ter, “The Pricing of Options on Debt Securities,” Journal of Financial andQuantitative Analysis, 15 (March 1980), 11-24.
> The way that the interest rate tree is used is explained later in the chapter.Interest Rate Derivatives: Models of the Short Rate 539
 
InterestrateN\ High interest ratehas negative trendoe Reversion[ levelLow interest rate/ has positive trendTime>
 
Figure 23.1 Mean reversion
Mean Reversion
The assumption that the short-term interest rate behaves like a stock price is a natural startingpoint but is less than ideal. One important difference between interest rates and stock prices is thatinterest rates appear to be pulled back to some long-run average level over time. This phenomenonis known as mean reversion. When r is high, mean reversion tends to cause it to have a negativedrift; when r is low, mean reversion tends to cause it to have a positive drift. Mean reversion isillustrated in Figure 23.1. The Rendleman and Bartter model does not incorporate mean reversion.
There are compelling economic arguments in favor of mean reversion. When rates are high, theeconomy tends to slow down and there is low demand for funds from borrowers. As a result, ratesdecline. When rates are low, there tends to be a high demand for funds on the part of borrowers andtates tend to rise.
23.4 THE VASICEK MODEL
 
 
In Vasicek’s model, the risk-neutral process for r is
dr = a(b —r)dt+odz
where a, b, and o are constants.? This model incor Sto a level b at rate a. Superimposed upon this * ‘pull” is a normally distributed stochastic term o dz.Vasicek shows that equation (23.2) can be used to obtain the following expression for the price
at time t of a zero-coupon bond that pays $1 at time T:
758°
P(t, T) = A(t, T)e BUDO (23.6)
* See O. A. Vasicek, ‘An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics,5 (L977), 177-88.540 CHAPTER 23
Yield on Yield onzero-coupon zero-couponbond bond
 
ee
Maturity Maturity
 
Yield onzeiu-cuupoll
bond
Maturity
 
Figure 23.2 Possible shapes of term structure when the Vasicek model is used
In this equation, r(t) is the value of r at time ¢,
 
