Generating Multi-factor Arbitrage-Free Scenario Trees with Global Optimization

Simulation models of economic, financial and business risk factors are widely used to assess risk exposures and support decisions. Extensive literature on scenario generation methods aims at describing some underlying stochastic processes with the least number of scenarios to overcome the "curse of dimensionality". There is, however, an important issue that is usually overlooked when one departs from the application domain of security pricing: the no-arbitrage restriction. We formulate a moment matching model to generate multi-factor scenario trees satisfying no-arbitrage restrictions as a global optimization problem. While general in its formulation the resultant model is nonconvex and can grow substantially even for a modest number of assets and scenarios. Exploiting the special structure of the problem we develop convex lower bounding techniques for its solution. Applications to some standard problems from the literature illustrate that this is a reliable approach to stochastic tree generation and is used to price a European basket option in complete and incomplete markets.