Theoretical average torwarrt-raic curve

Sample average forward-rale опте

4 0 8

Maturity in years

Figure 11.2. Sample and Theoretical Average Fonvard-Itale Curves

delation of the forward rate declines at rate ф. In the 1952 to 1991 period the standard deviation of the n-period forward rate barely declines al all willi maturity n, and this feature of the data can only be fit by an extremely persistent short-rate process. Backus and Zin (1994) discuss this problem in detail and suggest that a higher-order model which allows both transitory and persistent movements in the short rate can fit the term structure more successfully.

Parameter identification is somewhat more difficult in the square-root model. Here the moments given in equation (11.2.19) become

Corr[yi jyi., ] = Ф

Var[y ] = (1 -4

lim EKm-i -ywl = lim К[/и(- Vwl = {-ВВа2 - dV/21/t

-♦00 >l-> OO

E[y ] = / 1 -0V72).

(11.2.20)

where Ii is the limiting value of Ii from equation (I 1. 1.23). As before, we can identify 0 = 0.98 from the estimated first-order autocorrelation of the short rale, but now the other parameters of the model are simultaneously determined. One can of course estimate them by Generalized Method of Moments. The square-root model, like the homoskedastic model, produces an average forward-rate curve that approaches ils asymptote very slowly when the short rale is highly persistent; thus the model has many of the same empirical limitations as the homoskedastic model.

In summary, the single-factor affine-yield models we have described in ibis chapter are too restrictive lo lit the behavior of nominal interest rates, fhe latent-variable structure of the dala, the nature of die short rale process, and the shape of the average term .structure are all hard to lit with these models. In response to this researchers are exploring more general models, including affine-yield models in which die single state variable follows a higher-order ARMA process (Backus and Zin [1994]), affine-yield models with several state variables (Lougslaff and Schwartz [ 1992)), regime-switching models (Gray [1996], Naik and Lee [1994]), and GARCH models of interest rale volatility (Brenner, Harjes, and Kroner [1996]). No one model has yet emerged as the consensus choice for modeling the nominal term structure. We note however that Brown and Schaefer (1994) and Gibbous and Ramaswatny (1993) have achieved some success in filling simple models to prices of UK index-linked bonds and econometric forecasts of real relurns on US nominal bonds. Thus single-factor afline-yicld models may be more appropriate for modeling real interest rales than for modeling nominal interest rates.

11.3 Pricing Fixed-Income Derivative Securities

One of the main reasons for the explosion of interest in term-structure models is the practical need to price and hedge fixed-income derivative securities. In this section we show how icrni-structure models can be used in this context. Section 11.3.1 begins by discussing ways to augment standard term-structure models so that they fit the current yield curve exactly. Derivatives traders usually want to take this yield curve as given, and so they waul to use a pricing model that is fully consistent with all current bond prices. We explain the popular approaches of I lo and Lee (1986), Black, Dcrman, and Toy (1990), and Heath, Jarrow, and Morion (1992). Section 11.3.2 shows how lerni-struclure models can be used to price forward and futures contracts on fixed-income securities, while Section 11.3.3 explores option pricing in the context of a term-structure model.

I1.1.1 Filling tlie Current Term Structure Exactly

In <rciKT.il a model gives an exact lit to as many data points as it has parameters. The homoskedaslic single-factor model presented in Section 11.1, for example, has lour parameters, </>, />, a-, and Ц. Inevitably this model does not lil the whole term strtu lure exactly. To allow for this ihe empirical work ol the previous sec lion added error terms, rellecling model specification error and measurement error in bond prices.

In pricing fixed-income derivative securities it may he desirable to have a model that does (it the current term structure exactly. To achieve this, we can use the result of Cox, Ingersoll, and Ross (1985a) and Dybvig (HIS1.!) thai one can add independent lerm-struclure models together. Л simple approach, due originally to I lo and I .ее (1980), is to break observed forward rates / , into two components:

и = /:,+/!;, (hid

where is ihe forward rate implied by a standard tractable model and /,[, is the residual. The residual component is then attributed to a deterministic term-structure model. Since a deterministic process is independent of any stochastic process, the decomposition (I 1.3.1) is always legitimate. There is a corresponding decomposition of the stochastic discount factor,

mi = ,Vi + Vr (11.3.2)

ln a deterministic model, the absence of arbitrage requires ih;>i

it, = yL,r = !+ (ics.S)

Thus we are postulating that future stochastic discount (actors contain a deterministic component lhat is reflected in future short-term interest rates and current forward rates.

