Term-Structure Models

A term-structure is a function that relates a certain financial variable or parameter to its maturity. Prototypical examples are the term-structure of interest rates or zero-coupon bond prices. But there are also term-structures of option implied volatilities, credit spreads, variance swaps, etc. Term-structures are high-dimensional objects, which often are not directly observable. On the empirical side this requires estimation methods that are flexible enough to capture the entire market information. But flexibility often comes at the cost of irregular term-structure shapes and a great number of factors. Principal component analysis and parametric estimation methods can put things right. On the modeling side we find several challenging tasks. Bonds and other forward contracts expire at maturity where they have to satisfy a formally predetermined terminal condition. For example, a zero-coupon bond has value one at maturity, a European-style option has a predetermined payoff contingent on some underlying instrument, etc. Under the absence of arbitrage this has non-trivial implications for any dynamic term-structure model. As a consequence, various approaches to modeling the term-structure of interest rates have been proposed in the last decades, starting with the seminal work of Vasiček (J. Financ. Econ. 5:177–188, 1977). By arbitrage we mean an investment strategy that yields no negative cash flow in any future state of the world and a positive cash flow in at least one state; in simple terms, a risk-free profit. The assumption of no arbitrage is justified by market efficiency as a consequence of which prices tend to converge to arbitrage-free prices due to demand and supply effects.

A bond is a securitized form of a loan. Bonds are the primary financial instruments in the market where the time value of money is traded. This chapter provides the basis concepts of interest rates and bond markets. We start with zero-coupon bonds and define a number of related interest rates. We then look at market conventions and learn how caps, floors and swaptions are priced by market practice.

In our theoretical framework we often assume a term-structure for the continuum of maturities T. In other words, we assume that the forward or zero-coupon yield curve is given by a function of the continuous variable T. This should be seen as approximation of the reality, which comes along with finitely many (possibly noisy) market quote observations. In Chap. 11 we will model the term-structure of interest rates by choosing finitely many maturities. This is appropriate if we want to price a predetermined finite set of derivatives, such as caps and swaptions. However, as soon as more exotic derivatives be priced whose cash flow dates possibly do not match the predetermined finite time grid, one has to interpolate the term-structure. In this chapter, we learn some term-structure estimation methods. We start with a bootstrapping example, which is the most used method among the trading desks. We then consider more general aspects of non-parametric and parametric term-structure estimation methods. In the last part we perform a principal component analysis for the term-structure movements, which is the best-known dimension reduction technique in multivariate data analysis.

This chapter briefly recalls the fundamental arbitrage principles in a Brownian-motion-driven financial market. The basics of stochastic calculus are provided without proofs. Standard terminology is employed without further explanation. Readers are requested to consult one of the many text books on stochastic calculus. References are given in the notes section. The main pillars for financial applications are Itô’s formula, Girsanov’s change of measure theorem, and the martingale representation theorem.

The earliest stochastic interest rate models were models of the short rates. This chapter gives an introduction to diffusion short-rate models in general, and provides a survey of some standard models. Particular focus is on affine term-structures.

As we have seen in Chap. 5, short-rate models are not always flexible enough to calibrating them to the observed initial term-structure. In the late eighties, Heath, Jarrow and Morton (henceforth HJM) (Econometrica 60:77–105, 1992) proposed a new framework for modeling the entire forward curve directly. This chapter provides the essentials of the HJM framework.

In this chapter, we discuss two common types of term contracts: forwards, which are mainly traded over the counter (OTC), and futures, which are actively traded on many exchanges. The underlying is in both cases a T-claim  \(\mathcal{Y}\) , for some fixed future date T. This can be an exchange rate, an interest rate, a commodity such as copper, any traded or non-traded asset, an index, etc. We discuss interest rate futures and futures rates in a separate section and relate them to forward rates in the Gaussian HJM model.

Practitioners and academics alike have a vital interest in parameterized term-structure models. In this chapter, we take up a point left open at the end of Chap. 3, and exploit whether parameterized curve families φ(⋅,z), used for estimating the forward curve, go well with arbitrage-free interest rate models. According to Table 3.4, taken from the BIS document (Technical Documentation, Bank for International Settlements, Basle, March 1999), there is a rich source of cross-sectional data, that is, daily estimations of the parameter z, for the Nelson–Siegel and Svensson families. This suggests that calibrating a diffusion process Z for the parameter z would lead to an accurate factor model for the forward curve. Conditions for the absence of arbitrage can be formulated in terms of the drift and diffusion of Z and derivatives of φ. These conditions turn out to be surprisingly restrictive in some cases.

We have seen in Sects. 5.3 and 9.3 above that an affine diffusion induces an affine term-structure. In this chapter, we discuss the class of affine processes in more detail. Their nice analytical properties make them favorite for a broad range of financial applications, including term-structure modeling, option pricing and credit risk modeling.

Instantaneous forward rates are not so simple to estimate, as we have seen. One may want to model other rates, such as LIBOR, directly. There has been some effort in the years after the publication of HJM (Econometrica 60:77–105, 1992) in 1992 to develop arbitrage-free models of other than instantaneous, continuously compounded rates. The breakthrough came 1997, when the LIBOR market models were introduced by Miltersen et al. (J. Finance 52:409–430, 1997) and Brace et al. (Math. Finance 7(2):127–155, 1997) who succeeded in finding a HJM-type model inducing lognormal LIBOR rates. At the same time, Jamshidian (Finance Stoch. 1(4):290–330, 1997) developed a framework for arbitrage-free LIBOR and swap rate models not based on HJM. The principal idea of these approaches is to choose a different numeraire than the risk-free account (the latter does not even necessarily have to exist). Both approaches lead to Black’s formula for either caps (LIBOR models) or swaptions (swap rate models). Because of this they are usually referred to as “market models”.

So far bond price processes P(t,T) had the property that P(T,T)=1. That is, the payoff was certain, there was no risk of default of the issuer. This may be the case for treasury bonds. Corporate bonds, however, may bear a substantial risk of default. Investors should be adequately compensated by a risk premium, which is reflected by a higher yield on the bond.

In this chapter, we will briefly review the two most common approaches to credit risk modeling: the structural and the intensity-based approach. The structural approach models the value of a firm’s assets. Default is when this value hits a certain lower bound. This approach goes back to Merton’s (J. Finance 29(2):449–470, 1974) seminal corporate debt model. In the intensity-based approach, default is specified exogenously by a stopping time with given intensity process. This approach can be traced back to work of Jarrow, Lando and Turnbull in the early 1990s.