PUB - Publikationen an der Universität Bielefeld

We study the arbitrage opportunities in the presence of transaction costs in a sequence of binary markets approximating the fractional Black-Scholes model. This approximating sequence was constructed by Sottinen and named fractional binary markets. Since, in the frictionless case, these nuirkets admit arbitxage, we aim to determine the size of the transaction costs needed to eliminate the arbitrage from these models. To gain more insight, we first consider only 1-step trading strategies and we prove that arbitrage opportunities appear when the transaction costs are of order o(1/root N). Next, we characterize the asymptotic behavior of the smallest transaction costs lambda((N))(c), called "critical" transaction costs, starting from which the arbitrage disappears. Since the fractional Black-Scholes model is arbitrage-free under arbitrarily small transaction costs, one could expect that lambda((N))(c) converges to zero. However, the true behavior of lambda((N))(c) is opposed to this intuition. More precisely, we show, with the help of a new family of trading strategies, that lambda((N))(c) converges to one. We explain this apparent contradiction and conclude that it is appropriate to see the fractional binary markets as a large financial market and to study its asymptotic arbitrage opportunities. Finally, we construct a 1-step asymptotic arbitrage in this large market when the transaction costs are of order o(1/N-H), whereas for constant transaction costs, we prove that no such opportunity exists.