On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes

Abstract

This paper considers a sequence of discrete-time random walk markets with a safe and a single risky investment opportunity, and gives conditions for the existence of arbitrages or free lunches with vanishing risk, of the form of waiting to buy and selling the next period, with no shorting, and furthermore for weak convergence of the random walk to a Gaussian continuous-time stochastic process. The conditions are given in terms of the kernel representation with respect to ordinary Brownian motion and the discretisation chosen. Arbitrage and free lunch with vanishing risk examples are established where the continuous-time analogue is arbitrage-free under small transaction costs—including for the semimartingale modifications of fractional Brownian motion suggested in the seminal Rogers [Math. Finance 7 (1997) 95–105] article proving arbitrage in fBm models.