 
1 — eat)B(t, T) = ——— (23.7)and(Bt, T) -—T + tab — 07/2) oo BL, TYT)= - 8A(t, T) exp( 2 ha (23.8)When a =0, we have A(t, T) = T —t, and A(t, T) = expfo°(T ~ 0° /6].Using equation (23.4), we getRG, T) — — | tn AGT) + at Bi, Tyr) (23.9)T-t T~t
showing that the entire term structure can he determined as a function of r(t) once a, b, and a arechosen. Its shape can be upward sloping, downward sloping, or slightly “humped” (see Figure 23.2).
Valuing European Options on Zero-Coupon Bonds
Jamshidian has shown that options on zero-coupon bonds can be valued using Vasicek’s model.’The price at time zero of a European call option maturing at time T on a zero-coupon bond with
* See F. Jamshidian, “An Exact Bond Option Pricing Formula,” Journal of Finance, 44 (March 1989), 205-9.Interest Rate Derivatives: Models of the Short Rate 541
principal L isLP(O, s)N(h) — K P(G, T)N(h — op) (23.10)
where L is the bond principal, s is the bond maturity,
_ 1, LPO,s) , opha mMeOnKt 2—aT_ eo _ pas-T) l-eoP a (Ine ) 2a
and K is the strike price. The price of a European put option on the bond isKP(O, T)N(—h + op) — LP(0, s)N(—h) (23.11)When a = 0, we have op = o(s — T)VT.
Valuing European Options on Coupon-bearing Bonds
Jamshidian also shows that the prices of options on coupon-bearing bonds can be obtained fromthe prices of options on zero-coupon bonds in a one-factor model, such as Vasicek’s, where allrates are positively related to r. Consider a European call option with exercise price K andmaturity 7 on a coupon-bearing bond. Suppose that the bond provides a total of n cash flowsafter the option matures. Let the ith cash flow be c; and occur at time s; (1 <i <1; 5; > T).Define:
rx: Value of the short rate, r, at time 7 that causes the coupon-bearing bond price to equal thestrike price
K;: Value at time T of a zero-coupon bond paying off $1 at time s; when r = rx
When bond prices are known analytically as a function of r (as they are in Vasicek’s model), rg canbe obtained very quickly using an iterative procedure such as the Newton-Raphson method infootnote 2 of Chapter 5.
The variable P(T,s;) is the price at time T of a zero-coupon bond paying $1 at time s,;. Thepayoff from the option is therefore
max(0 Soe F(T, si) x)
i=]
Because all rates are increasing functions of r, all bond prices are decreasing functions of r. Thismeans that the coupon-bearing bond is worth more than K at time T and should be exercised ifand only ifr < rx. Furthermore, the zero-coupon bond maturing at time s; underlying the coupon-bearing bond is worth more than c;K; at time T if and only if r < rg. It follows that the payofffrom the option is
» ¢, max[0, P(T, s;) — Kj]i=l
This shows that the option on the coupon-bearing bond is the sum of » options on the underlyingzero-coupon bonds. A similar argument applies to European put options on coupon-bearingbonds.542 CHAPTER 23
Example 23.1 Suppose that a= 0.1, b=0.1, and o = 0.02 in Vasicek’s model with the initialvalue of the short rate being 10% per annum. Consider a three-year European put option with astrike price of $98 on a bond that will mature in five years. Suppose that the bond has a principal of$100 and pays a coupon of $5 every six months. At the end of three years, the bond can be regardedas the sum of four zero-coupon bonds. If the short-term interest rate is r at the end of the three years,the value of the bond is, from equation (23.6),
5A(3, 3.5)e 88? 4 5AG3, Ae POM” + 5A(3, 4.5)e FF” 4 105A(3, Se PO”Using the expressions for A(t, T) and B(t, T) in equations (23.7) and (23.8), we obtain5 x 0.9988e9 877" 4.5 x 0.995209 516" 15 x 0.989507!" 4 105 x 0.981907! 8127
To apply Jamshidian’s procedure, we must find rx, the value of r for which this bond price equals thestrike price of 98. An iterative procedure shows that rg = 0.10952. When r has this value, the valuesof the four zero-coupon bonds underlying the coupon-bearing bond are 4.734, 4.484, 4.248,and 84.535. The option on the coupon-bearing bond is therefore the sum of four options on zero-coupon bonds:
1. A three-year option with strike price 4.734 on a 3.5-year zero-coupon bond with a principal of 52. A three-year option with strike price 4.484 on a 4-year zero-coupon bond with a principal of 53. A three-year option with strike price 4.248 on a 4.5-year zero-coupon bond with a principal of 54. A three-year option with strike price 84.535 on a 5-year zero-coupon bond with a principal of 105
To illustrate the pricing of these options, consider the fourth. From equation (23.6), P(0, 3) = 0.7419and P(0, 5) = 0.6101. Also, op = 0.05445, h = 0.4161, L = 105, and K = 84.535, Equation (23.11)gives the value of the option as 0.8085. Similarly, the value of the first, second, and third options are,respectively, 0.0125, 0.0228, and 0.0314. The value of the option under consideration is therefore0.0125 + 0.0228 + 0.0314 + 0.8085 = 0.8752.
eo oko FAY oeI
23.5
 
 
In Vasicek’s model the short-term interest rate, r, can become negative. Cox, Ingersoll, and Rosshave proposed an alternative model where rates are always nonnegative. The risk-neutral processfor r in their model is
dr = a(b —n)dt + a./rdz
This has the same mean-reverting drift as the Vasicek model, but the standard deviation of thechange in the short rate in a short period of time is proportional to /r. This means that, as theshort-term interest rate increases, its standard deviation increases.
Cox, Ingersoll, and Ross show that, in their model, bond prices have the same general form as inVasicek’s model, that is,
P(t, T) = A(t, Tye BO
> See J. C. Cox, J. E. Ingersoll, and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica,53 (1985), 385-407.Interest Rate Derivatives: Models of the Short Rate 543
 