Although this procedure works well in any one period, there is nothing lo ensure thai il will be consistent from period lo period. A typical application of the approach sets yj( = 0, so that the current short rale is used as an input into the stochastic term-structure model without any adjustment for a deterministic component. Deterministic components of future short rates у , are then set to nou/eio values to (it the time ( term structure. When time / + n arrives, however, this procedure is repeated; now y\ is set lo /его and deterministic components of mote distant future short rates are made noii/cro to lit ihe lime / 4- n lerm structure. As Dybvig (1989) emphasizes, ibis lime inconsistency is troublesome although the procedure may work well for some purposes.

It is also important to understand lhat lining one set of assel prices exactly does not guarantee that a model will lit other asset prices accurately.

E,[r +1 -yu] = -(l/2)Var,[r +1 - yu]. (11.3.5)

Backus, Foresi, and Zin (1996) illustrate this problem as follows. They as*, sume that the homoskedastic single-factor model of subsection 11.1.1 holdi; with a mean-reverting short rate so ф < 1. They show that one can exactly i( fit the current term structure with a homoskedastic random walk model, a ; lognormalversionofHoandI.ee (1986). The model uses equation (ill.1.5), but replaces equation (11.1.3) with

x,+i = + +i + ?/+.. Ul.3.4)

where g,+, is a deterministic drift term that is specified at time i for all future dates / + i in order to fit the time t term structure of interest rates, and as before f,+, is a normally distributed shock with constant variance a2. Backus, Foresi, and Zin (1996) show that this model does not capture the conditional means or variances of future interest rates, and so it misprices options on bonds. Problem 11.1 works this out in detail. \

A somewhat more sophisticated procedure for fitting the term structure of interest rates specifies future deterministic volatilities of shcjrt rate movements, as well as future deterministic drifts. Black, Derman, ajid Toy (1990) do this in a lognormal model for the short rate. In the present model one can replace the constant variance of a2, with a detWmin-istically time-varying one-period-ahead conditional variance of+i. Backus, Foresi, and Zin (1996) show that if the true model is the mean-reverting homoskedastic model, the misspecified random walk model with deterministic volatilities and drifts can fit any two of the current term structure, the conditional means of future short rates, and the conditional variances of future short rates. However it still cannot fit all three of these simultaneously, and so it cannot correctly price a complete set of bond options. The lesson of this example is that fixed-income derivative security prices depend on the dynamic behavior of interest rates, so it is important to model interest rate dynamics as accurately as possible even if one is interested only in pricing derivative securities today.

A related approach that has become very popular is due to Heath, Jar-row, and Morton (1992). These authors start from the current forward-rate curve discussed in Chapter 10, and they suggest that one should specify a term structure of forward volatilities to determine the movements of future risk-adjusted forward rates. To understand this approach as simply as possible, suppose that interest-rate risk is unpriced, so there are no risk premia in bond markets and the objective process for forward rates coincides with the risk-adjusted process. In this case bonds of all maturities must have the same expected instantaneous return in a continuous-time setting, and the same expected one-period return in a discrete-time setting. That is, the one-period version of the pure expectations hypothesis of the term structure (PEH) holds, so from (10.2.6) of Chapter 10 we have

TJjc expected log excess return on a bond of any maturity over the one-period interest rate is minus one-half the variance of the log excess return.

