The Performance of Multi-Factor Term Structure Models for Pricing and Hedging Caps and Swaptions

Transcription

1 The Performance of Multi-Factor Term Structure Models for Pricing and Hedging Caps and Swaptions Joost Driessen Pieter Klaassen Bertrand Melenberg This Version: January, 2002 We thank Lars Hansen, Michael Johannes, Theo Nijman, Antoon Pelsser, Alessandro Sbuelz, and an anynomous referee for many valuable comments. We also thank participants of the EC 2 -conference, Madrid 1999, and the Derivatives Day, Tilburg 2000, as well as seminar participants at Maastricht University, University of Amsterdam, and Erasmus University of Rotterdam for their comments. Joost Driessen, Finance Group, Faculty of Economics and Econometrics, University of Amsterdam. Pieter Klaassen, Credit Risk Modeling Department, ABN-AMRO Bank Amsterdam, and Department of Financial Sector Management, Vrije Universiteit, Amsterdam. Bertrand Melenberg, Department of Econometrics and CentER, Tilburg University. Corresponding author: Joost Driessen, Finance Group, Faculty of Economics and Econometrics, University of Amsterdam, Roetersstraat 11, 1018 WB, Amsterdam, The Netherlands. Tel: jdriess@fee.uva.nl.

2 The Performance of Multi-Factor Term Structure Models for Pricing and Hedging Caps and Swaptions Abstract In this paper we empirically compare a wide range of different term structure models when it comes to the pricing and, in particular, hedging of caps and swaptions. We analyze the influence of the number of factors on the hedging and pricing results, and investigate which type of data!interest rate data or derivative price data! should be used to estimate the model parameters to obtain the best hedging and pricing results. We use data on interest rates, and cap and swaption prices from 1995 to The empirical results on the hedging of caps and swaptions show that, if the number of hedge instruments is equal to the number of factors, the multi-factor models outperform one-factor models in hedging caps and swaptions. However, if one uses a large set of hedge instruments, one-factor models perform as well as multi-factor models. In terms of pricing, we find that models with two or three factors imply better out-of-sample predictions of cap and swaption prices than one-factor models. Also, estimation on the basis of current derivative prices leads to more accurate out-of-sample prediction of cap and swaption prices than estimation on the basis of interest rate data. JEL Codes: G12, G13, E43. Keywords: Term Structure Models; Interest Rate Derivatives; Option Pricing; Hedging.

3 1 Introduction Many large financial institutions use term structure models to price and hedge interest rate derivative securities. The empirical evaluation of such models has received considerable attention in the academic literature. As outlined in Section 2, little empirical evidence exists of how well one- and multi-factor models perform in terms of hedging interest rate derivatives. The performance in terms of hedging may be quite different from the performance in terms of pricing. In a simulated two-factor economy, Canabarro (1995) shows that, although one-factor models may yield accurate derivative price predictions, these models lead to a poor hedge performance for interest rate options. The main goal of this paper is to empirically analyze the performance of a wide range of both one- and multi-factor models, not only in terms of pricing, but mainly focusing on the (delta-)hedging of caps and swaptions, using weekly data on US cap and swaption prices from 1995 until We consider one- to three-factor models from three classes of models. The first class consists of standard HJM-models with volatilities parameterized as in the generalized Vasicek (1977)- model or the Hull and White (1990)-model. In the multi-factor cases we allow the factors to be correlated. The second class consists of standard HJM-models with deterministic timehomogeneous volatilities, to be determined by Principal Components Analysis. The third class consists of Libor market models (or, discrete-tenor string models) with level dependent volatilities, where, given the levels, the volatilities are deterministic and time-homogeneous. The choice of models investigated is guided by their attractiveness from an empirical point of view, which make these models likely to be chosen in practice. We use panel data on prices of US caps and swaptions, with weekly observations from 1995 to Our empirical analysis consists of the following steps. First, we estimate the parameters of the models under consideration. In case of option-based estimation we estimate the parameters for every week in our dataset, using the cross-section of cap and swaption prices. In case of interest-rate-based estimation we estimate the parameters for every week using a time-series of interest rate changes and a rolling horizon of 39 weeks. Second, we analyze for each model and for both estimation strategies the accuracy of predicting the prices of caps and swaptions out-ofsample. Following Amin and Morton (1994) and Buhler et al. (1999) we use "time-varying" estimation, as there is some evidence for time-varying interest rate volatility in the literature. In addition to the "time-varying" estimation strategy, we also investigate as comparison an -1-

