  We study, from the perspective of large financial markets, the asymptotic
arbitrage opportunities in a sequence of binary markets approximating the
fractional Black-Scholes model. This approximating sequence was introduced by
Sottinen and named fractional binary market. The large financial market under
consideration does not satisfy the standard assumptions of the theory of
asymptotic arbitrage. For this reason, we follow a constructive approach to
show first that a strong type of asymptotic arbitrage exists in the large
market without transaction costs. Indeed, with the help of an appropriate
version of the law of large numbers and a stopping time procedure, we construct
a sequence of self-financing strategies, which leads to the desired result.
Next, we introduce, in each small market, proportional transaction costs, and
we construct, following a similar argument, a sequence of self-financing
strategies providing a strong asymptotic arbitrage when the transaction costs
converge fast enough to 0.
