Memorandum, Department of Economics, University of Oslo 20/2011
This paper considers a sequence of discrete-time random walk markets with a single risky asset, and gives conditions for the existence of arbitrage opportunities 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 examples are established where the continuous analogue is arbitrage-free under small transaction costs - including for the semimartingale modifications of fractional Brownian motion suggested in the seminal Rogers (1997) article proving arbitrage in fBm models.
Stock price model random walk Gaussian processes weak convergence free lunch with vanishing risk arbitrage transaction costs