4 estimation strategy where all parameters are constant throughout the sample. Therefore, we split up the sample in two, and use the second half of the sample for an out-of-sample analysis. Thirdly, for each model, we assess the hedging accuracy for caps and swaptions. We construct for each model strategies for delta-hedging caps and swaptions, assuming the validity of the underlying model, using discount bonds as hedge instruments. We calculate how much the variability of cap and swaption prices decreases if one rebalances the hedge portfolio every two weeks. The choice of hedge instruments is an important issue in our analysis. For each model, we consider two hedge strategies, factor hedging and bucket hedging. In case of factor hedging, as many hedge instruments as factors are used, while in case of bucket hedging each cap and swaption is hedged using bonds whose maturities correspond to all cash flow dates of the option. Factor hedging represents a hedge strategy that makes use of as few instruments as theoretically necessary, whereas bucket hedging probably comes closer to hedging as it is performed in practice for large books of derivative instruments. To help to judge the hedge performance, we also include several empirical hedging techniques, based on linear regression and in line with Boudoukh et al. (1995) and Collin- Dufresne and Goldstein (2001a), 1 which can serve as a comparison to these model-based hedging strategies. Assuming that the regression based models capture the true hedging relationship sufficiently well, we can investigate "how far one can get" using model-based hedging applying empirically attractive models. The empirical hedging results can be summarized as follows. If we use as many hedge instruments as the number of factors in the model (factor hedging), we find large differences between the one- and multi-factor models (with one exception). However, if we use for every cap and swaption a set of hedge instruments that corresponds to all cash flow dates of the cap or swaption (bucket hedging), the differences between the one- and multi-factor models disappear. Hence, the choice of the number of hedge instruments and the maturities of these hedge instruments seem to be more important than the particular model choice. These findings are quite robust: we obtain the same results for both interest-rate-based and option-based estimation, as well as for the constant parameter and time-varying parameter estimation approaches. Also, the differences between these approaches are quite small. In the case of a three-factor model with 1 We consider empirical hedging techniques based on linear regression, also including higher order terms to obtain sufficient flexibility; Boudoukh et al. (1995) obtain this flexibility by using Kernel-based nonparametric estimation techniques. -2-

5 option-based estimation, bucket hedging reduces the cap price variance with 79.8% and the swaption price variance with 86.9%, indicating that these models are quite succesfull in the hedging of price risk. Next to model-based hedging, we also investigate empirical hedging strategies. These strategies perform somewhat better than the model-based strategies. However, the difference is not large, indicating that the empirically attractive class of models with time-homogeneous volatility functions, combined with time-varying parameter estimation, is flexible enough to be able to yield acceptable results, assuming that the nonlinear regression models capture the true hedging relation sufficiently well. In terms of pricing, a three-factor model, applying option-based estimation, results in the best out-of-sample predictions for cap and swaption prices. In all cases, option-based estimation leads to better out-of-sample price predictions than interest-rate-based estimation, and the time-varying parameter approach yields much better results than the constant parameter approach. If one compares different models on the basis of interest-rate-based estimation only, the multi-factor models on average lead to worse predictions of cap and swaption prices than a one-factor model. This result corresponds to the results of Buhler et al. (1999), who use interest-rate-based estimation only, and implies that conclusions solely on the basis of interest-rate-based estimation might be premature. The remainder of this paper is organized as follows. In Section 2 we review the related literature and highlight our contribution. Section 3 briefly reviews the literature on HJM models and the pricing of caps and swaptions. Section 4 describes the data. In Section 5 we discuss the specification of the different models and the estimation results. In Section 6, we analyze the predictions of caps and swaption prices for the various models. In Section 7, we assess the hedging accuracy for caps and swaptions. Section 8 contains concluding remarks. 2 Contribution to Existing Literature A large part of the term structure literature has focused on the performance of term structure models in terms of the pricing of bonds, see, for example, Pearson and Sun (1994), Babbs and Nowman (1999), and Dai and Singleton (2000). More recently, interest rate derivative data are also used to empirically analyze these models, and this paper contributes to this part of the -3-

6 literature. To indicate in more detail our contribution to the existing empirical literature on interest rate derivatives, we distinguish four different aspects of these empirical analyses: (i) the models that are considered, (ii) the estimation approach, (iii) the choice of instruments, and (iv) the evaluation criterion. With respect to the term structure models, most articles use models that are specified in the Heath, Jarrow, and Morton (HJM, 1992) framework. 2 In this framework, current bond prices or interest rates are fitted exactly by construction, which is convenient when analyzing derivatives. The HJM-approach encompasses the Libor market models (LMM) 3 and the discrete-tenor string models. 4 Buhler et al. (2001) and Jagannathan et al. (2001) consider affine term structure models of Duffie and Kan (1996), in which the current term structure is not fitted exactly by construction. Some articles consider only one-factor models, but more recent articles show that these one-factor models are outperformed by multi-factor models. The models that we analyze are all specified following the Heath, Jarrow and Morton (HJM, 1992) approach. We consider up to three factors. We extend the existing literature by comparing several choices for the HJMvolatility functions, and by comparing two popular choices for the probability distribution of interest rates, Gaussian and lognormal. Second, in the existing literature essentially two estimation approaches are used: the optionbased estimation method (Amin and Morton (1994)), and the interest-rate-based estimation method (Buhler et al. (1999) and Jagannathan er al. (2000)). 5 Our paper extends these articles by applying both these estimation approaches, which enables us to analyze which type of estimation will result in the best out-of-sample hedging of derivative prices. A similar question has been studied by Chernov and Ghysels (2000) for the estimation of stochastic volatility models for equity prices. Moreover, in this way we can also assess the sensitivity of the conclusions of, for example, Amin and Morton (1994) and Buhler et al. (1999) to the specific estimation method 2 These include Flesaker (1993), Amin and Morton (1994), Moraleda and Vorst (1997), Buhler et al. (1999), Gupta and Subrahmanyam (2001), Longstaff et al. (2001), and Fan et al. (2001). 3 See Brace et al. (1997), Jamshidian (1997), and Miltersen et al. (1997). 4 See Santa-Clara and Sornette (2000) and Longstaff et al. (2001). 5 Buhler et al. (1999) call their methodology a global approach, as opposed to the local approach of Amin and Morton (1994). Global or interest-rate-based estimation is also applied by Moraleda and Vorst (1996), while the local or option-based estimation method is also used by Flesaker (1993), Moraleda and Vorst (1997), Gupta and Subrahmanyam (2001), and Fan et al. (2001). -4-

