A term structure model of interest rates with quadratic volatility

This study proposes a no-arbitrage term structure model that can capture the volatility of interest rates without sacrificing the goodness-of-fit to the cross-section and predictive ability about the level of interest rates. The key feature of the model is the covariance matrix of changes in factors, which is specified as quadratic functions of factors. The quadratic specification can capture intense volatility even with spanned factors, which is not the case for the affine specification. Furthermore, since the quadratic specification guarantees the positive definiteness of the covariance matrix without restricting the sign of factors, it allows for a flexible specification of the physical drift as does the Gaussian term structure model, contributing also to accurate level prediction.     

The volatility term structure in a lognormal process for the short rate

One-factor Markov models are widely used by practitioners for pricing financial options. Their simplicity facilitates their calibration to the intial conditions and permits fast computer Implementations. Nevertheless, the danger remains that such models behave unrealistically, if the calibration of the volatility is not properly done. Here, we study a lognormal process and investigate how to specify the volatility constraints in such a way that the term structure of volatility at future times, as implied by the short rate process, has a realistic and stable shape. However, the drifting down of the volatility term structure is unavoidable. As a result, there is a tendency to underestimate option prices.     

Consistency conditions for affine term structure models

Sufficient conditions for the application of the Feynman-Kac formula for option pricing for wide classes of affine term structure models in the jump-diffusion case are derived by generalizing earlier results for bond pricing in the pure-diffusion case The author is grateful to Mikhail Chernov and Darrel Duffie for useful discussions and suggestions.     

The value of convexity: a theoretical and empirical investigation

We explore from a theoretical and an empirical perspective the value of convexity in the US Treasury market. We present a quasi-model-agnostic approach that is rooted in the existence of some affine model capable of recovering with good accuracy the market yield curve and covariance matrix. As we show, at least one such model exists, and this is all we require for our results to hold. We show that, as a consequence, the theoretical ‘value of convexity’ purely depends on observable features of the yield curve, and on statistically determinable yield volatilities. We then address the question of whether the theoretical convexity is indeed correctly reflected in the shape of the yield curve. We present empirical results about the predictive power of a strategy based on the discrepancies between the theoretical and the predicted value of convexity. By looking at 30 years of data, we find that neither the strategy of being systematically long or short convexity (and immunized against ‘level’ and ‘slope’ risk) would have been profitable. However, a conditional strategy that looks at the difference between the ‘implied’ and the statistically estimated value of convexity would have identified extended periods during which the proposed approach would have delivered attractive Sharpe Ratios.     

On the equivalence of a class of affine term structure models

In specifying a finite factor model for the term structure of interest rates, one usually begins by modeling the dynamics of the underlying factors. In most cases, this is sufficient to completely determine the term structure model. However, a point that is often overlooked is that seemingly different specifications of the factor dynamics may generate indistinguishable term structure models, in the sense that they produce pathwise identical bond prices. Consequently, it is important to be able to determine, at the level of factor dynamics, the conditions under which the models they generate are indistinguishable. In the case of time-homogeneous affine term structure models (ATSMs), such conditions were first described in Dai and Singleton (J Finance 55:1943–1978, 2000). In this paper, we formalize and extend their results to a class of time-inhomogeneous ATSMs, and obtain a simple method for determining the indistinguishability of these models in terms of the underlying factor dynamics.      

The term structure of inflation expectations

We use information in the term structure of survey-based forecasts of inflation to estimate a factor hidden in the nominal yield curve. We construct a model that accommodates forecasts over multiple horizons from multiple surveys and Treasury real and nominal yields by allowing for differences between risk-neutral, subjective, and objective probability measures. We establish that model-based inflation expectations are driven by inflation, output, and one latent factor. We find that this factor affects inflation expectations at all horizons but has almost no effect on the nominal yields; that is, the latent factor is hidden. We show that this hidden factor is not related to either current and past inflation or the standard set of macro variables studied in the literature. Consistent with the theoretical property of a hidden factor, our model outperforms a standard macro-finance model in its forecasting of inflation and yields.     

