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An Optimal Control Problem

Some light can be shed on the nature of this problem, however, and its solution. What follows is an approach that provides some middle ground. It is a dynamic optimal control problem that carries less baggage relative to Leland, that is, the asset dynamics are less involved and so are the differentials to be solved for, but it has a well-defined objective function whose solution (the level of rebalancing) provides a significant amount of insight into the rebalancing problem. It is also implementable, as will be illustrated empirically at the end of this chapter.

First, let's begin with a statement of the problem. Once again, assume that a trader wishes to hold risky assets in target proportions but for whom divergence between target levels and actual (market) levels create increases in portfolio risk (as measured using VaR). For expositional simplicity, assume a single risky asset (the analysis will generalize to the multiple asset case). Define the following:

Divergence from target weights:

Portfolio value at time t:

Portfolio risk at time t:

Rebalancing ...