7 chosen. The third aspect of studies on interest rate derivatives concerns the instruments that are analyzed. Flesaker (1993) and Amin and Morton (1994) consider short-maturity options on shortmaturity Eurodollar futures. Buhler et al. (1999) analyzes short-maturity options on longmaturity German government bonds, Gupta and Subrahmanyam (2001) use cap price data, and Fan et al. (2001) use swaption price data. In line with Jagannathan et al. (2001) and Longstaff et al. (2001), we use data on both caps and swaptions. There is a clear advantage of using both caps and swaptions data. These derivative prices contain much information, because they contain both short- and long-maturity options, ranging from 1 month to 10 years, and these options are written on both single interest rates (caps) and combinations of interest rates of different maturities (swaptions). This variety in instruments enables us to analyze in detail both the entire volatility structure of interest rates and the correlations between these interest rates. Finally, an important fourth aspect of empirical research on option prices is the evaluation criterion that is used. Most articles focus on the fit on option prices or the prediction of option prices. 6 As mentioned above, the main contribution of this paper is to evaluate models in terms of their performance in hedging caps and swaptions. 7 Few articles have examined the hedging of interest rate derivatives, exceptions being Boudoukh et al. (1995), who study the hedging of mortgage-backed securities, and the recent studies by Gupta and Subrahmanyam (2001) who investigate the performance of hedging caps and floors in one- and two-factor models, and Fan et al. (2001) who investigate the hedging of swaptions. Our paper is different from these hedging studies because (i) we investigate the hedging of both caps and swaptions, (ii) we analyze the influence of different estimation strategies,(iii) we compare model-based hedging with empirical, regression-based hedge strategies, and (iv) we compare a wide range models. 3 Term Structure Models In this section we briefly review the HJM approach to modeling the term structure of (forward) 6 These include Flesaker (1993), Amin and Morton (1994), Moraleda and Vorst (1997), Buhler et al. (1999), Jagannathan et al. (2001), and Longstaff et al. (2001). 7 For equity options, several articles examine the effectiveness of delta-hedging, for example, Dumas et al. (1998). -5-

8 interest rates, either in terms of instantaneous forward rates, or in terms of LIBOR-forward rates. Let f(t,t) denote the instantaneous forward interest rate at time t for riskless and instantaneous borrowing or lending at date T. The key to the HJM approach is to start with modeling the processes of these instantaneous forward interest rates, given the current instantaneous forward rate curve f(0,t) : df(t,t) ' µ f (t,t, )dt % j K i'1 f,i (t,t, )dw i (t). (1) Here W are K factors, being independent Brownian Motions 8 i (t), i'1,..,k, µ f (t,t, ) is the drift function, and f,i (t,t, ) is the volatility function of factor i; T represents the state of nature. In the general set-up, both the drift function and the volatility functions can be quite general, and only have to satisfy weak regularity conditions. The process presented in (1) is under the "true" probability distribution. HJM (1992) show that in an arbitrage-free economy, the resulting drift function µ f (t,t, ) of the forward rates under the equivalent martingale measure, with the money- market account as numeraire, is completely determined by the volatility functions in (1), i.e., µ f (t,t, ) ' j K i'1 T f,i (t,t, ) m t f,i (t,u, )du. (2) This implies that for the pricing and hedging of interest rate derivatives, only the volatility functions need to be specified and estimated. Instead of instantaneous forward rates, one could model forward Libor rates, which are defined as L (t,t) ' &1 (P(t,T)/P(t,T% )&1) with P(t,T) the time t price of a zero coupon bond maturing at time T, and with * the payment period, equal to 3 months in our application. Under the condition 8 The assumption that the Brownian motions are independent is a normalization. In the given framework, a model with correlated Brownian motions can be rewritten as a model with independent Brownian motions, by modifying the volatility functions. -6-

9 T% mt L f,i (t,u, )du ' (t,t) 1% L (t,t) L,i (t,t, ), (3) it turns out that the process for the forward Libor rates becomes equal to dl (t,t) '... dt % j K i'1 L,i (t,t, )L (t,t)dw i (t), (4) for some drift term, left here unspecified, and given the current forward Libor rates L (0,T), see Brace et al. (1997), Jamshidian (1997), and Miltersen, Sandmann, and Sonderman (1997). As discussed by these authors, in an arbitrage free economy, and under the equivalent so-called forward measure, the Libor forward rate becomes dl (t,t) ' j K i'1 F L,i (t,t, )L (t,t)dwi (t), (5) where the superindex F indicates Brownian Motions under the forward measure. Again, for the pricing and hedging of interest rate derivatives, only the volatility functions (of the forward Libor rates) need to be specified and estimated. This approach is actually equivalent to the discretetenor string approach of Longstaff et al. (2001) (see Kerkhof and Pelsser (2001)). So, our starting point is either equation (1) for instantaneous forward rates or equation (5) for forward Libor rates. However, in this paper, we only analyze models with time-homogeneous, deterministic volatility functions, i.e., volatility functions that only depend on the dates t and T and, thus, not on T, where the dependence on T and t is through their difference T-t, and where, in the lognormal case, we allow for a level effect in the volatilities. The reason for this choice is twofold. First, estimation of time-inhomogeneous volatility functions from historical interest rate data is at the least very difficult. Secondly, a time-inhomogeneous volatility function can lead to a very unrealistic pattern for the future volatility of the spot rate. The assumption of deterministic volatility functions together with time-homogeneity implies -7-

