Common Correlation Structures for Calibrating the Libor Model

Abstract

In 1997 three papers that introduced very similar lognormal diffusion processes for interest rates appeared virtually simultaneously. These models, now commonly called the 'LIBOR models' are based on either lognormal diffusions of forward rates as in Brace, Gatarek & Musiela (1997) and Miltersen, Sandermann & Sondermann (1997) or lognormal diffusions of swap rates, as in Jamshidian (1997). The consequent research interest in the calibration of the LIBOR models has engendered a growing empirical literature, including many papers by Brigo and Mercurio, and Riccardo Rebonato. The art of model calibration requires a reasonable knowledge of option pricing and a thorough background in statistics - techniques that are quite different to those required to design no-arbitrage pricing models. Researchers will find the book by Brigo and Mercurio (2001) and the forthcoming book by Rebonato (2002) invaluable aids to their understanding. I aim to provide an accessible account of some interesting problems in LIBOR model calibration, but the ideas are complex, so notational complexities are reduced to a minimum. We take fresh look at three important modelling decisions when calibrating the LIBOR model: the parameterization of the correlation matrix for semi-annual forward rates; the use of principal component analysis in the orthogonal transform of the log normal forward rate model; and the iterations for recovering caplet volatilities from 'flat' cap volatilities. The first section provides the briefest of introductions to the lognormal forward rate version of the LIBOR model. Section two considers the calibration of the model to the cap market, where it is shown that the iteration of caplet volatilities from the 'flat' cap volatilities quoted in the market, should be performed by equating a vega weighted sum of caplet volatilities to the cap volatility. Section 3 discusses the more difficult problem of calibration to the swaption market. Two full rank parsimonious parameterizations of the semi-annual correlation matrix are specified, where correlations between annual forward rates are determined by the same parameters. Section 4 reconsiders calibration to the swaption market where the rank of forward rate correlation matrices is reduced by setting all but the three largest eigenvalues to zero. Rebonato (1999a), Rebonato and Joshi (2001) Hull and White (1999, 2000) and Longstaff, Santa-Clara and Schwartz (1999) have all found that this technique is useful for performing the simulations that are necessary for pricing path dependent options. The implication of zeroing eigenvalues is a transformation of the lognormal forward rate model where each forward rate is driven by three orthogonal factors that are derived from a principal component analysis. We show that, in fact, it is the common principal components model of Flury (1988) that should be used. That is, the same eigenvectors should be calibrated to all swaptions of the same tenor, and we advocate the use of market data rather than historical data for the calibration of these common eigenvectors.

Keywords: Calibration, Correlation, Common Principal Component Analysis, LIBOR Model, Log Normal Forward rate Model, Forward Rates, Interest Rate Models

JEL Classification: G13