A fractional binary market is a binary model approximation for the fractional Black-Scholes model, which Sottinen constructed with the help of a Donsker-type theorem. In a binary market the non-arbitrage condition is expressed as a family of conditions on the nodes of a binary tree. We call "arbitrage points" the nodes which do not satisfy such a condition and "arbitrage paths" the paths which cross at least one arbitrage point. In this work, we provide an in-depth analysis of the asymptotic proportion of arbitrage points and arbitrage paths. Our results are obtained by studying an appropriate rescaled disturbed random walk.

Abstract. We study the arbitrage opportunities in the presence of transaction costs in a sequence of binary markets approximating the fractional BlackScholes 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/ √ N ). Next, we characterize the asymptotic behavior of the smallest transaction costs λ, 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 λ 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.

Abstract. We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for large financial markets with small proportional transaction costs λn on market n in terms of contiguity properties of sequences of equivalent probability measures induced by λn-consistent price systems. These results are analogous to the frictionless case, compare [16], [23]. Our setting is simple, each market n contains two assets with continuous price processes. The proofs use quantitative versions of the Halmos-Savage Theorem, see [24], and a monotone convergence result of nonnegative local martingales. Moreover, we present an example admitting a strong asymptotic arbitrage without transaction costs; but with transaction costs λn > 0 on market n (λn → 0 not too fast) there does not exist any form of asymptotic arbitrage.

Abstract. The goal of this work is to study binary market models with transaction costs, and to characterize their arbitrage opportunities. It has been already shown that the absence of arbitrage is related to the existence of λ-consistent price systems (λ-CPS), and, for this reason, we aim to provide conditions under which such systems exist. More precisely, we give a characterization for the smallest transaction cost λc (called "critical" λ) starting from which one can construct a λ-consistent price system. We also provide an expression for the set M(λ) of all probability measures inducing λ-CPS. We show in particular that in the transition phase "λ = λc" these sets are empty if and only if the frictionless market admits arbitrage opportunities. As an application, we obtain an explicit formula for λc depending only on the parameters of the model for homogeneous and also for some semi-homogeneous binary markets.

We study, from the perspective of large financial markets, the asymptotic arbitrage (AA) 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 AA. For this reason, we follow a constructive approach to show first that a strong AA (SAA) exists in the frictionless case. 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 trading strategies leading to the desired result. Next, we introduce, in each small market, proportional transaction costs, and we show that a slight modification of the previous trading strategies leads to a SAA when the transaction costs converge fast enough to 0.