Fractional white noise calculus and applications to finance

The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick-) It (o) over cap type of stochastic differential equations driven by fractional Brownian motion B-H(t); 1/2 < H < 1. We show that if we use an It (o) over cap type of stochastic integration with respect to B-H(t) (as developed in Ref. 8), then the corresponding It (o) over cap fractional Black-Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our It (o) over cap fractional Black-Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).