the forward rate for the period between 10.0 years and 10.25 years?
A trader wishes to compute the price of a !-year American call option on a 5-year bond with aface value of 100. The bond pays a coupon of 6% semiannually and the (quoted) strike price ofthe option is $100. The continuously compounded zero rates for maturities of 6 months, | year,2 years, 3 years, 4 years, and 5 years are 4.5%, 5%, 5.5%, 5.8%, 6.1%, and 6.3%. The best-fitreversion rate for either the normal or the lognormal model has been estimated as 5%.
A l|-year European call option with a (quoted) strike price of 100 on the bond is activelytraded. Its market price is $0.50. The trader decides to use this option for calibration. Use theDerivaGem software with 10 time steps to answer the following questions.
a. Assuming a normal model, imply the parameter o from the price of the European option.
b. Use the parameter o to calculate the price of the option when it is American.
c. Repeat (a) and (b) for the lognormal model. Show that the model used does not significantly
affect the price obtained providing it is calibrated to the known European price.
d. Display the tree for the normal model and calculate the probability of a negative interest rate
occurring.
e. Display the tree for the lognormal model and verify that the option price is correctly
calculated at the node where, with the notation of Section 23.12, i=9 and j= —-l.Use the DerivaGem software to value | x 4, 2 x 3, 3 x 2, and 4x | European swap options to
receive Aloatine and say fred. Assume that the 1-. 2-. 2-. 4. and S-_vear interest rates arc 2%,receive floating and pay nxca. Assume that the i-, <-, 5-, 4, ana 5-year interest rates are oi,
3.5%, 3.8%, 4.0%, and 4.1%, respectively. The payment frequency on the swap is semiannualand the fixed rate is 4% per annum with semiannual compounding. Use the lognormal modelwith a = 5%, o = 15%, and 50 time steps. Calculate the volatility implied by Black’s model foreach option.
Verify that the DerivaGem software gives Figure 23.11 for the example considered. Use thesoftware to calculate the price of the American bond option for the lognormal and normal modelswhen the strike price is 95, 100, and 105. In the case of the normal model, assume that a = 5% ando =1%. Discuss the results in the context of the heaviness of tails arguments of Chapter 15.
Modify Sample Application G in the DerivaGem Application Builder software to test theconvergence of the price of the trinomial tree when it is used to price a two-year call option onafive-year bond with a face value of 100. Suppose that the strike price (quoted) is 100, the couponrate is 7% with coupons being paid twice a year. Assume that the zero curve is as in Table 23.2.Compare results for the following cases:
a. Option is European; normal model with o = 0.01 and a = 0.05
b. Option is European; lognormal model with o = 0.15 and a = 0.05
c. Option is American; normal model with o =-0.01 and a = 0.05
d. Option is American; lognormal model with o = 0.15 and a = 0.05CHAPTER 24
INTEREST RATE DERIVATIVES:MORE ADVANCED MODELS
The interest rate models discussed in Chapter 23 are widely used for pricing instruments when thesimpler models in Chapter 22 are inappropriate. They are easy to implement and, if used carefully,can ensure that most nonstandard interest rate derivatives are priced consistently with activelytraded instruments such as intcrest ratc caps, European swap options, and European bond options.Two limitations of the models are:
1. They involve only one factor (i.e., one source of uncertainty).
2. They do not give the user complete freedom in choosing the volatility structure.
As explained in Section 23.13, an analyst can use the models so that they provide a perfect fit tovolatilities observed in the market today, but there is no way of controlling future volatilities. Thefuture pattern of volatilities is liable to be quite different from that observed in the market today.
This chapter discusses multifactor term structure models and presents some general approachesto building term structure models that give the user more flexibility in specifying the volatilityenvironment. The models require much more computation time than the models in Chapter 23. Asaresult, they are often used for research and development rather than routine pricing.
This chapter also covers the mortgage-backed security market in the United States anddescribes how some of the ideas presented in the chapter can be used to price instruments inthat market.
24.1. TWO-FACTOR MODELS OF THE SHORT RATE
 
