A parsimonious model for generating arbitrage-free scenario trees

Abstract

Simulation models of economic, financial and business risk factors are widely used to assess risks and support decision-making. 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 requirement that is usually overlooked when one departs from the application domain of security pricing: the no-arbitrage condition. We formulate a moment matching model to generate multi-factor scenario trees for stochastic optimization satisfying no-arbitrage restrictions with a minimal number of scenarios and without any distributional assumptions. The resulting global optimization problem is quite general. However, it is non-convex and can grow significantly with the number of risk factors, and we develop convex lower bounding techniques for its solution exploiting the special structure of the problem. Applications to some standard problems from the literature show that this is a robust approach for tree generation. We use it to price a European basket option in complete and incomplete markets.

Keywords:

• Scenario trees
• Global optimization
• Convex lower bounding
• Stochastic programming
• Pricing in incomplete markets

Notes

No potential conflict of interest was reported by the authors.

1 We leave out the literature on simulations for security pricing, see, e.g. Glasserman (2004), which focuses on a specific problem and hence may take advantage of specific stochastic process structures; we take up this issue in the application section.

2 According to theory, risk neutral probabilities have to be strictly greater than zero.

3 Note that lower bounds of —and also of , —are strictly greater than zero. There is no rule to determine them and we use 1E04.

4 The error tolerances for the heuristic and are set to 1E03 (default value) and to 5E02. We choose these values after an exploratory phase, where we tried different tolerance values and picked values such that the heuristic converges for all test problems.

5 The symbols of the columns correspond to the equations of Problem 1. For readability, we omit the errors for the normalization constraints for and , which are in the range 3.26E6 to 9.33E7.