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

机译：关于Wick产品和分数Black-Scholes模型的注释

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摘要

In some recent papers (Elliott and van der Hoek 2003; Hu and Oksendal 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 Oksendal 2004). 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 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 Oksendal (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".
机译：在最近的一些论文中（Elliott和van der Hoek 2003; Hu和Oksendal 2003），提出了分数Black-Scholes模型作为经典Black-Scholes模型的改进（另请参见Benth 2003; Biagini等人2002; Biagini and Oksendal 2004）。这些分数Black-Scholes模型的共同点是，驱动布朗运动被分数布朗运动代替，而Ito积分被Wick积分代替，并且已经证明这些分数Black-Scholes模型没有套利。这些缺乏套利完全性的结果与文献中的许多早期结果相矛盾，这些结果证明分数式Black-Scholes模型（及相关模型）实际上将允许套利。本文的目的是通过指出上述论文中使用的自筹资金交易策略的定义和/或投资组合价值的定义没有合理的经济解释来解决这一矛盾。这些论文中的结果在经济上没有意义。特别是，我们表明，在Elliott和van der Hoek（2003）的框架中，幼稚的买入并持有策略通常不符合“自负盈亏”的条件。我们还表明，在Hu and Oksendal（2003）中，由正数价格为正数的股票组成的投资组合可能具有负概率“负”。

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