Binary Markets Under Transaction Costs

Abstract

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.

Strong asymptotic arbitrage in the large fractional binary market

“…Sottinen proves in [14] that, for N sufficiently large, the N -fractional binary market admits arbitrage. From the results in [4], we conclude that the smallest transaction cost, λ N c , needed to eliminate the arbitrage in the N -fractional binary market is strictly positive. Moreover, from [5], we know that Section 3 and Section 4 the main results are concentrated.…”

“…According to [23,Theorem 5.3], the markets in this sequence also allow for arbitrage opportunities, which persist even under sufficiently small transaction costs (see [7]). Moreover, in [6] it is shown that the smallest transaction cost, λ N c , needed to eliminate the arbitrage in the N -period fractional binary market (called N -fractional binary market) is asymptotically close to 1.…”

Critical Transaction Costs and 1-Step Asymptotic Arbitrage in Fractional Binary Markets

“…In addition, if we decompose a multi-step binary market in 1-step sub-markets, one can obtain the following lower bound for λ c (see [5,Proposition 3.1]):…”

“…In the friction case, the arbitrage condition was studied in [5], where the authors provide a characterization of the smallest transaction costs, called "critical" transaction costs and denoted by λ c , needed to remove arbitrage opportunities, i.e λ c = inf{λ ∈ [0, 1] : there is no λ-arbitrage}.…”

Asymptotic proportion of arbitrage points in fractional binary markets

“…As mentioned by Sottinen, one may expect that the arbitrage disappears when transaction costs are taken into account. This latter problem was treated in its most generality in [4], where a characterization of the smallest transaction cost (called "critical" transaction costs) starting from which the arbitrage is eliminated is provided. However, since the parameters of the model depend on time and space, this characterization does not give a closed-form solution, but reduces to solving an optimization problem in a binary tree.…”