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European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Using the Black–Derman–Toy interest rate model for portfolio optimization Alex Weissensteiner *
Department of Banking and Finance, University of Innsbruck, Universitätsstraße 15, 6020 Innsbruck, Austria
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No-arbitrage interest rate models are designed to be consistent with the current term structure of interest rates. The diffusion of the interest rates is often approximated with a tree, in which the scenariodependent fair price of any security is calculated as the present value of the risk-neutral expectation by backward induction. To use this tree in a portfolio optimization context it is necessary to account for the so-called ‘‘market price of risk”. In this paper we present a method to change the conditional probabilities in the Black–Derman–Toy model to the physical (or real) measure, including the market price of risk, and explore the economic implications for expected spot rates and for expected bond returns. Ó 2009 Elsevier B.V. All rights reserved.
Article history: Received 12 August 2008 Accepted 24 April 2009 Available online 3 May 2009 Keywords: Finance Stochastic programming Asset/liability management Change of measure Portfolio optimization
1. Introduction No-arbitrage interest rate models are used to price interest-sensitive securities under a risk-neutral measure. Depending on the number of sources of uncertainty one can distinguish one-factor models from two- or multi-factor approaches. In one-factor models a shock to the short rate is transmitted equally through all maturities, i.e., the rates for all maturities move in the same direction due to perfect correlation (see, e.g. Brigo and Mercurio, 2006). This simplistic feature is a clear drawback when pricing payoffs which depend on the joint distribution of interest rates with different maturities, e.g. for swaptions. In such cases, to incorporate a more realistic evolution for the instantaneous covariance structure, twoor multi-factor models are used. In general, if the resulting model is beyond analytical tractability prices are calculated by backward induction of a calibrated interest rate tree. In this paper we focus, without loss of generality, on the onefactor (Black et al., 1990) lattice model (BDT). Beginning with Mulvey and Zenios (1994) and Golub et al. (1995) this model is frequently used by the operations research community in portfolio optimization tasks as a discrete approximation of the interest rate uncertainty. Having only two successor nodes the BDT-model generates comparatively smaller scenario trees, which allow for computational tractability of problems with many decision stages. We examine the relation of this interest rate model to the local expectation hypothesis (LEH). As shown by Cox et al. (1981) and Chance and Rich (2001), the one-period-ahead forward price of any security is equal to the expected spot price at this time under the risk-neutral probabilities. The forward price more than one * Tel.: +43 512 5077556; fax: +43 512 5072846. E-mail address: email@example.com 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.04.020
period ahead does not, however, equal the expected spot price. Further, the forward rate, regardless of the number of periods ahead, does not equal the expected spot rate. The whole tree, with the corresponding interest rates and asset prices in each node, represents the uncertainty of the future. Under the risk-neutral or equivalent martingale measure, which is used for pricing, the expected return of each security is the same and equal to the spot interest rate. In the real world, however, investors typically require compensation for bearing (interest rate) risks, i.e., they...