Ebook Dynamic Portfolio Replication Using Stochastic Programming
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.
Here we analyze instances of the daily tracking problem involving a specified target portfolio of 144 European options on the S?P 500 index of different strikes and maturities within a one year planning horizon, using the technique of dynamic stochastic programming (DSP). Thus over the horizon considered in our experiments the values of both the target and dynamic tracking portfolios will exhibit a Bermudan path-dependency at rebalance points of the latter. Gondzio, Kouwenberg ? Vorst (1998) have studied the use of DSP techniques to implement the Black-Scholes dynamic hedging strategy for a single European 0ption—a simple special case of the tracking problem studied here. The present paper is to our knowledge the first application of DSP to a realistically large portfolio containing instruments which mature within the planning horizon.
Over the last few years leading edge risk management practice has evolved from current mark-to-market to one period forward VaR and mark-to-future techniques (Jamshidian ? Zhu 1997, Chisti 1999). When such static methodologies are applied over long horizons to target portfolios containing instruments with maturities within the horizon, they take no account of changes in portfolio composition due to instruments maturing—for replication of such portfolios dynamic trading strategies are required which may be found optimally using dynamic stochastic programming techniques. DSPs are a form of stochastic dynamic programming problem—but solved by mathematical programming techniques which allow very large numbers of decisions and high dimensional data processes at a smaller number of natural decision points or stages (such as option maturity dates) than the fine timestep typically considered in traditional dynamic programming problems. For practical purposes these latter are restricted by the large number of timesteps to only a few state and decision variables—Bellman's curse of dimensionality. DSPs are multi-stage stochastic programming problems for which the term dynamic signifies that the underlying uncertainties are being modelled as evolving in continuous time and the corresponding scenarios approximating the data process paths are to be simulated with a much finer timestep (here daily) than the (multi-day) interval between decision points.
In this paper we demonstrate that the full use of such an approach—which we term dynamic portfolio replication—provides a better solution with respect to alternative definitions of tracking error to the daily tracking problem than simpler approaches. We also confirm in our context the general view in the literature (cf. Dempster et al., 2000) regarding other DSP problems that the scenario trees required for such an approach reduce tracking error when initial branching is high.
The paper is organized as follows. In Section 2 we describe the DSP approach more fully and review the relevant recent literature. Section 3 discusses in detail the process of constructing dynamic trading strategies using DSP. We describe our particular problem and how the DSP approach is applied in this situation in Section 4. In Section 5 we report a number of numerical tests to compare the daily tracking performance of our dynamic trading strategies with simpler hedges including the portfolio delta hedge. Section 6 concludes and discusses current research directions. A brief description of a prototype implementation of the methods of this paper in Algorithmics software is given in an appendix.
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