but the functions B(t, T) and A(t, T) are different:
WerT-9 — 1)B(t, T) = ————___—_ > —_(vy tay(e"F9 — 1) + 2y(aty(T-H/2 2ab/o”AC.T) = { 2ye'
\(y Faye" ~ 1) + 2y)
with y= Ja? +207, Upward-sloping, downward-sloping, and slightly humped yield curves arepossible. As in the case of Vasicek’s model, the long rate, R(t, T), is linearly dependent on r(t). Thismeans that the value of r(¢) determines the level of the term structure at time ¢. The general shapeof the term structure at time ¢ is independent of 7(t), but does depend on t.
Cox, Ingersoll, and Ross provide formulas for European call and put options on zero-couponbonds. These involve integrals of the noncentral chi-square distribution. European oplions oncoupon-bearing bonds can be valued using Jamshidian’s approach in a similar way to thatdescribed for Vasicek’s model.
23.6 TWO-FACTOR EQUILIBRIUM MODELS
 
A number of researchers have investigated the properties of two-factor equilibrium models. Forexample, Brennan and Schwartz have developed a model where the process for the short ratereverts to a long rate, which in turn follows a stochastic process.° The long rate is chosen as theyield on a perpetual bond that pays $1 per year. Because the yield on this bond is the reciprocal ofits price, It6’s lemma can be used to calculate the process followed by the yield from the processfollowed by the price of the bond. The bond is a traded security. This simplifies the analysisbecause the expected return on the bond in a risk-neutral world must be the risk-free interest rate.
Another two-factor model, proposed by Longstaff and Schwartz, starts with a general equi-librium model of the economy and derives a term structure model where there is stochasticvolatility.’ The model proves to be analytically quite tractable.
23.7 NO-ARBITRAGE MODELS
 
 
The disadvantage of the equilibrium models presented in the preceding few sections is that they donot automatically fit today’s term structure. By choosing the parameters judiciously, they can bemade to provide an approximate fit to many of the term structures that are encountered in practice.But the fit is not usually an exact one and, in some cases, there are significant errors. Most tradersfind ihis unsatisfactory. Not uni¢asunably, they argue that ihey cai have very little confidence inthe price of a bond option when the model does not price the underlying bond correctly. A 1% error
in the price of the underlying bond may lead to a 25% error in an option price.
® See M. J. Brennan and E. S. Schwartz, “A Continuous Time Approach to Pricing Bonds,” Journal of Banking andFinance, 3 (Suly 1979), 133-55; M. J, Brennan and E. 8. Schwartz, “An Equilibrium Model of Bond Pricing and aTest of Market Efficiency,” Journal of Financial and Quantitative Analysis. 21, no. 3 (September 1982), 301-29.
7 See F. A. Longstaff and E. 8, Schwartz, “Interest Rate Volatility and the Term Structure: A Two Factor GeneralEquilibrium Model,” Journal of Finance, 47, no. 4 (September 1992), 1259-82.544 CHAPTER 23
A no-arbitrage model is a model designed to be exactly consistent with today’s term structure ofinterest rates. The essential difference between an equilibrium and a no-arbitrage model is thereforeas follows. In an equilibrium model, today’s term structure of interest rates is an output. In ano-arbitrage model, today’s term structure of interest rates is an input.
In an equilibrium model, the drift of the short rate (i.e., the coefficient of dt) is not usually afunction of time. In a no-arbitrage model, this drift is, in general, dependent on time. This isbecause the shape of the initial zero curve governs the average path taken by the short rate in thefuture in a no-arbitrage model. If the zero curve is steeply upward sloping for maturities between t,and t), r has a positive drift between these times; if it is steeply downward sloping for thesematurities, r has a negative drift between these times.
It turns out that some equilibrium models can be converted to no-arbitrage models by includinga function of time in the drift of the short rate.
23.8 = THE HO-LEE MODEL
 