10 a Gaussian distribution for instantaneous forward rates (equation (1)), or a lognormal distribution for forward Libor rates (equation (5)). Of course, in case of Gaussian models positive interest rates are not guaranteed. However, for realistic parameter values, the probability of negative interest rates is small for Gaussian models (see Rogers (1997)). Thus, the models that we analyze have the following form. In terms of instantaneous forward rates, we consider as models df(t,t) ' µ f (t,t, )dt % j K i'1 f,i (T&t)dW i (t), (6) The implied drift function under the equivalent martingale measure becomes (2), with f,i (t,t, ) replaced by f,i (T&t), so that µ f (t,t, ) ' µ f (t,t). In terms of forward Libor rates, we consider as models dl (t,t) '... dt % j K i'1 L,i (T&t)L (t,t)dw i (t), (7) The implied drift function under the equivalent forward measure disappears, see (5). The models in the classes as given by (6) and (7), that we will consider, differ through the number of factors K and the specification of each volatility f,i (T&t) and L,i (T&t) as function of T-t. In the sequel, we shall refer to the models in class (6) as the Gaussian models and to the models in class (7) as the Lognormal models. Given the specification of these models, the pricing formulas for caps and swaptions that we use in estimation, hedging, and prediction, are readily available from the financial literature. For the sake of completeness, we present these formulas in the appendix. -8-

11 4 Caps and Swaptions Data We use two US data sets 9 for our analysis: one data set containing money-market and swap rates and the other data set containing implied Black (1976) volatilities of caps and swaptions. From January 1994 until June 1999 we have weekly data on US money-market rates with maturities of 1, 3, 6, 9, and 12 months, and data on US swap rates with maturities ranging from 2 to 15 years. All weekly observations are on the Monday of each week. These interest rate data are used to construct the forward interest rate curve at each Monday in the dataset. We need these forward interest rates for two reasons. First, when pricing derivatives with HJM models, the initial forward interest rate curve is an input to the HJM model. Second, one way to estimate the parameters of the HJM volatility functions is based on the variances and covariances of historical forward rate changes of different maturities. However, when constructing the forward interest rate curves, one should be aware of a trade off, as noted by Buhler et al. (1999). In principle, for the pricing of derivatives at one day, one would like to fit the price of the underlying instrument perfectly. On the other hand, because estimates for forward interest rates turn out to be sensitive to small differences between money market or swap rates of nearly the same maturity, a perfect fit of all underlying money market and swap rates generally leads to unreasonably high estimates for the volatilities of historical forward rate changes. Therefore, we impose some smoothness conditions on the shape of the forward interest rate curve, as described in, for example, Bliss (1997). Thus, we parameterize the price of a zero-coupon bond maturing at T at date t as follows P(t,T) ' exp( 1 (T&t)% % d (T&t) d % j s j'1 d%j max(0,t&t&k j )d ) (8) In our application we choose d equal to 3 and the number of knot points s equal to 2, with equal to 2 years and k 2 equal to 4 years. The parameters $ are estimated for each Monday by minimizing the sum of squared relative differences between the observed money-market and swap rates and the corresponding money-market and swap rates as implied by (8). In Table 1, we present some summary statistics on the fit. It follows that the average absolute error is 0.39% k 1 set 9 The data are provided by ABN-AMRO Bank, Amsterdam, the Netherlands. -9-

12 for money-market rates and 0.23% for swap rates, which is equivalent to, respectively, 2.1 and 1.1 basis points, which seems satisfactory. Notice that from (8) we can obtain both the instantaneous forward interest curve f(t,t) and the Libor forward interest curve L (t,t) (with * representing a three-month period). The derivatives data that we use are weekly quotes, again on each Monday of the week, for the implied Black (1976) volatilities of at-the-money-forward US caps and swaptions, from January 2, 1995 to June 7, In total, this renders 232 weekly time-series observations on 63 instruments. The caps have maturities ranging from 1 to 10 years, and their payoffs are defined on 3-month interest rates. The 1-year cap consists of 3 caplets with maturities of 3, 6, and 9 months, and the 10-year cap consists of 39 caplets, with maturities ranging from 3 months to 9 years and 9 months. The other caps are constructed in a similar way. The strike of each cap is equal to the corresponding swap rate with quarterly compounding. Caps are quoted in the market by Black implied volatilities. Given the underlying forward interest rate curve, there is a one-toone correspondence between the cap implied volatility and the price of a cap. In Table 2 we provide some summary statistics on the implied volatilities of the caps. Although these implied volatilities cannot be interpreted directly as volatilities of single interest rates, because a cap consists of several caplets, we can still conclude that there is some evidence for a hump shaped volatility structure, which is in line with Amin and Morton (1994), and Moraleda and Vorst (1997). More empirical evidence for hump shaped volatility structures will be given later in this paper. A swaption is characterized both by the option maturity and the swap maturity. In our data, the option maturities range from 1 month to 5 years, while the swap maturities range from 1 to 10 years. We do not include prices of swaptions with total maturities (defined as the sum of the option maturity and the swap maturity) longer than 11 years, because the implied volatilities of these swaptions are not always updated in our data. The strike of an at-the-money swaption is equal to the corresponding forward swap rate. Hence, given the underlying forward interest rate curve, there is a one-to-one correspondence between swaption implied volatilities and swaption prices. In Tables 3 and 4, we provide summary statistics for the swaption implied Black volatilities. Again, there is some informal evidence for a hump shaped volatility structure. We also see that the variability over time in the swaption implied volatilities is somewhat lower than for cap implied volatilities. -10-

