Arbitrage-free scenario trees for financial optimization

This paper presents a method which is designed to generate arbitrage-free scenario trees representing multivariate return distributions. We derive bounds on expected excess returns required to achieve this objective. Our approach is embedded in the setting of arbitrage pricing theory (APT), and asset returns are assumed to be driven by orthogonal factors. We derive no-arbitrage bounds for the least possible number of scenarios (i.e. the smallest dimension of the discrete state-space) necessary to match the first two moments and to exclude arbitrage at the outset. This not only safeguards against the curse of dimensionality: Numerical results from solving two-stage asset allocation problems show that highly accurate results can be obtained with the smallest possible scenario tree.