Path-dependent scenario trees for multistage stochastic programmes in finance

Abstract

The formulation of dynamic stochastic programmes for financial applications generally requires the definition of a risk–reward objective function and a financial stochastic model to represent the uncertainty underlying the decision problem. The solution of the optimization problem and the quality of the resulting strategy will depend critically on the adopted financial model and its consistency with observed market dynamics. We present a recursive scenario approximation approach suitable for financial management problems, leading to a minimal yet sufficient representation of the randomness underlying the decision problem. The method relies on the definition of a benchmark probability space generated through Monte Carlo simulation and the implementation of a scenario reduction scheme. The procedure is tested on an interest rate vector process capturing market and credit risk dynamics in the fixed income market. The collected results show that a limited number of scenarios is sufficient to capture the exposure of the decision maker to interest rate and default risk.

Keywords:

• Stochastic programming
• Financial applications
• Scenario generation
• Monte Carlo methods
• Interest credit risk

Acknowledgements

The authors acknowledge the support given by research projects PRIN2005, ‘Credit risk measurement and control for portfolios of defaultable securities’ (No. 2005139555_002sci. to V. Moriggia), and PRIN2007 ‘Optimization of stochastic dynamic systems with applications to finance’ (No. 20073BZ5A5sci. to G. Consigli). The comments of two anonymous referees substantially improved the initial version of this article.