13 5 Model Specification and Estimation In this section we describe in further detail the models in the classes, given by (6) and (7), and the way we estimate these models. We start with the class of Gaussian models (6). The different models in the time-homogeneous Gaussian HJM class that we consider arise from the number of factors that is included (one, two, or three) and the particular functional shape of the volatility function corresponding to each factor. Largely inspired by models that are proposed and analyzed in the existing literature on interest rate models, we choose to analyze two types of specifications for the volatility function in the Gaussian HJM-class, namely a purely parametric one, and one based upon Principal Components Analysis (PCA): (I) Parametric One, Two, and Three Factor Models f,i (T&t) ' i e & i (T&t), i'1,2,3 (II) PCA One-, Two-, and Three-Factor Models f,i (T&t) ' g i (T&t), i'1,2,3 In case of (I) i, i, i'1,2,3, are unknown real-valued parameters. In the one-factor variant we have the Generalized Vasicek (1977)-model, or, equivalently, the one-factor Hull and White (1990)-model; the Ho and Lee (1986)-model is obtained if 1 '0. The Brownian Motions that drive the volatility functions in the parametric models (I) are allowed to be correlated. Notice that, to fit this model in the framework discussed in Section 3, the correlated Brownian Motions can be transformed into independent Brownian Motions (with appropriately modified volatility functions). Dai and Singleton (2000) show that (within the class of affine models) correlation between factors is necessary to generate hump shaped volatility functions. The instantaneous correlations will be referred to by the symbol D, where the sub-indices will refer to the factors. In case of (II) g 1, g 2,andg 3 are unknown functions of the time to maturity T-t. The one-factor PCA model is obtained if the functions g 2 and g 3 vanish, and the two-factor PCA is obtained if the function g 3 vanishes. The functions g 1, g 2, and g 3 in the PCA models will be estimated using principal components analysis (PCA). The use of principal components analysis to estimate HJM-volatility functions was proposed initially by Heath, Jarrow, and Morton (1990), and has been applied to interest rate data by, for example, Litterman and Scheinkman (1991), Knez, Litterman, and Scheinkman (1994), Moraleda and Vorst (1996), and -11-

14 Buhler et al. (1999). In the sequel, we shall refer to the class of models (I) as the Gaussian parametric models, and to the class (II) as Gaussian PCA models. In case of the class of Lognormal models (7), we also consider one, two, and three factor models. In this case, we only analyze the volatility functions based upon Principal Components Analysis (PCA), similar to the time-homogeneous Gaussian HJM class, see also Longstaff et al. (2001). Thus, we have (III) PCA One-, Two-, and Three-Factor Models L,i (T&t) ' h i (T&t), i'1,2,3 We investigate two estimation strategies. In the first estimation strategy, we estimate the model keeping the parameters constant throughout the estimation period. To be able to perform an out-of-sample analysis, the estimation period will be the first half of the sample; the second half of the data will then be used for out-of-sample evaluations. Following Amin and Morton (1994), and Buhler et al. (1999), we also consider a different estimation approach. In this case, we do not restrict the parameters to be constant over the entire valuation period, as there is some evidence for time-varying interest rate volatility in the literature 10. One could try to model time-varying volatility using a stochastic-volatility approach (see, for example, Hull and White (1987) for the case of stocks, and Collin-Dufresne and Goldstein (2001a, 2001b) for general affine term structure models). Theoretically the stochastic volatility approach is to be preferred, but, given that one does not know the market price of volatility risk, tests of stochastic volatility models cannot be separated from tests on the assumption about the market price of volatility risk. Also, as noted by Amin and Morton (1994), using a model without stochastic volatility, but with market-implied or time-varying volatility parameters may work as well in practice, especially when options are not too far from at-themoney (see Hull and White (1987)). To be able to compare the results for time-varying estimation with the constant parameter case, we start the time-varying estimation procedure in the second half of the sample, rendering 117 estimations in the time-varying case. As discussed in the next two subsections, for both the constant parameter case and the timevarying parameter case, we consider estimation on the basis of interest rate data, and estimation on the basis of option price data. 10 See, for example, Ball and Torous (1999). -12-

15 5.1 Interest-Rate-Based Estimation of Volatility Functions We will use the term interest-rate-based estimation for estimation strategies that are solely based on interest rate time series data. In this subsection we first describe the interest rate-based estimation for the class of Gaussian models. Then we briefly discuss the interest rate based estimation for the class of Lognormal models. Because in case of the class of Gaussian models the (forward) interest rates are normal in our modeling framework, and because under the equivalent martingale measure Q (with the money market as numeraire) the drifts of interest rates are determined by the variances and covariances of interest rates, we only need to estimate the volatility functions of the models, which can be achieved by estimating the variances and covariances of the forward interest rates. As mentioned above, in case of constant parameters we use the first half of the sample for estimation. In case of time-varying parameters, we use a rolling horizon estimation strategy to account for time-varying behaviour of interest rate volatility. In line with Buhler et al. (1999), we use a rolling horizon of 9 months (39 weeks) 11. As already mentioned, we start the timevarying estimation procedure in the second half of the sample. and For the model-class (II), we use principal components analysis to estimate the functions g 1, g 2,. The approach uses the fact that for Gaussian PCA models, which has independent g 3 factors, the covariance matrix of instantaneous forward rate changes is given by Cov[df(t,T i ),df(t,t j )] ' j K k'1 g k (T i &t)g k (T j &t)dt. (9) By approximation, this relationship also holds for forward rate changes over small time periods, in our case weekly changes 12. We choose a finite number of forward rate maturities, construct a covariance matrix of weekly forward rate changes for these forward rate maturities and determine the first three principal components of this covariance matrix. This renders estimates 11 We have also used exponential smoothing to give recent observations a higher weight. However, this does not improve the pricing of caps and swaptions in general. This is in contrast to the results of Bali and Karagozoglu (1999) for Eurodollar futures options. 12 This approximate relationship is only exact if the drift of forward rate changes is equal to zero. For weekly forward rate changes, the drift term is very small relative to the volatility of forward rate changes. -13-

16 of the volatility functions g i at the forward rate maturities T&t that are used; we linearly interpolate between these points to obtain the entire volatility function. This approach implies that the volatility function of the one-factor model is the same as the volatility function of the first factor of a two- or three-factor model. We use a set of forward rates with forward rate maturities from 0 up to 11 years, with quarterly intervals. In the upper panel of Figure 1, we plot the average of the volatility functions of the PCA Gaussian models. The shapes for these three factors can be interpreted as level, steepness, and curvature. These shapes are also found by, for example, Litterman and Scheinkman (1991). As we use a rolling horizon, the estimated volatility functions change weekly, but the shapes of these volatility functions turn out to be quite constant over time. On average, the first three factors explain about 97.8 % of the variation in forward interest rates. The first factor explains on average 83.7%, the second factor 10.1%, and the third factor 4.0%. For the models with parametric volatility functions in specification (I), a principal component analysis is not directly applicable. Therefore, we choose a different approach that is approximately based on the same information as used with the principal components analysis. More precisely, we use the generalized method of moments (GMM, Hansen (1982)) to estimate the parameters of the volatility functions, using both variances and covariances of forward rate changes as moment restrictions. These moment restrictions are similar to those in equation (9), but in this case also involve the correlation between the factors kl Cov[df(t,T i ),df(t,t j )] ' j K K j k'1 l'1 kl k e & k (T i &t) l e & l (T j &t) dt. (10) The following moment restrictions are used: the variances of forward rate changes with forward maturities of 3 months, 1, 3, 5, 7, and 10 years, and the covariance of the change in the forward rate with 3-month forward maturity with the changes in forward rates with forward maturities of 1, 3, 5, 7, and 10 years. This yields 11 moment restrictions. Again, a rolling horizon of 9 months is used for this GMM estimation. In Table 5 we provide information on the parameter estimates. We report the parameter estimates averaged over all weekly estimations, together with the corresponding standard error. In case of the two- and three-factor models, the parameter estimates result in a hump-shaped volatility. The given ordering of the factors corresponds to a first factor representing a level effect -14-

17 and a second factor representing a steepness effect. The correlations and their standard errors suggest that there is some negative correlation between the factors, in line with Dai and Singleton (2000). In the class (7) the forward Libor-rate models are lognormally distributed. Similarly to the Gaussian-case, we only need to estimate the volatility functions of the models, which can be achieved analogously to the Gaussian framework. So, applying principal components, we estimate the functions h 1, h 2, and h 3, using the fact that the covariance matrix of the forward Libor rate changes is given by Cov[dlogL (t,t i ),dlogl (t,t j )] ' j K k'1 h k (T i &t)h k (T j &t)dt. (11) By approximation, this relationship also holds for forward Libor rate changes over small time periods, in our case weekly changes. In the lower panel of Figure 1, we plot the average of the volatility functions of the Lognormal PCA models. The shapes for these three factors are quite similar to the Gaussian PCA-models, with again as interpretation level, steepness, and curvature. On average, the first three factors explain in this case about 96.9% of the variation in forward interest rates. The first factor explains on average 82.8%, the second factor 10.4%, and the third factor 3.6%. 5.2 Option-Based Estimation of Volatility Functions A different way to estimate the parameters of the volatility functions of the different models is to use option price data. We shall call this estimation strategy option-based estimation. For the Gaussian models with PCA volatility functions, option-based estimation is performed as follows. To obtain a parsimonious specification of the volatility function and facilitate the interpretation of the factors, we choose to maintain the shape of the volatility functions, as estimated with principal components analysis based on the interest rate data. We model the volatility function of each factor g i as g i ' i ĝ i, where ĝ i denotes the estimated g i according to the interest-rate-based estimation method, and where ( i is an unknown parameter. These parameters i are estimated by minimizing the weighted sum of squared relative pricing errors -15-

