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.     

A Class of Gaussian Hybrid Processes for Modeling Financial Markets

This paper proposes a one-factor model of financial markets using a class of Gaussian process that can be decomposed into a Brownian motion and an Ornstein–Uhlenbeck process. It is shown that this “hybrid” process is obtained as a continuous-time scaling limit of the differenced first-order autoregressive integrated moving average (ARIMA(1,1,1)) process. Parameter estimations using an ARIMA(1,1,1) framework and its variance ratio test show the accuracy of the proposed model. Construction of the one-factor commodity futures price model is presented as an application. A multidimensional extension of the hybrid process is also presented in the Appendix.     

On a general class of one-factor models for the term structure of interest rates

We propose a general one-factor model for the term structure of interest rates which based upon a model for the short rate. The dynamics of the short rate is described by an appropriate function of a time-changed Wiener process. The model allows for perfect fitting of given term structure of interest rates and volatilities, as well as for mean reversion. Moreover, every type of distribution of the short rate can be achieved, in particular, the distribution can be concentrated on an interval. The model includes several popular models such as the generalized Vasicek (or Hull-White) model, the Black-Derman-Toy, Black-Karasinski model, and others. There is a unified numerical approach to the general model based on a simple lattice approximation which, in particular, can be chosen as a binomial or -nomial lattice with branching probabilities .     

Bond pricing in a hidden Markov model of the short rate

Abstract. We consider a diffusion type model for the short rate, where the drift and diffusion parameters are modulated by an underlying Markov process. The underlying Markov process is assumed to have a stochastic differential driven by Wiener processes and a marked point process. The model for the short rate thus falls within the category of hidden Markov models.     

Yield curve modeling and forecasting using semiparametric factor dynamics

Using a dynamic semiparametric factor model (DSFM) we investigate the term structure of interest rates. The proposed methodology is applied to monthly interest rates for four southern European countries: Greece, Italy, Portugal and Spain from the introduction of the Euro to the recent European sovereign-debt crisis. Analyzing this extraordinary period, we compare our approach with the standard market method – dynamic Nelson–Siegel model. Our findings show that two nonparametric factors capture the spatial structure of the yield curve for each of the bond markets separately. We attributed both factors to the slope of the yield curve. For panel term structure data, three nonparametric factors are necessary to explain 95% variation. The estimated factor loadings are unit root processes and reveal high persistency. In comparison with the benchmark model, the DSFM technique shows superior short-term forecasting in times of financial distress.     

Monetary policy,bond risk premia,and the economy

Within an affine model of the term structure of interest rates, where bond yields get driven by observable and unobservable macroeconomic factors, parameter restrictions help identify the effects of monetary policy and other structural disturbances on output, inflation, and interest rates and decompose movements in long-term rates into terms attributable to changing expected future short rates versus risk premia. When estimated, the model highlights a broad range of channels through which monetary policy affects risk premia and the economy, risk premia affect monetary policy and the economy, and the economy affects monetary policy and risk premia.     

The fed and the term structure: Addressing simultaneity within a structural VAR model

This paper applies a new identification approach to estimate the contemporaneous relation between the term structure and monetary policy within a VAR framework. To achieve identification, we combine high-frequency Treasury futures and fed funds futures data with the VAR methodology. Results indicate that policy actions have a slope effect in the yield curve. We also find that the Fed responds to Treasury yields and that this response is stronger for the short and intermediate rates and less aggressive for long-yields. All estimated parameters are significant and robust to various model specifications.     

Optimal portfolio choice in the bond market

We consider the Merton problem of optimal portfolio choice when the traded instruments are the set of zero-coupon bonds. Working within a Markovian Heath–Jarrow–Morton model of the interest rate term structure driven by an infinite-dimensional Wiener process, we give sufficient conditions for the existence and uniqueness of an optimal trading strategy. When there is uniqueness, we provide a characterization of the optimal portfolio as a sum of mutual funds. Furthermore, we show that a Gauss–Markov random field model proposed by Kennedy [Math. Financ. 4, 247–258(1994)] can be treated in this framework, and explicitly calculate the optimal portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities of a certain function of the market parameters.     

Analyzing interest rate risk: Stochastic volatility in the term structure of government bond yields

We propose a Nelson–Siegel type interest rate term structure model where the underlying yield factors follow autoregressive processes with stochastic volatility. The factor volatilities parsimoniously capture risk inherent to the term structure and are associated with the time-varying uncertainty of the yield curve’s level, slope and curvature. Estimating the model based on US government bond yields applying Markov chain Monte Carlo techniques we find that the factor volatilities follow highly persistent processes. We show that yield factors and factor volatilities are closely related to macroeconomic state variables as well as the conditional variances thereof.     

New No-arbitrage Conditions and the Term Structure of Interest Rate Futures

Interest rate futures are basic securities and at the same time highly liquid traded objects. Despite this observation, most models of the term structure of interest rate assume forward rates as primary elements. The processes of futures prices are therefore endogenously determined in these models. In addition, in these models hedging strategies are based on forward and/or spot contracts and only to a limited extent on futures contracts. Inspired by the market model approach of forward rates by Miltersen, Sandmann, and Sondermann (J Finance 52(1); 409–430, 1997), the starting point of this paper is a model of futures prices. Using, as the input to the model, the prices of futures on interest related assets new no-arbitrage restrictions on the volatility structure are derived. Moreover, these restrictions turn out to prevent an application of a market model based on futures prices.     

Interest rate risk of German financial institutions: the impact of level,slope, and curvature of the term structure

We investigate here the sensitivity of the equity values of a large sample of German financial institutions to movements in the term structure of interest rates. While similar approaches rely on a single interest rate factor only, we quantify the exposure to changes in level, slope, and curvature, which are the driving factors of term structure changes. Our main findings are: (i) banks and insurances are exposed to level and curvature changes but only marginally to slope movements; (ii) the interest rate risk exposure depends on the banking sector investigated; (iii) level and curvature changes are priced in the cross-section of stock returns.

Stochastic duration and fast coupon bond option pricing in multi-factor models

Generalizing Cox, Ingersoll, and Ross (1979), this paper defines the stochastic duration of a bond in a general multi-factor diffusion model as the time to maturity of the zero-coupon bond with the same relative volatility as the bond. Important general properties of the stochastic duration measure are derived analytically, and the stochastic duration is studied in detail in various well-known models. It is also demonstrated by analytical arguments and numerical examples that the price of a European option on a coupon bond (and, hence, of a European swaption) can be approximated very accurately by a multiple of the price of a European option on a zero-coupon bond with a time to maturity equal to the stochastic duration of the coupon bond. This revised version was published online in June 2006 with corrections to the Cover Date.     

Arbitrage bounds for the term structure of interest rates

This paper proposes a methodology for simultaneously computing a smooth estimator of the term structure of interest rates and economically justified bounds for it. It unifies existing estimation procedures that apply regression, smoothing and linear programming methods. Our methodology adjusts for possibly asymmetric transaction costs. Various regression and smoothing techniques have been suggested for estimating the term structure. They focus on what functional form to choose or which measure of smoothness to maximize, mostly neglecting the discussion of the appropriate measure of fit. Arbitrage theory provides insight into how small the pricing error will be and in which sense, depending on the structure of transaction costs. We prove a general result relating the minimal pricing error one incurs in pricing all bonds with one term structure to the maximal arbitrage profit one can achieve with restricted portfolios.