Statistical arbitrage in the multi-Asset black-scholes economy

In this study, we consider the statistical arbitrage definition given in Hogan, S, R Jarrow, M Teo and M Warachka (2004). Testing market efficiency using statistical arbitrage with applications to momentum and value strategies, Journal of Financial Economics, 73, 525-565 and derive the statistical arbitrage condition in the multi-Asset Black-Scholes economy building upon the single asset case studied in Göncü, A (2015). Statistical arbitrage in the Black Scholes framework. Quantitative Finance, 15(9), 1489-1499. Statistical arbitrage profits can be generated if there exists at least one asset in the economy that satisfies the statistical arbitrage condition. Therefore, adding a no-statistical arbitrage condition to no-Arbitrage pricing models is not realistic if not feasible. However, with an example we show that what excludes statistical arbitrage opportunities in the Black-Scholes economy, and possibly in other complete market models, is the presence of uncertainty or stochasticity in the model parameters. Furthermore, we derive analytical formulas for the expected value and probability of loss of the statistical arbitrage portfolios and compute optimal boundaries to sell the risky assets in the portfolio by maximizing the expected return with a constraint on the probability of loss.