Ho and Lee proposed the first no-arbitrage model of the term structure in a paper in 1986.8 Theypresented the model in the form of a binomial tree of bond prices with two parameiers: the shori-rate standard deviation and the market price of risk of the short rate. It has since been shown thatthe continuous-time limit of the model is
dr = @(t)dt +adz (23.12)
where o, the instantaneous standard deviation of the short rate, is constant and 9(2) is a function oftime chosen to ensure that the model fits the initial term structure. The variable 6(t) defines theaverage direction that r moves at time t. This is independent of the level of r. Interestingly, Ho andLee’s parameter that concerns the market price of risk proves to be irrelevant when the model isused to price interest rate derivatives. This is analogous to risk preférences being irrelevant in thepricing of stock options.
The variable 0(t) can be calculated analytically. It is
A(t) = F,(0, 1) + 07t (23.13)
where the F(0, t) is the instantaneous forward rate for a maturity t as seen at time zero and thesubscript t denotes a partial derivative with respect to t. As an approximation 4(t) equals F,(0, t).This means that the average direction that the short rate will be moving in the future is approxi-mately equal to the slope of the instantaneous forward curve. The Ho—Lee model is illustrated inFigure 23.3. The slope of the forward curve defines the average direction that the short rate ismoving at any given time. Superimposed on this slope is the normally distributed randomoutcome.
In the Ho-Lee model, zero-coupon bonds and European options on zero-coupon bonds can bevalued analytically. The expression for the price of a zero-coupon bond at time t in terms of theshort rate is
P(t, T) = A(t, Te OT (23.14)
5 See T.S. Y. Ho and S.-B. Lee, ‘Term Structure Movements and Pricing Interest Rate Contingent Claims,” Journalof Finance, 41 (December 1986), 1011-29.interest Rate Derivatives: Models of the Short Rate 545,
4 Short . poo
rate
   
Initial forward curve
 
Time
Figure 23.3 The Ho-Lee model
whereinate. Dat D) pp 7212 POD ree — pe(t, 7) PO, 1) ( ) at 37 tl )
 
 
In these equations, time zero is today. Times t and T are general times in the future with T > 1. Theequations therefore define the price of a zero-coupon bond at a future time t in terms of the shortrate at time t and the prices of bonds today. The latter can be calculated from today’s termstructure.
For the remainder of this chapter, we will denote the dt-period interest rate at time t by R(t) orjust R. From equation (23.14) we can show (see Problem 23.22) that
P(t, T) = A(t, Te ROT (23.15)whereP(O,T) T- t P(O, t +82)
2PO.) 7 BOD 1o°(T — OT — f) - 51] (23.16)
In A(t, T) = In
 
In practice, we usually compute bond prices in terms of R rather than r and so equation (23.15) ismore useful than equation (23.14). Equations (23.15) and (23.16) involve only bond prices at timezero, not partial derivatives of these prices. They demonstrate that we do not require the initial zerocurve to be differentiable in applications of the model.The price at time zero of a call option that matures at time T on a zero-coupon bond maturing attime s isLP(0, s)N(A) — K P(O, T)N(h — op)546 CHAPTER 23
where L is the principal of the bond, K is its strike price,
_ 1, EPO,s) opham O.nK + 2and
op =o(s—T)VTThe price of a put option on the bond isKP(0, T)N(—h + op) — LP(O, s)N(—h)
This is essentially the same as Black’s model for pricing bond options in Section 22.2. The bondprice volatility is o(s — T) and the standard deviation of the logarithm of the bond price at time Tis op. As explained in Section 22.3, an interest rate cap or floor can be expressed as a portfolio ofoptions on zero-coupon bonds. It can therefore be valued analytically using the Ho-Lee model.
The Ho-Lee model is an analytically tractable no-arbitrage model. It is easy to apply andprovides an exact fit to the current term structure of interest rates. One disadvantage of the modelis that it gives the user very little flexibility in choosing the volatility structure. The changes in allspot and forward rates during a short period of time have the same standard deviation. A relateddisadvantage of the model is that it has no mean reversion. Equation (23.12) shows that, regardlessof how high or low interest rates are at a particular point in time, the average direction in whichinterest rates move over the next short period of time is always the same.
23.9 THE HULL-WHITE MODEL
 
