On the Information in the Interest Rate Term Structure and Option Prices

We examine whether the information in cap and swaption prices is consistent with realized movements of the interest rate term structure. To extract an option-implied interest rate covariance matrix from cap and swaption prices, we use Libor market models as a modelling framework. We propose a flexible parameterization of the interest rate covariance matrix, which cannot be generated by standard low-factor term structure models. The empirical analysis, based on US data from 1995 to 1999, shows that option prices imply an interest rate covariance matrix that is significantly different from the covariance matrix estimated from interest rate data. If one uses the latter covariance matrix to price caps and swaptions, one significantly underprices these options. We discuss and analyze several explanations for our findings.     

An Empirical Comparison of Alternative Models of the Short-Term Interest Rate

We estimate and compare a variety of continuous-time models of the short-term riskless rate using the Generalized Method of Moments. We find that the most successful models in capturing the dynamics of the short-term interest rate are those that allow the volatility of interest rate changes to be highly sensitive to the level of the riskless rate. A number of well-known models perform poorly in the comparisons because of their implicit restrictions on term structure volatility. We show that these results have important implications for the use of different term structure models in valuing interest rate contingent claims and in hedging interest rate risk.     

Implied volatilities,stochastic interest rates,and currency futures options valuation: an empirical investigation

Different models of pricing currency call and put options on futures are empirically tested. Option prices are determined using different models and compared to actual market prices. Option prices are determined using historical as well as implied volatility. The different models tested include both constant and stochastic interest rate models. To determine if the model prices are different from the market prices, regression analysis and paired t-tests are performed. To see which model misprices the least, root mean square errors are determined. It is found that better results are obtained when implied volatility is used. Stochastic interest rate models perform better than constant interest rate models.     

A statistical comparison of the short-term interest rate models for Japan,U.S., and Germany

We propose a simple and practical model selection method for continuous time models. We apply the method to several continuous time short-term interest rate models using discrete time series data of Japan, U.S. and Germany. All the models can be easily estimated from discrete observations, and their performances can be evaluated in a uniform statistical framework. The models that allow dependence of volatility on the level of interest rates tend to perform well empirically. The degree of volatility dependence on the interest rate levels seems to be different across the countries. For the German data, we observe that a model with nonlinear drift performs better than the best linear drift model.     

A model of discontinuous interest rate behavior,yield curves,and volatility

This paper develops an equilibrium model in which interest rates follow a discontinuous (generalized) gamma process. The gamma process has finite variation, takes an infinite number of “small” jumps in every interval, and includes the Wiener process as a limiting case. The gamma interest rate model produces yield curves that closely resemble those of diffusion models. But in contrast to diffusion models, the curvature of the yield curve does not directly depend on the true volatility of the interest rate process, but instead depends on a different risk-neutral volatility. The gamma model appears to fit the distribution of interest rates changes and the jump characteristics of interest rate paths. Empirical tests reject a diffusion model of interest rates in favor of the more general gamma model because daily interest rate innovations are highly leptokurtic. The author appreciates comments from George Constantinides, Jon Ingersoll, Herbert Johnson, Ray Rishel, and an anonymous referee, computational assistance from Kerry Back and Saikat Nandi, and support from Atlantic Asset Management. Any errors are the responsibility of the author.     

Econometric models of the Eleventh District Cost of Funds Index

The Eleventh District Cost of Funds Index (COFI) is a popular index for pricing adjustable-rate mortgages. COFI is calculated from the interest expenses incurred by thrifts when raising funds. It is a mixture of current and past interest rates on many different financial instruments. COFI can be modelled well with simple econometric models. Commonly used, simple COFI models are compared using a method developed by Hendry (1989). Some of these models, which appear to fit the data well, have nonrobust parameters, significant serial correlation, and heteroscedastic errors. These poorly specified models may lead to systematic mispricing of COFI mortgages. Once a robust econometric model is chose, the lagged adjustment of COFI to movements in interest rates can be incorporated into mortgage pricing models.     

Interest Rate Futures Options and Interest Rate Options

This paper derives pricing models of interest rate options and interest rate futures options. The models utilize the arbitrage-free interest rate movements model of Ho and Lee. In their model, they take the initial term structure as given, and for the subsequent periods, they only require that the bond prices move relative to each other in an arbitrage-free manner. Viewing the interest rate options as contingent claims to the underlying bonds, we derive the closed-form solutions to the options. Since these models are sufficiently simple, they can be used to investigate empirically the pricing of bond options. We also empirically examine the pricing of Eurodollar futures options. The results show that the model has significant explanatory power and, on average, has smaller estimation errors than Black's model. The results suggest that the model can be used to price options relative to each other, even though they may have different expiration dates and strike prices.     

Is the Real Interest Rate Stable?

Univariate time-series models for consumption, nominal interest rates, and prices each appear to have a single unit root before 1979. If nominal interest rates have a unit root but inflation and inflation forecast errors do not, then ex ante real interest rates have a unit root and are therefore nonstationary. This deduction does not depend on the properties of the unobservable ex post observed real return, which combines the ex ante real interest rate and inflation-forecasting errors. The unit-root characteristic of real interest rates is puzzling from at least two perspectives: many models imply that the growth rate of consumption and the real interest rate should have similar time-series characteristics; also, nominal returns for other assets (e.g., stocks and bonds) appear to have different times-series properties from those of treasury bills.     

Are there nonlinearities in short‐term interest rates?

The present paper investigates the characteristics of short‐term interest rates in several countries. We examine the importance of nonlinearities in the mean reversion and volatility of short‐term interest rates. We examine various models that allow the conditional mean (drift) and conditional variance (diffusion) to be functions of the current short rate. We find that different markets require different models. In particular, we find evidence of nonlinear mean reversion in some of the countries that we examine, linear mean reversion in others and no mean reversion in some countries. For all countries we examine, there is strong evidence of the need for the volatility of interest rate changes to be highly sensitive to the level of the short‐term interest rate. Out‐of‐sample forecasting performance of one‐factor short rate models is poor, stemming from the inability of the models to accommodate jumps and discontinuities in the time series data.     

On the valuation of compositions in Lévy term structure models

We derive explicit valuation formulae for an exotic path-dependent interest rate derivative, namely an option on the composition of LIBOR rates. The formulae are based on Fourier transform methods for option pricing. We consider two models for the evolution of interest rates: an HJM-type forward rate model and a LIBOR-type forward price model. Both models are driven by a time-inhomogeneous Lévy process.     

The Effect of Money Shocks on Interest Rates in the Presence of Conditional Heteroskedasticity

Most current empirical work finds no evidence that money shocks lower interest rates. We show that these nonresults are mainly due to a failure to model the conditional heteroskedasticity of interest rates. Autoregressive conditional heteroskedasticity (ARCH) models find a significant liquidity effect where ordinary least squares (OLS) models do not. The existence of a liquidity effect is found using different models and sample periods when ARCH models are used in estimation, but never when OLS is employed.     

VaR limits for pension funds: an evaluation

In a series of papers during the last ten years an interest rate theory with models which are driven by Lévy or more general processes has been developed. In this paper we derive explicit formulas for the correlations of interest rates as well as zero coupon bonds with different maturities. The models considered in this general setting are the forward rate (HJM), the forward process and the LIBOR model as well as the multicurrency extension of the latter. Specific subclasses of the class of generalized hyperbolic Lévy motions are studied as driving processes. Based on a data set of parametrized yield curves derived from German government bond prices we estimate correlations. In a second step the empirical correlations are used to calibrate the Lévy forward rate model. The superior performance of the Lévy driven models becomes obvious from the graphs.     

Gaussian risk models with financial constraints

In this paper, we investigate Gaussian risk models which include financial elements, such as inflation and interest rates. For some general models for inflation and interest rates, we obtain an asymptotic expansion of the finite-time ruin probability for Gaussian risk models. Furthermore, we derive an approximation of the conditional ruin time by an exponential random variable as the initial capital tends to infinity.     

