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Arbitrage in a discrete version of the Wick-Fractional Black Scholes model

Bender, C.; Elliott, R.
Fonte: Inst Operations Research Management Sciences Publicador: Inst Operations Research Management Sciences
Tipo: Artigo de Revista Científica
Publicado em //2004 Português
Relevância na Pesquisa
66.32%
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 ½ < H < 1 the result is applied to a discrete version of the (Wick-)fractional Black-Scholes market.; Christian Bender and Robert J. Elliott

Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus

Bender, Christian; Parczewski, Peter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 08/10/2010 Português
Relevância na Pesquisa
46.3%
We approximate the solution of some linear systems of SDEs driven by a fractional Brownian motion $B^H$ with Hurst parameter $H\in(\frac{1}{2},1)$ in the Wick--It\^{o} sense, including a geometric fractional Brownian motion. To this end, we apply a Donsker-type approximation of the fractional Brownian motion by disturbed binary random walks due to Sottinen. Moreover, we replace the rather complicated Wick products by their discrete counterpart, acting on the binary variables, in the corresponding systems of Wick difference equations. As the solutions of the SDEs admit series representations in terms of Wick powers, a key to the proof of our Euler scheme is an approximation of the Hermite recursion formula for the Wick powers of $B^H$.; Comment: Published in at http://dx.doi.org/10.3150/09-BEJ223 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)