Asymptotic proportion of arbitrage points in fractional binary markets

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.