Strong asymptotic arbitrage in the large fractional binary market

TL;DR

This paper investigates the existence of strong asymptotic arbitrage opportunities in large fractional binary markets, especially when approximating the fractional Black-Scholes model, considering both frictionless and transaction cost scenarios.

Contribution

It demonstrates the presence of strong asymptotic arbitrage in large fractional binary markets without transaction costs and with diminishing transaction costs, using constructive probabilistic methods.

Findings

Strong asymptotic arbitrage exists without transaction costs.

Arbitrage opportunities persist when transaction costs decrease rapidly.

Constructive strategies are developed for both scenarios.

Abstract

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.