 
In a paper published in 1990, Hull and White explored extensions of the Vasicek model thatprovide an exact fit to the initial term structure.” One version of the extended Vasicek model that
they consider isdr = [0(t) — ar] dt + o dz (23.17)
ar=a(*- ) dt odsa
where a and o are constants. This is known as the Hull-White model. It can be characterized as theHo-Lee model with mean reversion at rate a. Alternatively, it can be characterized as the Vasicekmodel with a time-dependent reversion level. At time t the short rate reverts to O(t)/a at rate a. TheHo-Lee model is a particular case of the Hull-White model with a =0.
The model has the same amount of analytic tractability as Ho-Lee. The function 6(t) can becalculated from the initial term structure:
or
2O(t) = F,(0, t) + aF(0, 1) + 5 — ey (23.18)
The last term in this equation is usually fairly small. If we ignore it, the equation implies that thedrift of the process for r at time ¢ is F,(0, t) + a[F(0, t) — r]. This shows that, on average, r follows
° See J. Hull and A. White, “Pricing Interest Rate Derivative Securities,” Review of Financial Studies, 3, no. 4 (1990),573-92.Interest Rate Derivatives: Models of the Short Rate 547
Shortrate
   
Initial forward curve
Figure 23.4 The Hull-White model
the slope of the initial instantaneous forward rate curve. When it deviates from that curve, it revertsback to it at rate a. The model is illustrated in Figure 23.4.Bond prices at time t in the Hull-White model are given by
 
 
P(t, T) = A(t, T)e MO (23.19)
where
}— earnB(t, T) = -_—_—— (23.20)aandl
In A(t, T) = In an 2 Bt, ren e D i (et — eM) (e** — 1) (23.21)
Equations (23.19), (23.20), and (23.21) define the price of a zero-coupon bond at a future time f interms of the short rate at time t and the prices of bonds today. The latter can be calculated fromtoday’s term structure.
As in the case of the Ho—Lee model, it is more relevant to relate P(t, T) to R(t), the St-period rateat time t, We can show (see Problem 23.23) that
P(t, T) = A(t, Tye BPR (23.22)wherePO,T)  B,T) , PO,t+8t) &
On Bt, T)[B(t, T) — BUt, t + 80)]
In A(t, T) =1nA T= Oy Barto” PO.) 4a
 
(23.23)548 CHAPTER 23
andBt, T)
BUT) = Ba e+ on
(23.24)
As with the Ho—Lee model, these equations do not require the initial zero curve to be differenti-able.The price at time zero of a call option that matures at time T on a zero-coupon bond maturing attime s isLP(0, s)N(A) — KP(O, T)N(h — op) (23.25)
where L is the principal of the bond, K is its strike price,
1 LPQO,s) opn= In 50.NK* 2
a ee 1—e-%4Top = [le We
The price of a put option on the bond is
and
KP(0, T)N(—h + ap) — LP(0, s)N(—A) (23.26)
These option pricing formulas are the same as those given for the Vasicek model in equations (23.10)and (23.11). They are also equivalent to using Black’s model as described in Section 22.2. Thevariable op is the standard deviation of the logarithm of the bond price at time T, and the volatilityfor the bond used in Black’s model is op//T. As explained in Section 22.3, an interest rate cap orfioor can be expressed as a portfolio of options on zero-coupon bonds. It can iherefore be vaiuedanalytically using the Hull-White model.
The volatility structure in the Hull-White model is determined by both a and a. The model canrepresent a wider range of volatility structures than Ho—Lee. The volatility at time t of the price ofa zero-coupon bond maturing at time T is
*(1 —e MF)
The instantaneous standard deviation at time t of the zero-coupon interest rate maturing at time T is
oO
_ ear)aren °
 
and the instantaneous standard deviation of the T-maturity instantaneous forward rate is ce“? ~”.
These functions are shown in Figure 23.5. The parameter o determines the short rate’s instant-aneous standard deviation. The reversion rate parameter, a, determines the rate at which bondprice volatilities increase with maturity and the rate at which interest rate standard deviationsdecline with maturity. When a = 0, the model reduces to Ho—Lee, and zero-coupon bond pricevolatilities are a linear function of maturity with the instantaneous standard deviations of bothspot and forward rates being constant.Interest Rate Derivatives: Models of the Short Rate 549
Bond price Spot ratevolatility standarddeviation
  
 
Maturity Maturity
(a) (b)
Forward ratestandarddeviation
 
Maturity
(c)
Figure 23.5 Volatility structure in the Hull-White model
23.10 OPTIONS ON COUPON-BEARING BONDS
 
 
In Section 23.4 we showed how, when the Vasicek equilibrium model is used, we can expressan option on a coupon-bearing bond as a portfolio of options on zero-coupon bonds (seeExample 23.1). This section shows how we can do the same for the Ho—Lee and Hull-Whitemodels.
In the case of Vasicek’s model, we calculated the value of the short rate, r = ry, for which thecoupon-bearing bond price equaled the strike price. We then argued that the option on thecoupon-bearing bond was equivalent to a portfolio of options on the zero-coupon bondscomprising the coupon-bearing bond. The strike price of each option is the value of the corres-ponding zero-coupon bond when r = rx.
We could follow exactly the same procedure for the Ho—Lee and Hull-White models as for theVasicek model. But it is more convenient to work with the 4t-period rate, R, than with theinstantaneous short rate, r. We then never need to calculate partial derivatives of P(0,t) with550 CHAPTER 23
 
respect to t.!° A convenient choice for dt is the time between the maturity of the option and the firstsubsequent coupon on the underlying bond.
We calculate a value of R for which the coupon-bearing bond’s price equals the strike price.Suppose this is Rx. The option on the coupon-bearing bond is equivalent to a portfolio of optionson the zero-coupon bonds comprising the coupon-bearing bond. The strike price of each option isthe value of the corresponding zero-coupon bond when R= Rx.
Example23.2 Suppose that in the Hull-White model a = 0.1 and o = 0.015 and we wish to value a3-month European put option on a 15-month bond that pays a coupon of 12% semiannually. Wesuppose that both the bond principal and the strike price are 100. The continuously compounded zerorates for maturities of 3 months, 9 months, and 15 months are 9.5%, 10.5%, and 11.5%, respectively.The option under consideration is an option on a portfolio of two zero-coupon bonds. The first zero-coupon bond has a maturity of 9 months and a principal of $6. The second zero-coupon bond has amaturity of 15 months and a principal of $106.
Define R as the value of the 6-month rate at the maturity of the option. The value of the first zero-coupon bond underlying the option is 6e~*“°°. The value of the second zero-coupon bond is, fromequation (23.22),
106A(0.25, 1.25)e7 802512908
where the 5¢ in equation (23.22) equals 0.5. In this case, B(0.25, 0.75) = 0.4877 and B(0.25, 1.25) =0.9516, so that equation (23.24) gives B(0.25, 1.25) = 0.9756. Also, from equation (23.23),A(0.25, 1.25) = 0.9874. Let Rx be the value of R for which the coupon-bearing bond price equalsthe strike price. It follows that
be *RK 4. 106A(0.25, 1.25) O25129R — 100
This can be solved using an iterative procedure such as Newton-Raphson to give Ry = 10.675%.When & has this value, the zero-coupon bond maturing at time 0.75 is worth 5.68814 and the zero-coupon bond maturing at time 1.25 is worth 94.31186. The option on the coupon-bearing bond istherefore equivalent to:1. A European put option with a strike price of 5.68814 on a zero-coupon bond maturing attime 0.75 with a principal of 6; and2. A European put option with a strike price of 94.31186 on a zero-coupon bond maturing attime 1.25 with a principal of 106.Equation (23.26) gives the prices of these options as 0.01 and 0.43. The price of the option on thezero-coupon bond is therefore 0.44.
As explained in Section 22.4, a European swap option can be viewed as an option on a