economic point of view. The reasons are given. As a conclusion, the authors claim that in general no reasons against FBM in finance and their criticism does not extend to all the financial interpretations of models with FBM.


91G99 Actuarial science and mathematical finance
91B24 Microeconomic theory (price theory and economic markets)
60H05 Stochastic integrals
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