Arbitrage and sampling uncertainty in financial stochastic programming models

Asset liability management (ALM) is an important and challenging problem for institutional investors and financial intermediaries. The requirement to fulfill its liablilities constrains the institutional investor in its asset allocation possiblilites. We formulate an ALM model for pension funds as a multistage stochastic programming model. Relevant variables such as future inflation rates, stock retruns, and bond yields are unknown. This is incorporated in the ALM model by means of an event tree, which represents the expected development of the economic variables as well as the corresponding uncertainty. The event tree is constructed by sampling from a time series model for the variables, and is therefore subject to sampling uncertainty. Moreover, for the event tree to be realistic, it is required not to exhibit arbitrage opportunies. In ths paper we examine the effect of sampling uncertainty and the structure of the event tree on the optimal policies. Furthermore, we consider the effect of random sampling and the tree structure on the probability of arbitragefree trees. We also compare the optimal solutions to the ALM problem for trees with an without arbitrage. For these purposes, we consider data from a Dutch pension fund.