Now recall the relation between an n-pcriod-ahead 1-pcriod log forward r;tc / , and log bond prices, given as (10.1.8) in Chapter 10: /, = />, -pr+i.t- This implies that the change from time t lo time I + 1 in a forward rate for an investment to be made at time ( + n is

/n-l.l+l - fnl = (/i.-U+l - /V<+l) - (P l - pn+\.i)

- rll./+l ~~ r,l+l./+l

= (r .(+i -yu) ~ (V+l.i+t - yu)- (11.3.6)

Tsil

- V,i,(+l J\it vn+l.t-t-i Jill-

iking expectations of (11.3.6) and using (11.3.5), we find that

E,[/ i +i-/ ,] = (Jj (Var,[r +i + , -yu] - Var,[r .(+i - yu]) -

(11.3.7)

T le conditional variances of future excess bond relurns determine the expected changes in forward rates, and these expected changes together with tbie current forward-rate curve determine the forward-rate curves and yield curves that arc expected to prevail at every date in the future. Similar properties hold for the risk-adjusted forward-rate process even when interest-rate risk is priced.

Heath,Jarrow, and Morton (1992) exploit this insight in a continuous-time setting and show how it can be used to price fixed:income derivative securities. It is still important to model interest-rate dynamics accurately in this framework, but now the parameters of the model arc expressed as volatilities; many participants in the markets for fixed-income securities find it easier to work with these parameters than with the parameters that govern short-rate dynamics and interest-rate risk prices in traditional models. A drawback of this approach, however, is that the implied process for the short-term interest rate is generally extremely complicated.

11.3.2 Forwards and Futures

A particularly simple kind of derivative security is a forward contract. An л-period forward contract, negotiated al time I on an underlying security with price .S,+ al lime I + n, specifies a price at which the security will he purchased at time I + n. Thus ihe forward price, which we write C> is determined at time I but no money changes hands until time J + n.H

Cox, Ingersoll, and Ross (1981b) show that the forward price C, t is the lime t price of a claim to a payoff of .V/+ , at lime t+n. Equivalendy, G, P ,

The n-period forward rale defined in Section 1(1.1.1 of Chapter 10 is the yield она forward contract to buy a zero-coupon bond with maturity dale / + и + I at time / + n.

is the price of a claim to a payoff of ,V,+ . Intuitively, the / , terms appear because no money need be paid until lime / + n; thus the purchaser of a forward contract has the use of money between I and / + n. Cox, Ingersoll, and Ross establish this proposition using a simple arbitrage argument. They consider the following investment strategy: At time /, lake a long position in l/Pu, forward contracts and pul G , into n-pcriod bonds. By doing this one can purchase G ,/Pt bonds. Ihe (layoff from this strategy al lime I + n is

r[Sl+ -C.lll]+(- = (11.3.8)

Hi I III I III

where the first term is the profit or loss on the forward contracts and the second term is die payoff on the bonds. Since this investment strategy costs G , at time С and pays S,+ /P , al time I + n, the proposition is established. It can also be slated using stocliastic-discount-laclor notation as

G, = ЕЛА/ .,+ ,Л,+ /Л,(1, (11.3.9)

where the n-period stochastic discount factor M ,+ is the product of n successive one-period stochastic discount factors: M ,H = Af,+ ... /Vf(+ .

A futures contract differs from a forward contract in one important respect: It is marked to market <?&c\\ period during the life of die contract, so that the purchaser of a futures contract receives the futures price increase or pays the futures price decrease each period. Because of these margin payments, futures pricing-unlike forward pricing-generally involves more than just die two periods / and / -f n} If we write Ihe price of an -period futures contract as then wc have

H , = E,[M,+ l llH-U+i/Pu]. (11.3.10)

This can be established using a similar argument lo lhat of Cox, Ingersoll, and Ross. Consider the following investment strategy: At time t, take a long position in \/P\i futures contracts and put H , into one-period bonds. By doing this one can purchase HMjP\, bonds. At time t + 1, liquidate die futures contracts. The payoff from this strategy at time I + 1 is

[ - + ,-Я ,] + = ±i±i. (11.3.11)

к it u

Ihe lifasmy-boml and treasury-note futures contracts nailed on the Chicago Board of Trade also have a number of special option features that affect their prices. Л trader with a shot! position can choose lo deliver on any day wilhin the senlemeni month and can choose to deliver a number of alternative bonds. The short trader also has a wild card option lo announce delivery at a particular days settlement price any lime in the six hours after that price is determined. The discussion here abstracts Irom ihese option lealuies: see Hull (ISM, Chapter A) lor an introduction to them.