Critical transaction costs and 1-step asymptotic arbitrage in fractional binary markets

TL;DR

This paper investigates how transaction costs influence arbitrage opportunities in fractional binary markets, revealing that the critical costs for arbitrage do not vanish but tend to one, contrasting with the continuous fractional Black-Scholes model.

Contribution

It introduces a new analysis of asymptotic arbitrage in fractional binary markets, showing the critical transaction costs converge to one and constructing explicit arbitrage strategies.

Findings

Arbitrage appears when transaction costs are smaller than order 1/√N.

Critical transaction costs converge to one as market size increases.

Asymptotic arbitrage exists for costs smaller than order 1/N^H.

Abstract

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.