  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 markets admit arbitrage,
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 $o(1/\sqrt{N})$. Next, we characterize the
asymptotic behavior of the smallest transaction costs $\lambda_c^{(N)}$, 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 $\lambda_c^{(N)}$ converges to
zero. However, the true behavior of $\lambda_c^{(N)}$ is opposed to this
intuition. More precisely, we show, with the help of a new family of trading
strategies, that $\lambda_c^{(N)}$ 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.
