A Dual Characterisation of Regulatory Arbitrage for Expected Shortfall

We study portfolio selection in a one-period financial market with an Expected Shortfall (ES) constraint. Unlike in classical mean-variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation regulatory arbitrage and show that the presence or absence of regulatory arbitrage for ES is intimately linked to the fine structure of equivalent martingale measures (EMMs) for the discounted risky assets. More precisely, we prove that the market does not admit regulatory arbitrage for ES at confidence level α if and only if there exists an EMM Q ≈ P such that ll dQ/dP ll_∞ < 1/α