 
In recent years there have been a number of attempts to extend the models introduced in Chapter 23so that they involve two or more factors. Examples are in papers published by Duffie and Kan andHull and White.! The Hull-White approach involves a similar idea to the equilibrium modelsuggested by Brennan and Schwartz that we described briefly in Section 23.6. It assumes that therisk-neutral process for the short rate, r, is
df(r) = (0) +u —af(n]dt +o, dz, (24.1)
' See D. Duffie and R. Kan, “A Yield-Factor Model of Interest Rates,” Mathematical Finance, 6, no. 4 (1996),379-406; J. Hull and A. White, “Numerical Procedures for Implementing Term Structure Models II: Two-FactorModels,” Journal of Derivatives, 2, no. 2 (Winter 1994), 37-48.
571572 CHAPTER 24
0.60
0.40
   
0.20Maturity (years)0.00
0 2 4 6 8 10
Figure 24.1 Example of volatility term structure that can be produced by the two-factor Hull-Whitemodel: f(r) =r, a= 1, b=0.1, 0, = 0.01, op = 0.0165, p = 0.6
 
 
where w has an initial value of zero and follows the processdu = —bu dt + 07 dz,
As in the one-factor models considered in Chapter 23, the parameter 6(t) is chosen to make themodel consistent with the initial term structure. The stochastic variable uw is a component of thereversion level of r and itself reverts to a level of zero at rate b. The parameters a, b, o,, and o> areconstants and dz, and dz) are Wiener processes with instantaneous correlation p.
This model provides a richer pattern of term structure movements and a richer pattern ofvolatility structures than the one-factor models considered in the Chapter 23. Figure 24.1 shows theforward rate standard deviations that are produced using the model when f(r) =r, a= 1,6 =0.1,o, = 0.01, a) = 0.0165, and » = 0.6. With these parameters, the model exhibits, at all times, a“humped” volatility structure similar to that observed in the market (see Figure 22.3). Thecorrelation structure implied by the model is also plausible.
When f(r) =r, the model is analytically tractable. The price at time t of a zero-coupon bondthat provides a payoff of $1 at time T is
P(t, T) = A(t, T) exp[— B(t, T)r — C(t, T)u] (24.2)where
1Bt. T)=—[l-e
1 | 1Ci. T= -a(T—t) er yt(N= Tab ba—»)° + ob
 
and A(t, T) is as given in Appendix 24A.The prices c and p at time zero of European call and put options on a zero-coupon bond areInterest Rate Derivatives: More Advanced Models 573
given byc= LP(0,s)N(h) 7 KP(0, T)N(h — op)
p= KP(0, T)N(—h + ap) — LP(O, s)N(—h)
where 7 is the maturity of the option, s is the maturity of the bond, K is the strike price, L is thebond’s principal,I In LP(0,s) | op
at 5
op P(O,T)K 2
h=
and gp is as given in Appendix 24A. Because this is a two-factor model, an option on a coupon-bearing bond cannot be decomposed into a portfolio of options on zero-coupon bonds as describedin Section 23.10. However, we can obtain an approximate analytic valuation by calculating the firsttwo moments of the price of the coupon-bearing bond and assuming the price is lognormal.
Constructing a Tree
To construct a tree for the model in equation (24.1), we simplify the notation by defining x = f(r),so thatdx = [0(t) + u — ax) dt +0, dzwithdu = —bu dt + 02 dz
Assuming a # 6 we can eliminate the dependence of the first stochastic variable on the second bydefining
 
 
ver b-aso thatdy = (A(t) — ayl dt + 03 dz;du = —bu dt + 02d23where2_ 2 a 205152oj =o, +— Fa +bay baa
and dz; is a Wiener process. The correlation between dz, and dz; is
pa + 9n/(b — a)03
Hull and White explain how an approach similar to that in Section 20.8 can be used to develop athree-dimensional tree for y and w on the assumption that 4(t) = 0 and the initial values of y and uare zero. A methodology similar to that in Section 23.12 can then be used to construct the final treeby increasing the values of y at time i 5 by a;. In the case f(r) = r, an alternative approach is to usethe analytic expression for 6(¢), given in Appendix 24A.
Rebonato gives some examples of how the model can be calibrated and used in practice.
> See R. Rebonato, Interest Rate Option Models, 2nd edn., Wiley, Chichester, 1998, pp. 306-8.574 CHAPTER 24
24.2 THE HEATH, JARROW, AND MORTON MODEL
 
 
 
