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Arbitrage in a Discrete Version of the Wick-Fractional Black-Scholes Market

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany
Haskayne School of Business, Scurfield Hall, University of Calgary, 2500 University Drive, Calgary, Alberta, Canada T2N 1N4
bender{at}wias-berlin.de
relliott{at}ucalgary.ca

We consider binary market models based on the discrete Wick product instead of the pathwise product and provide a sufficient criterion for the existence of an arbitrage. This arbitrage is explicitly constructed in the class of self-financing one-step buy-and-hold strategies, (i.e., the investor holds shares of the stock only at one time step). Using coefficients obtained from an approximation of a fractional Brownian motion with Hurst parameter < H < 1 the result is applied to a discrete version of the (Wick-)fractional Black-Scholes market.

Key Words: arbitrage; binary market models; discrete Wick products; fractional Brownian motion
History: Received: March 28, 2003; revision received: June 16, 2003;revision received: January 7, 2004;