Affine Term Structure Models and the Forward Premium Anomaly

One of the most puzzling features of currency prices is the forward premium anomaly : the tendency for high interest rate currencies to appreciate. We characterize the anomaly in the context of affine models of the term structure of interest rates. In affine models, the anomaly requires either that state variables have asymmetric effects on state prices in different currencies or that nominal interest rates take on negative values with positive probability. We find the quantitative properties of either alternative to have important shortcomings.     

Multi-Factor Cox-Ingersoll-Ross Models of the Term Structure: Estimates and Tests from a Kalman Filter Model

This paper presents a method for estimating multi-factor versions of the Cox-Ingersoll-Ross (1985b) model of the term structure of interest rates. The fixed parameters in one, two, and three factor models are estimated by applying an approximate maximum likelihood estimator in a state-space model using data for the U.S. treasury market. A nonlinear Kalman filter is used to estimate the unobservable factors. Multi-factor models are necessary to characterize the changing shape of the yield curve over time, and the statistical tests support the case for two and three factor models. A three factor model would be able to incorporate random variation in short term interest rates, long term rates, and interest rate volatility.     

Junp-Diffusion Interest Rate Process: An Empirical Examination

We investigate a jump-diffusion process, which is a mixture of an O-U process used by Vasicek (1977) and a compound Poisson jump process, for the term structure of interest rates. We develop a methodology for estimating the jump-diffusion model and complete an empirical study in comparing the model with the Vasicek model, for the US money market interest rates. The results show that when the short-term interest rate is low, both models predict an upward sloping term structure, with the jump-diffusion model fitting the actual term structure quite well and the Vasicek model overestimating significantly. When the short-term interest rate is high, both models predict a downward sloping term structure, with the jump-diffusion model underestimating the actual term structure more significantly than the Vasicek model.     

BOND RETURNS,DISCRETE STOCHASTIC PROCESSES,AND DURATION

A particular duration measure may correspond to many different stochastic processes that generate fluctuations in the term structure of interest rates. There does not exist a one-to-one correspondence between the duration measure and an underlying stochastic process. In particular, durations derived from disequilibrium processes also correspond to equilibrium processes. Furthermore, it is shown that multi-factor discrete models of bond returns may also correspond to multi-duration models of bond returns.     

Are central banks' projections meaningful?

Central banks' projections – i.e. forecasts conditional on a given interest rate path – are often criticized on the grounds that their assumptions are inconsistent with the existence of a unique equilibrium in many forward-looking models. The present paper describes three alternative approaches to constructing projections that are not subject to the above criticism, using the New Keynesian model as a reference framework. The three approaches are shown to generate different projections for inflation and output, even though they imply an identical path for the interest rate. The latter result calls into question the meaning and usefulness of such projections.     

Simple Monetary Policy Rules Under Model Uncertainty

Using stochastic simulations and stability analysis, the paper compares how different monetary policy rules perform in a moderately nonlinear model with a time-varying NAIRU. Rules that perform well in linear models but implicitly embody backward-looking measures of real interest rates (such as conventional Taylor rules) or substantial interest rate smoothing perform very poorly in models with moderate nonlinearities, particularly when policymakers tend to make serially-correlated errors in estimating the NAIRU. This challenges the practice of evaluating policy rules within linear models, in which the consequences of responding myopically to significant overheating are extremely unrealistic.     

On alternative interest rate processes

In this paper alternative interest rate processes are estimated for Denmark, Germany, Sweden, and the UK, using the generalized method of moments (GMM). In line with the study by Chan, Karolyi, Longstaff, and Sanders (1992) on US data, there seems to be a positive relation between interest rate level and volatility for some countries. In contrast to their study, it is found that mean-reversion plays an important role for the specification of the interest rate dynamics. The results seem to be robust to the use of different moment conditions, and simulations of the estimated models reveal that they are fairly able to capture non-fitted moments as well. In addition, there is evidence of a structural change in the Danish interest rate process in August 1985, which may be due to a change in monetary policy. The small sample properties of the GMM estimators are also studied through simulations.