Journal
INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS
Volume 6, Issue 1, Pages 1-32Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0219025703001110
Keywords
fractional Brownian motions; fractional white noises; chaos expansion; Wick calculus; Fractal Black-Scholes market; arbitrage; price formula; replicating portfolio
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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).
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