Robust statistical arbitrage strategies

Abstract

We investigate statistical arbitrage strategies when there is ambiguity about the underlying time-discrete financial model. Pricing measures are assumed to be martingale measures calibrated to prices of liquidly traded options, whereas the set of admissible physical measures is not necessarily implied from market data. Our investigations rely on the mathematical characterization of statistical arbitrage, which was originally introduced by Bondarenko [Statistical arbitrage and securities prices. Rev. Financ. Stud., 2003, 16, 875–919]. In contrast to pure arbitrage strategies, statistical arbitrage strategies are not entirely risk-free, but the notion allows one to identify strategies which are profitable on average, given the outcome of a specific σ-algebra. Besides a characterization of robust statistical arbitrage, we also provide a super-/sub-replication theorem for the construction of statistical arbitrage strategies for path-dependent options. In particular, we show that the range of statistical arbitrage-free prices is, in general, much tighter than the range of arbitrage-free prices.