Are Interest Rate Derivatives Spanned by the Term Structure of Interest Rates?

Abstract

THE JOURNAL OF FIXED INCOME 75 T he U.S. dollar LIBOR market includes both interest rates and interest rate options. We ask whether a common finite-dimensional system spans both types of instruments. We find that the options market exhibits factors seemingly independent of the underlying yield curve. There are three common factors in LIBOR and swap rates, yet these factors can explain only a little over half of the variation in the swaption implied volatilities. Three additional factors are needed to capture the movement of the implied volatility surface.Our interest in this issue arises because we see two groups of practitioners in the fixedincome market take sharply different approaches in applying interest rate models. Those trading the yield curve take an approach quite similar to academicians who work on equilibrium or structural models. They choose a two-or threefactor structural model, estimate the model parameters from the time series data, and then on each day choose the level of the state variables (factors) to fit the current term structure. The discrepancies between the fitted yield curve and the market prices are perceived as potential trading opportunities.Interest rate options traders take the yield curve as a given, with minimal or no smoothing. To fit the observed yield curve perfectly, they often allow some of the model parameters to be time-inhomogeneous. The model is then solved based on no-arbitrage conditions as in Heath, Jarrow, and Morton [1992], assuming some term structure for the volatilities. Even so, they may need to recalibrate the model every day, changing the parameters to match the newly observed implied volatility surface.The fact that yield curve traders apply a low-dimensional structural model implies that a finite set of state variables are sufficient to capture most of the common variations on the yield curve. The idiosyncratic part is potentially due to transient technical dislocations. Yet the full yield curve fitting practice in the options market implies that option traders do not want to be exposed to the idiosyncratic risks in the interest rate market, potentially because they have their own independent risks to deal with.Our empirical findings confirm such conjectures. Similar to Litterman and Scheinkman [1991], we identify three common interest rate factors that explain 99.50% of the variation on the yield curve. When applied to swaption implied volatilities, however, these three factors explain only 59.48% of the variation in that market. Simulation analysis further indicates that the low percentage is not an artifact of the principal components analysis technique we use, but rather a true feature of the data.To adequately explain the variation in the implied volatility surface, we need three additional volatility factors. These volatility factors are independent of the three interest rate factors, yet they are crucial in explaining the movement of interest rate options. Together with the three interest rate factors, they explain 97.62% of the variation in the implie...

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Nonparametric Estimation of State-Price Densities Implicit in Interest Rate Cap Prices

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“…This assumption implies that xed income markets are complete and that DTSMs can simultaneously price bonds and interest rate options. However, Collin-Dufresne and Goldstein (2002), Heidari and Wu (2003), and Li and Zhao (2006) document the existence of systematic stochastic volatility factors in interest rate derivatives markets that cannot be spanned by bond market factors. These studies demonstrate the limitations of existing DTSMs and suggest that models with USV factors are needed to price interest rate options.…”