Interest Rate Modelling

The initial formulation of Vasicek’s model is very general, with the short-term interest rate being described by a diffusion process. An arbitrage argument, similar to that used to derive the Black–Scholes option pricing formula [8], is applied within this broad framework to determine the partial differential equation satisfied by any contingent claim. A stochastic representation of the bond price results from the solution to this equation. Vasicek then allows more restrictive assumptions to formulate the specific model with which his name is associated.

Cox, Ingersoll and Ross (CIR) view the problem of interest rate modelling as one in “general equilibrium theory” [18]. Anticipation of future events, risk preferences, other investment alternatives and consumption preferences all affect the term structure. CIR make use of a general equilibrium asset pricing model to endogenously determine the stochastic process followed by the shortterm interest rate and the partial differential equation satisfied by the value of any contingent claim. Bond prices are then determined as solutions to this partial differential equation, contingent on the underlying short-term interest rate.

Brennan and Schwartz (BS) [10] challenge the primary assumption of many models. That is: all information about future interest rates is contained in the current instantaneous short-term interest rate and hence the prices of all default-free bonds may be represented as time-dependent functions of this instantaneous rate only. They point out that this is not an accurate representation of reality and propose an interest rate model based on the assumption that the whole term structure can be expressed as a function of the yields of the longest and shortest maturity default-free bonds.

Longstaff and Schwartz (LS) [38] developed a two-factor model of the term structure based on the framework of Cox, Ingersoll and Ross [18] discussed in Chapter 2. The two factors are the short-term interest rate and the instantaneous variance of changes in this rate (volatility of the short-term interest rate). Therefore the prices of contingent claims reflect the current levels of the interest rate and its volatility. The choice of interest rate volatility as the second state variable is supported by the fact that volatility is a key variable in contingent claim pricing.

The term structure of interest rates is embedded in the macro-economic system and is related to various economic factors. For this reason, Langetieg [36] proposes a model that can accommodate an arbitrary number of economic variables. The model is essentially an extension of Vasicek’s term structure model [50], studied in Chapter 1, with multiple sources of uncertainty.

Ball and Torous (BT) [4] propose an equilibrium methodology to value contingent claims on risk-free zero coupon bonds. The resulting closed-form valuation formula is independent of investor preferences and eliminates the need for numerical estimations of utility-dependent factors.

The Vasicek [50] and CIR [18] models, studied in Chapters 1 and 2 respectively, allow all interest rate contingent claims to be valued in a consistent manner, but involve unobservable parameters and do not provide a perfect fit for the current interest rate term structure.

Black, Derman and Toy (BDT) [6] make use of a binomial tree approach to model interest rates in a discrete time framework. The model has one fundamental factor, the short-term interest rate, which is used to determine all rates and security prices. The current term structure of interest rates and related volatilities are used to construct a binomial tree of possible shortterm interest rates in the future. Since an interest rate sensitive security is characterised by its payoff at expiry, the constructed tree of possible interest rates is used to determine the current price of a security by means of an iterative procedure.

The discrete time Black, Derman and Toy model [6], discussed in Chapter 8, makes provision for two time-dependent factors: the mean short-term interest rate and the short-term interest rate volatility. The continuous time equivalent of the model clearly shows that the rate of mean reversion is a function of the volatility. This is equivalent to future short-term interest rate volatilities being fully determined by the observed volatility term structure. This dependence makes it impossible to specify these two factors independently.

Models studied in the previous chapters specify the movement of the shortterm interest rate and thereby endogenously determine the form of term structure (including its initial value). Ho and Lee (HL) [27] developed a model which takes as input, the initial interest rate term structure and derives its subsequent stochastic evolution. Hence the theoretical zero coupon bond prices (that is, those produced by the model) will be exactly consistent with those observed in the market.

Heath, Jarrow and Morton (HJM) [25] present a unifying framework for term structure models. This framework introduces a formal elegance and generality to the interest rates modelling problem. It shows that the absence of arbitrage results in a link between the volatility of discount bonds and the drift of forward rates. In fact, in the risk-neutral world, the forward rate drift is completely determined by the specification of the discount bond volatility function. Previously developed models can be shown to be special cases of this general framework.

All the models examined thus far have been based on instantaneous short-term or forward interest rates. This implies that the fundamental building blocks, that is default-free bonds, are assumed to be continuous (or smooth) with respect to the tenor. Even the discrete time models such as Ho and Lee [27] (see Chapter 10) and Black, Derman and Toy [6] (see Chapter 8), which make use of a discrete set of discount bonds, assume these are extracted from an underlying continuum of default-free bonds. Such a continuum of default-free discount bonds is not actually traded, nor does the associated continuum of instantaneous shortterm or forward interest rates exist.

In §7.3 we examine the pricing of contingent claims within the HW-extended Vasicek framework. The time t price of a European call option, with expiry time T, t, T ∈ [0, T^*] and strike price X, on a zero coupon bond of maturity s is given by¹:

In Chapter 8 we examined the formulation of the BDT model within a (discrete time) binomial lattice as well as its continuous time equivalent. The short-term interest rate process takes the form¹:

The HJM framework demonstrates that when the money market account is used as numeraire, forward rates are not martingales but have a non-zero drift term fully determined by the forward rate volatility. Hence, only the forward rate volatilities are needed to price and hedge interest rate contingent claims. To begin an implementation of HJM, the form of the forward rate volatility function must be specified. Consider the differential form of equation (11.32), the forward rate process under the martingale measure¹: