where the first term is the mark-lo-inai kel payment on tlie futures contracts purchased at time I and the second term is the payoff on the bonds. He-cause Ihe futures contracts are marked to market, the entire position can he liquidated al lime /+ 1 without generating any further cash flows at time I + n. Since this investment strategy costs //, at time t and pays , и at lime Ml, we have shown lhat (11.3.10) holds. Furthermore, we can solve (11.3.10) forward to lime /+ n, using the fact that + = .S,+ , to obtain

, = F-,

ii.h а -Ч+и/II Y/+I

(11.3.12)

Comparing equations (1 1.3.0) and (11.3.12), we can see that there are some circumstances where forward contracts and futures contracts written on the same underlying asset with the same maturity have equal prices. First, il bond prices are not random then absence of arbitrage requires that

i - П ы1 Ум/. Sl> / = Чщ- This means that forward and futures prices are equal in any model with a constant interest rate. Second, if there is only one period to maturity then P, = / and again (. , = ,. Since futures contracts are marked to market daily the period here must be one day. so this result is of limited interest. Third, if the price of the underlying asset is not random, then forward and futures prices both equal the underlying asset price. To see this, note that if .S,+ = V, a constant, then (11.3.0) becomes

( , = К.Л/ .м 17 /1 = ( l ,)V.l[M,u+ ) = V, (I 1.3.13)

since/ , = К, I Л/ ,м . Under the same conditions Hlt = V, and we can show that , = Iif . = Ibecause (11.3.10) becomes

, = F., Л/,., i V/Ii,] = n7/ )F.,[M,+,] = I. (11.3.14)

I bus , = Ilor all n, so forward and futures prices are equal.

Mote generally, however, forward and futures prices may differ. In the case where the underlying asset is an n + т-period zero-coupon bond at time I, which will be a r-period bond at time / 4- n, we can write the forward price as С and the futures price as , ,. The forward juice is easy to calculate in this case:

( = (I ,) K,M .M /,. I = />, + ,/7> ,. (11.3.15)

When r = 1 die yield on this forward contract is the forward rale defined in Section 10.1.1 of Chapter If): / , = l/Cj ,.

The futures price must be calculated recursively from equation (I 1.3.10). In a particular term-strut tine model one can do the calculation explicitly and solve lot the relation between forward and futures prices.

In general equation (11.3.10) must be evaluated using numerical methods, but it simplifies dramatically in one special case. Suppose that M .,+ and .S,+ are jointly lognormal conditional on time / information, with conditional expectations of their logs ц, and д and conditional variances land covariance of their logs a, , , or. and ami. All these moments may depjend on / and n, but we suppress this for notational simplicity. Then we have

E,[M .,+ <>,+ I > X]

о mm 4 <7jj 4 2

= exp I fiui 4-/1,4-

л + о- ,. *+gi (11.3.17)

Е,[Л/ (

v., ( втт\ (Ц-т0,т-х\ ,0,0,

> X] = exp I Цт + - J Ф I---J. (11.3.18)

where Ф() is the cumulative distribution function of a standard normal random variable, and x = log(X).

Equations (11.3.17) and (11.3.18) hold for any lognormal random variables M and S and do not depend on any other properties of these variables.

iu4lic notation here (Idlers lioni the notation used in Chapter tl. There I, is used lor the ttudei King security price, hut here we reserve / lor /ern-coupnn bond prices and use Л, lor a generic security jMice.

11 these resit I is were derived hv Rubinstein (I(.l7(i); see also I luangaud t.it/eulierger t IMHH).

Problem 11.2 is to do this for the homoskedastic single-factor model devel oped in Section 11.1.1. The problem is to show that the ratio of forward to futures prices is constant in that model, and that it exceeds one so that % forward prices are always greater than futures prices.

11.3.3 Option Pricing in a Term-Structure Model

Suppose one wants to price a European call option written on an underlying security with price .S,.° If the option has n periods to expiration and exercise price X, then its terminal payoff is Max(.S,+ - X, 0). It can be priced like any other и-period asset using the м-period stochastic discount factor М ., = Л/,+ ... M,+ . Writing the option price as C ,(X), we have j

C ,(X) = E,[M, + Max(.V,+ -X,0>]

= E([M .,+ .V,+ I X,+ > X]

-XE,[Af ,+ I Sl+ > X]. (П.Я.16)

Го get the standard option pricing formula of Black and Scholes (197.3), we need two further assumptions. First, assume that the conditional variance of the underlying security price и periods ahead, tr , is proportional lo >>: <7 = tier2 for some constant a1. Second, assume that the term structure is (lat so that / , = e~ for some constant interest rate r. With these additional assumptions, (11.3.21) yields the Black-Scholes formula,1-

,. Л, - x + ir + a2/!)), c ,(x) = л,ф

-Хг-Ф(±-Я/2) У (4.3.22)

For fixed-income derivatives, however, the extra assumptions needed to gel die Black-Scholes formula (11.3.22) are nol reasonable. Suppose lhat ihe asset on which the call option is written is a zero-coupon bond which currently has н4-г periods to maun ily. If the option has exercise price Aand и periods to expiration, the options payoff al expiration will be Max(/,. -

-Ol onil.se, tin any given it wc ran always dclinc n = a.J it anil i = -/ ,/ so llial ll ic 1U.U k-.Scltnles lotniiil.i applies lor llial и. I lie assumptions given are needed tor Ilie lllai k-.Sjlioles lot inula In apply In all n willi die same i and a.

.V. 0). fhe relevant bond price al expiration is the г-pe riot I bond price since the maturity of the bond shrinks over lime. In a term-structure model the conditional volatility of the r-period bond price it periods ahead is not generally n times the conditional volatility of ihe (к + r - 1 {-period bond price one period ahead. Also, of course, the term structure is generally not Hal in a term-structure model.

To get closed-form solutions for interest-rale derivatives prices we need a term-structure model in which bond prices and stochastic discount factors are conditionally lognormal alall horizons; that is, we need die homoskedastic single-factor model of Section 11.1.1 or some nuillilactor generalization of it. In the single-factor model we can use the option pricing formula (11.3.21) with the following inputs: .V, = /,Hr., = exp(-A fr - /j +t x,), l i = exp(-A - ti xt), and

ст = Var, [/;,. ] = Var,[-/lr - II, ,v,+

., (1 - </>r)V-(l - 0- )

= Й Var,tx l = LtTT-iW1- <П-3-23> (1 - ф)-( I - ф1)

This expression for ais docs not grow linearly with п. I Ience if one uses the Black-Scholes formula (11.3.22) and calculates implied volatility, the implied volatility will depend on the maturity of the option; there will be a term structure of implied volatility that will depend on the parameters of the underlying term-structure model. Jamshidian (1989) presents a continuous-lime version of ibis result, and Turnbull and Milne (1991) derive il in discrete time along wilh numerous results lor other types of derivative securities. Option pricing is considerably more difficult in a square-root model, but Cox, Ingersoll, and Ross (1985a) present some useful results.

Investment professionals often want to price options in a way that is exactly consistent with the current term structure of interest rales. To do this, we can break the н-pcriod stochastic discount laclor into two components:

Af + = Л/,:.,+ <M. . (11.3.24)

where, as in Section 11.3.1, the -component is stochastic while the b-component is deterministic. There is a corresponding decomposition of bond prices for any maturity j: /у, = P t /. Then it is easy to show that

с лх) = е,[м /+и </+, Max(/;;,+ /, ,+ - x, o)j

= < . Pt.l+MK,+H Max(/r ,+ - .\7/r .,+, ())]

= l>t,i>i.,+ c:,miU >

= C, (ZWi ) ( .3.25)

Bui wc know Irom asscl pricing theory that ihe underlying security price S, must satisfy

