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 amount:

In what follows, we ignore explicit time subscripting. Value at risk (VaR) changes as market weights diverge from target levels. Define VaR then as:

and where z_α is the critical value corresponding to a α percent left-tail loss under the standard normal distribution (for example, 2.5 percent of the time, z is less than –1.96). Thus, VaR at time t is the product of the period t estimated portfolio risk σ, dollar exposure , and tracking error . We consequently define VaR as the risk function, that is, it is the minimization of this risk due to tracking error that is to be minimized (portfolio risk à la Markowitz has already been addressed at portfolio inception). This risk is reduced due to rebalancing at cost:

We wish to minimize the net loss:

with control variable . More specifically, we wish to choose a time path for the control (rebalancing) that minimizes the loss function along its entire path between periods and . The periods t₀ and t₁ are arbitrary. Since this occurs for a specific interval of time , we then minimize discounted losses. This can be represented by:

subject to:

, which is a Wiener process.

Here, is a differential that describes the evolution of the weights; part is due to rebalancing and part to market-induced stochastic movements in asset prices (the Wiener process). The final step in the setup is to set the end point (transversality) conditions. We require that the weights at equal , and at , weights be . If , then the target does not change.

The solution technique involves first setting up the current period Hamiltonian:

Differentiating with respect to the state variable (w) and the control variable produces two first-order conditions:

It is straightforward to show that the second-order conditions (with respect to ) satisfy a minimum. The first equation can be solved to show that:

while from the second first-order condition, it is the case that:

Solving:

where is the constant of integration. Substituting this solution into from before and solving yields:

Recalling that , then substituting it for optimal solves the optimal path for the actual weights as:

The last step is to impose the endpoint restrictions that solve for the constants of integration and . (We do this by solving and individually for and .) Doing this and substituting the values for these constants back into w and yield the optimal paths:

It is then easy to check that is correct, since and , that is, the state variable satisfies the transversality conditions. The control variable is of particular interest since it is the value of this variable in each period that is set to minimize the net cost of additional risk induced by divergence of w from (, on the other hand, is the optimal path for the state variable).