No-Arbitrage Bounds for Scenarios and Financial Optimization

We derive no-arbitrage bounds for expected excess returns to generate scenarios used in financial optimization. The bounds allow to distinguish three regions: one where arbitrage opportunities will never exist, a second where arbitrage may be present, and a third, where arbitrage opportunities will always exist. No-arbitrage bounds are derived in closed form for a given covariance matrix using the least possible number of scenarios. The same setting is also used in an algorithm to generate discrete scenarios and trees. Numerical results from solving two-stage asset allocation problems indicate that even for minimal tree size very accurate results can be obtained.