A preference free partial differential equation for the term structure of interest rates

The objectives of this paper are two-fold: the first is the reconciliation of the differences between the Vasicek and the Heath-Jarrow-Morton approaches to the modelling of term structure of interest rates. We demonstrate that under certain (not empirically unreasonable) assumptions prices of interest-rate sensitive claims within the Heath-Jarrow-Morton framework can be expressed as a partial differential equation which both is preference-free and matches the currently observed yield curve. This partial differential equation is shown to be equivalent to the extended Vasicek model of Hull and White. The second is the pricing of interest rate claims in this framework. The preference free partial differential equation that we obtain has the added advantage that it allows us to bring to bear on the problem of evaluating American style contingent claims in a stochastic interest rate environment the various numerical techniques for solving free boundary value problems which have been developed in recent years such as the method of lines.     

Estimating affine multifactor term structure models using closed-form likelihood expansions

We develop and implement a technique for closed-form maximum likelihood estimation (MLE) of multifactor affine yield models. We derive closed-form approximations to likelihoods for nine Dai and Singleton (2000) affine models. Simulations show our technique very accurately approximates true (but infeasible) MLE. Using US Treasury data, we estimate nine affine yield models with different market price of risk specifications. MLE allows non-nested model comparison using likelihood ratio tests; the preferred model depends on the market price of risk. Estimation with simulated and real data suggests our technique is much closer to true MLE than Euler and quasi-maximum likelihood (QML) methods.     

American bond option pricing in one-factor dynamic term structure models

Arbitrage-tree pricing of American options on bonds in one-factor dynamic term structure models is investigated. We re-derive a general decomposition result which states that the American bond option premium can be split into the value of an otherwise equivalent European option and anearly exercise premium. This extends earlier work on American equity options by e.g. Kim (1990), Jamshidian (1992) and Carr, Jarrow, and Myneni (1992) and parallels recent work by Jamshidian (1991, 1992, 1993) and Chesney, Elliott, and Gibson (1993). We examine a Gaussian class of special cases in some detail and provide a variety of numerical valuation results.An earlier version of the paper was entitled American Bond Option Pricing in One-Factor Spot Interest Rate Models.I am grateful for many helpful comments from two anonymous referees, the participants of the Second Nordic Symposium on Contingent Claims Analysis in Finance held in Bergen, Norway in May of 1994 and from the participants of the EIASM Doctoral Tutorial held in connection with the 1994 EFA annual meeting in Bruxelles. I am particularly indebted to Krishna Ramaswamy for his help and advice during my stay as visiting doctoral fellow at the Wharton School of the University of Pennsylvania. Financial support from the Aarhus University Research Foundation (Grants # E-1994-SAM-1-1-72 & E-1995-SAM-1-59), the Danish Social Science Research Council, and the Danish Research Academy is gratefully acknowledged. All errors and omissions are my own.     

Stock options and capital structure

We show that value-maximizing CEOs compensated with stock options prefer debt to equity. Our pecking order result does not depend on managerial risk aversion, managerial firm-specific human capital, or asymmetric information. Moreover, our result holds at least weakly regardless of the distribution of firm cash flows and strictly as long as the support of the cash flow distribution is big enough to bring all features of the stock option contract into play with positive probability JEL Classification Numbers: G0, G3 An earlier version of this paper was completed while Page was visiting CERMSEM at Paris 1 and the University of Warwick. Page gratefully acknowledges the support and hospitality of CERMSEM, Paris 1 and Warwick. Page also gratefully acknowledges financial support from the Department of Economics, Finance, and Legal Studies and the Culverhouse College of Business at the University of Alabama. Both authors are grateful to seminar participants in the Financial Markets Group Workshop at LSE for many helpful comments and both authors are especially grateful to an anonymous referee whose detailed and insightful comments led to substantial improvements in the paper     

Macroeconomic releases and the interest rate term structure

We extract two systematic economic factors from a wide array of noisy and sparsely observed macroeconomic releases, and link the dynamics and market prices of the two factors to the interest rate term structure. The two factors predict 77.9–82.1% of the daily variation in LIBOR and swap rates from one month to 10 years. Shocks on inflation-related releases have large, positive impacts on interest rates of all maturities, leading to parallel shifts of the yield curve, but shocks on output-related releases have larger impacts on the short rate than on the long rate, thus generating a slope effect.     

