In this paper we construct periodically rebalanced dynamic trading strategies for a portfolio containing a small number of tradable instruments which daily tracks the value of a large 'target' portfolio over a long period of time. A successful solution to this 'tracking' problem is practically useful in a number of financial applications. One such is the investment problem of index-tracking, where the target portfolio consists of the constituent assets of some equity or bond index such as the FTSE-100 or the EMBI+ (see e.g. Worzel, Vassiadou Zeniou ? Zenios, 1994) but here we will address a more complex problem involving nonlinear instruments from a risk management perspective. For risk managers the resolution of the tracking problem can be seen as a way of reducing extensive and complex daily value at risk (VaR) calculations for the target portfolio to the simpler task of evaluating the VaR of the tracking portfolio—we refer to this application as portfolio compression. In an investment bank the target portfolio would typically be very large.

The approach can also be useful as a hedging tool—which extends Black-Scholes delta hedging replication for a single option to a typical large portfolio of options or other derivatives of various maturities by finding a dynamic replication strategy for the target portfolio. A practical application might involve using a collection of liquid instruments to track the value of a much less liquid target.