18 for each week in our dataset. 13 As a consequence, the shape of each factor volatility function is the same as for interest-rate-based estimation, and only the volatility of the factor itself can be different for option-based estimation. For the Gaussian models with parametric volatility functions given in (I), option-based estimation is performed in a way similar to the PCA-version: each volatility function (i=1,2,3) is multiplied by a scale parameter ( i, and these scale parameters are estimated using nonlinear least squares on the option prices. In other words: we re-estimate the parameters F i (which we redefine as ( i ) in specification (I), keeping the other parameters fixed (including the correlations across factors). We estimate the parameters by minimizing the sum of squared relative differences between observed prices of caps and swaptions and the corresponding prices for caps and swaptions as implied by the model. In case of time-varying parameters, this minimization is performed for each week in our dataset separately, and, thus, this can lead to parameter estimates that differ from week to week (recall that we use the option prices that are observed on the Monday of each week). In case of constant parameters, option prices of all weeks in the first half of the sample are used for estimation. In case of the Lognormal models we only consider models with PCA-volatility functions. The approach followed here is equivalent to the Gaussian models with PCA-volatility functions. 14 The averaged option-based parameter estimates of all models are given in Table 6. In case of the Gaussian and Lognormal PCA models the scale parameter estimates are on average all close to one, in line with what one expects to find if the models give a reasonably accurate description of reality. However, in the parametric case, the scale estimates of the second and third factor strongly deviate from one, indicating that these models may contain some misspecification. 5.3 Correlation Analysis In Figure 2 we plot the average correlations of the three-month forward interest rate with the three-month forward rates of different forward rate maturities, implied by Gaussian PCA models 13 Because we observe many more swaption than cap prices, namely 56 vs. 7, we put a higher weight on caps than on swaptions. To be precise, all swaptions with a fixed option maturity (there are 9 different option maturities) have a total weight of one, and each cap has a weight of one. 14 Notice that this is almost the same methodology as Longstaff et al. (2001). The only difference is that these authors use the correlation matrix for the principal component analysis, and then estimate the scale parameters, whereas we use the covariance matrix for the principal component analysis. -16-

19 with time-varying parameters, using either interest-rate-based estimation or option-based estimation. The graph shows that the difference between the correlations of the two- and threefactor models is quite large. Hence, although the third factor only explains 4.0% of the total variation in forward rates, it strongly affects correlations between interest rates. This graph also shows that the model correlations when option-based estimation is used, are somewhat lower than the correlations implied by interest-rate-based estimation. 6 Conditional Prediction of Derivative Prices Evaluation of term structure models is quite often performed in terms of pricing accuracy. In this paper, our main aim is to evaluate term structure models in terms of hedging. But for the sake of completeness and comparison, we will briefly discuss in this section the conditional prediction of derivative prices. To measure how well a given model conditionally predicts derivative prices, our procedure is as follows. In the time-varying parameter case, the parameters of a model are estimated at each trading day in our dataset (starting in the second half of our sample), given information up to this day, using either interest-rate-based estimation or option-based estimation. Then, after J weeks from this trading day, we value the caps and swaptions using the appropriate parameter estimates and the term structure after J weeks and compare the implied prices of the caps and swaptions with the observed prices. This procedure is then repeated for all weeks in the second half of the dataset. In case of constant parameters, we use the first half of the sample to estimate the parameters, and evaluate the predictions for the second half of the sample. This conditional prediction methodology is also performed in Amin and Morton (1994), who refer to this as pricing options with lagged volatility. We will choose J equal to two (weeks), reflecting a risk horizon that is commonly used in bank risk management. 15 We start with the results for caps. In Table 7, we present the prediction results for caps in terms of average absolute prediction errors. To avoid overloading, we do not include the standard 15 Notice that this provides a fair comparison between the option-based estimation method and interest-ratebased estimation method, because we compare the out-of-sample fit of derivative prices. If we would compare the fit of derivative prices at the day at which parameters are estimated, option-based estimated models would always have a better fit than interest-rate-based estimated models. -17-

20 errors of the predictions in this and subsequent comparable tables. In Table 7, all standard errors lie between 3 and 4%. We see that with only one exception (the two-factor Parametric model acording to specification (I)) the absolute prediction errors are smaller when allowing for timevarying parameters; in particular, for the PCA (Gaussian and Lognormal) models the difference between constant and time-varying parameters is considerable. This supports the existing evidence for time-varying interest rate volatilities (and correlations). When comparing interest-based and option-based estimation, we see that with time-varying parameters, the option-based prediction results are better. In all cases the performance of the parametric models are quite poor, indicating that this class of models may not be flexible enough to obtain acceptable price predictions. On the other hand, the multi-factor PCA (Gaussian and Lognormal) models yield average absolute pricing errors of around 7%, which is somewhat smaller than the pricing errors reported by Longstaff et al.(2001), and much smaller than the pricing errors reported by Jagannathan et al. (2000). So, in case of pricing caps the conclusion seems clear: when considering up to three-factor models, multi-factor Gaussian or Lognormal PCA models in combination with option-based estimation with time-varying parameters perform best. When only interest-based estimation is considered, the multi-factor Gaussian PCA models yield acceptable results. In Figure 3, we plot the average and average absolute cap prediction errors for the three-factor models and option-based estimation with time-varying parameters. It is clear that there are maturity effects in these pricing errors. The 1-year cap is overpriced, all other caps are underpriced. It turns out that all models (except the Parametric one-factor model) generate a hump shaped volatility structure, but the observed hump in cap implied Black volatilities is too pronounced to obtain an exact fit. The average absolute size of the prediction errors is almost constant over all caps. Therefore, all caps contribute to the pricing errors of caps reported in Table 7. In Table 8, we give the corresponding prediction results for swaptions. The standard errors, which are not reported (see above), all lie between 2.5 and 3.5%. Again, similar to the predictions in terms of cap prices, the models with time-varying parameters perform overall much better than the same model with constant parameters. When turning to a comparison between interest-based and option-based estimation and, considering the varying parameter outcomes, we see that the option-based approach in all cases shows a better prediction performance in terms of absolute pricing errors than the corresponding interest-based approach. The multi-factor models are able -18-