Quadratic term structure models in discrete time

This paper extends the results on quadratic term structure models in continuous time to the discrete time setting. The continuous time setting can be seen as a special case of the discrete time one. Discrete time quadratic models have advantages over their continuous time counterparts as well as over discrete time affine models. Recursive closed form solutions for zero coupon bonds are provided even in the presence of multiple correlated underlying factors, time-dependent parameters, regime changes and “jumps” in the underlying factors. In particular regime changes and “jumps” cannot so easily be accommodated in continuous time quadratic models. Pricing bond options requires simple integration and model estimation does not require a restrictive choice of the market price of risk.     

Calibration of Gaussian heath, Jarrow and Morton and random field interest rate term structure models

We show how the recently introduced Gaussian random field interest rate term structure models can be calibrated accurately and quickly to market caps and swaptions prices. We show how the calibrated random field model can be approximated arbitrarily closely with a multi-factor Gaussian Heath, Jarrow and Morton model. We argue that the Gaussian random field model is easier to calibrate and can be used as an indirect way to calibrate multi-factor Gaussian Heath, Jarrow and Morton interest rate models.This work was carried out when the author was at the Financial Options Research Centre, Warwick Business School, University of Warwick. I wish to thank Stewart Hodges for many helpful discussions and comments and Martin Cooper of Tokai Bank, Europe, for supplying the data used in this paper. I also wish to thank the Economic and Social Research Council and FORC for funding. An earlier version of this paper entitled Multi-Factor Gaussian HJM Approximation to Kennedy and Calibration to Caps and Swaptions Prices was presented at the 9th Annual Conference, FORC., Warwick Business School, September 1996. Another version also appears in the author's Ph.D. thesis. I am grateful to the helpful comments provided by Marti Subrahmanyam and the two anonymous referees. All errors are my own and the views expressed in no way reflect the opinion of my employer.     

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.     

Time-variation in term premia: International survey-based evidence

Using a large, previously unexplored data set of survey-based interest rate forecasts that covers a broad range of countries, this paper re-examines the expectations hypothesis of the term structure of interest rates. We find that survey-based interest rate forecasts outperform not only a random walk forecast, but also outperform forecasts from forward rates. When using these superior survey-based forecasts in a modified expectations hypothesis test, the expectations hypothesis is rejected for fewer countries, at lower significance levels, and has greater explanatory power than when using a traditional forward rates-based test. We furthermore document strong time-variation in the term premia, which is an important reason why the traditional expectations hypothesis test is rejected so frequently. This time-variation seems to arise from the changing attitudes towards risk among market participants and as a compensation for the change in liquidity in the term structure. Finally, we find that generalizing findings from earlier U.S. studies to other countries may lead to bias in the true economic relationships in these countries.     

Managing the risk of loan prepayments and the optimal structure of short term lending rates

Loan pricing is an extremely important aspect of bank operations because loans are typically over two-thirds of bank assets. Many researchers have analyzed the theoretical and empirical impact of how different factors should and do affect fixed rate loan rates and loan prepayments. However, a theoretical decision making model for maximizing expected profit in a declining rate environment has not been developed. After describing the conditions for the optimal loan rate, we develop numerical solutions for it under varying conditions. The varying conditions include the trend in interest rates, volatility of interest rates, and loan maturity. We thank Yen Low and Hamed Bagherpour for their assistance.