л, = к,[л; ,+ л-,+ ) = esp( + ,t, + a+ 0 + 2ff ), ,n.;u<

We also know that the price of an n-pcriod zero-coupon bond, / must ilisfy

/, = К, I I = exp ir , + . (11 .3.21

Using (11.3.19) and (11..4.20) to.simplify (11.3.17) and (11.3.18), and substituting into (11.3.10), we get an expression for the price of a call option when the underlying security is jointly lognormal with the multiperiod stochastic discount factor:

, + o*m, - x + т \ Л<, 4- (T - л

<: ,(X> = .s,ф ----- - xф

j i , ut -.)(/ Ш t II II .1 I OllltS

where (. is thecall option prico dial would prevail if ihe stochastic discount lactor were Л1 . In oilier words options can be priced using die stochastic term-siriK ture model, using the deterministic model only to adjust the exercise price and the final solution for the option price. This approach was first used by I lo and I .ее (1980); however as Dybvig (1989) points out, I lo and I .ее choose as their n-modcl the single-factor homoskedaslic model with ф = 1, which has numerous unappealing properties. Шаек, Derman, and Toy (1990), 1 leath, Jarrow, and Morton (1992), and Hull and White (1990a) use similar approaches with different choices for the cmiiocIcI.

11.4 Conclusion

In ibis chapter we have thoroughly explored a tractable (hiss of hue rest-rate models, the so-called aflinc-yield models. In these models log bond yields are linear in stale variables, which simplifies the analysis of the term structure of interest rates and of fixed-income derivative securities. We have also seen that affme-yield models have some limitations, particularly in describing the dynamics of the short-term nominal interest rate. There is accordingly great interest in developing more flexible models that allow for such phenomena as multiple regimes, nonlinear mean-reversion, and serially correlated interest-rale volatility, and that fully exploit the information in the yield curve.

As the term-structure literature moves forward, it will be important to integrate it with the rest of the asset pricing literature. We have seen that term-structure models can be viewed as time-series models for the stochastic discount factor. The research on stock returns discussed in Chaptci 8 also seeks to characterize the behavior of the stochastic discount factor. Ry combining the information in the prices of stocks and fixed-income securities it should be possible lo gain a better understanding of the economic forces lhat determine the prices of financial assets.

Problems-Chapter 11

11.1 Assume that the homoskedaslic lognormal bond pricing model given by equations (11.1.3) and (I 1.1.5) holds with ф < \.

11.1.1 Suppose von fit the current term structure of interest rates using a random walk model augmented by deterministic drill terms, equation (11 .3.4). Derive an expression relating the drift terms to the state variable л/ and the parameters of the true bond pricing model.

11.1.2 (lompare ihe expected future log shorl rates implied by the true bond pricing model and the random walk model with deterministic drifts.

Problems

11.1.3 Compare the time / conditional variances of log bond prjices al time I 4- 1 implied by the true bond pricing model and the random walk

model with deterministic drifts. i

11.1.4 Compare the prices of bond options implied by the true! bond pricing model and the random walk model with deterministic drifts.

Note: This question is based on Backus and Zin (1994).

11.2 Define (>, , to be the price at time / of an n-period forward contract on a zero-coupon bond which matures al time / 4- n 4- r. Define Htlll to be the price at time I of an n-period futures contract on the same zero-coupon bond. Assume thai the homoskedastic single-factor term-structure model of Section 11.1.1 holds.

11.2.1 Show thai both the log forward price g and the log futures price /(, , are affme in the state variable x,. Solve for the coefficients determining these prices as functions of the term-structure coefficients A and B .

11.2.2 Show lhat the ratio of forward to futures prices is constant and greater than one. Give some economic intuition for this result.

11.2.3 For the parameter values in Section 11.2.2, plot the ratio of forward prices to futures prices as a function of maturity n.

Nole: This question is based on Backus, Foresi, and Zin (1996).