21 to yield an average absolute pricing error between 8 and 9%, which is reasonable when taking into account the bid-ask spread of around 6% typically found in the swaptions market (see Longstaff et al. (2001)). The pricing errors are also smaller than the pricing errors reported by Jagannathan et al. (2000). So, in case of pricing swaptions, the conclusion seems to be that option-based estimation with time-varying parameters, in combination with multi-factor models, is preferred, where the model class turns out to be of less importance. In case of interest-based estimation none of the model classes is able to yield acceptable results, when considering up to three factors. So here interest-based estimation is not to be recommended. In Figures 4a and 4b, we plot the prediction errors of the three-factor Gaussian PCA model. It follows that swaptions with short option or short swap maturities have the highest prediction errors. Also, the models generally overprice swaptions with short swap maturities. For the joint prediction of cap and swaption prices, the three-factor Gaussian PCA model, applying time-varying option-based estimation, has the best performance, as it is the only model that is not significantly outperformed by any other model in predicting cap or swaption prices, while the model outperforms any other model either in predicting cap prices or predicting swaption prices, or both, in a statistically significant way Hedging Caps and Swaptions In this section, we empirically investigate the size of hedging errors of delta-hedging strategies, and we analyze the differences between the hedging errors of the different models. In particular, we focus on the differences between one- and multi-factor models, the difference between interest-based and option-based estimation, and the difference between hedge strategies. 7.1 Setup of Hedge Strategies For each week in the second half of our data set, we estimate the models applying either optionbased or interest-rate-based estimation. We calculate for each cap and swaption the deltas (as 16 We have performed a pairwise statistical comparison of the prediction errors of each pair of models, as performed by Buhler et al. (1999). For the sake of brevity these results are available on request. -19-

22 implied by the model under consideration) with respect to certain hedge instruments (to be presented), and construct for each derivative instrument a delta-hedged portfolio. After two weeks, we compute the change in the value of this hedge portfolio, using the observed prices for the hedge instruments and the derivatives. 17 This procedure is repeated for each week. This gives us 115 (partly overlapping) time-series observations on hedging errors for each cap and swaption. We measure the accuracy of a hedging strategy by calculating for each cap and swaption the ratio of the variance 18 of these 115 hedging errors and the variance of two-week changes in an unhedged investment in the particular derivative instrument. This latter variance is model independent. Thus, the ratio of variances measures how much of the variability in the derivative instrument is removed by a delta-hedging strategy. If we would hedge continuously, using the correct model, the hedge-portfolio would have zero variance. Because we only hedge discretely, and because the models we analyze are approximations to reality, we will observe a positive hedging error variance. We implement two hedging strategies, factor hedging and bucket hedging. Factor hedging is based on the fact that in a K-factor model, K different instruments (together with the moneymarket account) are theoretically sufficient to replicate every derivative instrument in continuous time. Furthermore, the same K hedging instruments can be used for all derivatives. If the model describes the interest rate movements correctly, the choice of hedging instruments is irrelevant as long as the prices of the instruments can be inverted to all K factors. We choose a zero-coupon bond with 6 months maturity as the hedge instrument for all one-factor models, zero-coupon bonds with maturities of 6 months and 10 years for all two-factor models and zero-coupon bonds with maturities of 6 months, 3 years and 10 years for all three-factor models. This choice of instruments is based on the fact that (i) the first factor of an interest rate model is often chosen to be related to the short interest rate, (ii) the second factor is often associated with the spread 17 We only observe the prices of at-the-money caps and swaptions at each trading day. Clearly, an option that is at-the-money at a particular trading day will not be exactly at-the-money two weeks later. To be able to calculate the price of an off-market cap or swaption after these 2 weeks, we assume that there is no implied Black volatility smile, i.e., we assume that the observed implied Black volatility for a cap or a swaption is the same for all strike rates. As the changes in the at-the-money strike rate in 2 weeks are not very large, this assumption seems reasonable and not in favour of any particular model. 18 Fan et al. (2001) argue that the bias of a hedge strategy, i.e. the average change in the hedge portfolio value, should also be considered. It turns out that, for our dataset and hedge strategies, the bias of the hedge portfolios is essentially constant across models (and small). Therefore, our conclusions would not change if we would use the second moment instead of the variance to measure hedge effectiveness. -20-