We now present a model of the yield curve expressed in terms of the processes followed byinstantaneous forward rates. We continue to use the traditional risk-neutral world.
Notation
We will adopt the following notation:
P(t, T): Price at time ¢ of a zero-coupon bond with principal $1 maturing at time T
Q,: Vector of past and present values of interest rates and bond prices at time t thatare relevant for determining bond price volatilities at that time
v(t, T, Q,): Volatility of P(t. T)f(t, T,, T): Forward rate as seen at time t for the period between time 7; and time 7)F(t, T): Instantaneous forward rate as seen at time ¢ for a contract maturing at time Tr(t): Short-term risk-free interest rate at time t
dz(t): Wiener process driving term structure movements
The variable F(t, T) is the limit of f(t, 7, T + dt) as dt tends to zero.
Processes for Zero-Coupon Bond Prices and Forward Rates
When we are assuming just one factor, the risk-neutral process for P(t, T) has the formdP(t, T) = r(t)P(t, T)dt + w(t, T, 2,) P(t, T) z(t) (24.3)
The expected return is r(t) because a zero-coupon bond is a traded security providing no income.As its argument Q, indicates, the volatility, v, can, in the most general form of the model, be anywell-behaved function of past and present interest rates and bond prices. Because a bond’s pricevolatility declines to zero at maturity, we must have*
v(t, t, Q,) = 0
From equation (5.1), the forward rate, f(t, T,, 7), can be related to zero-coupon bond prices asfollows:r) = Inf P(t, Ti) — Inf P(t, T)]
i, 7), 7 (24.4)142 TT, UFrom equation (24.3) and [t6’s lemma,v(t, Tr, 21)"dIn{P(t, T))] = («9 - er) dt + W(t, Ty, 2) dz(2)
> The v(t, t, 2,) = 0 condition is equivalent to the assumption that all discount bonds have finite drifts at all times. Ifthe volatility of the bond does not decline to zero at maturity, an infinite drift may be necessary to ensure that thebond's price equals its face value at maturity.Interest Rate Derivatives: More Advanced Models 575
and
dt + v(t, Ty, 2,) dz(t)
dIn[P(t, T)] = (x9- v(t, fe |
so thatot, D, QY — ot, T, 2) at $e Ti, &) = Wt Th, 2)
b)=Me TT) 1h =T) TT,
 
dot) (24.5)
Equation (24.5) shows that the risk-neutral process for f depends solely on the v’s. It depends on rand the P’s only to the extent that the v’s themselves depend on these variables.
When we put 7; = T and 7, = T + 6T in equation (24.5) and then take limits as 67 tends to zero,f(t, T,, Ty) becomes F(t, T), the coefficient of dz(t) becomes v7(t, T, 2,), and the coefficient of dtbecomes
a[v(t, T, Q1 Ble 7 2)"T _ 4 7 pope, 7,2)oTwhere the subscript to v denotes a partial derivative. It follows thatdF(t,T) = ot. T, Q)vr(t, T, 2, dt — vy(t, T, 2, d2(t) (24.6)
Once the function v(t, 7, 2,) has been specified, the risk-neutral processes for the F(t, T)’s areknown.Equation (24.6) shows that there is a link between the drift and standard deviation of an
instantaneous forward rate. Heath, Jarrow, and Morton (HJM) were the first to point this out.4
Stantaneous forward rate, Heath, Jarrow, and Morton UaJM) were the rst HS OW
Integrating v,(t, Tt, Q;) between t = ft and t = T, we obtainTv(t, T, Q,) — v(t, t, Q,) = | v,(t, t, Q,) dttBecause v(t, t, Q,) = 0, this becomes7u(t, T, Q,) = | v(t, T, 2) dtt
If m(t, T, Q,) and s(t, 7, Q,) are the instantaneous drift and standard deviation of F(t, T), so thatdF(t,T) = m(t, T, 2,)dt + s(t, T, Q,) dz
it follows from equation (24.6) thatTm(t, T, 2) = s(t, T. a) s(t, t, 2) dt (24.7)t
The Process for the Short Rate
We now consider the relationship between the models we have developed in this section and themodels of the short rate considered in Chapter 23. Consider the one-factor continuous-time model
4 See D. Heath, R. Jarrow, and A. Morton, “Bond Pricing and the Term Structure of Interest Rates: A NewMethodology,” Econometrica, 60, no. 1 (1992), 77-105.576 CHAPTER 24
Figure 24.2 A nonrecombining tree such at that arising from the general HJM modelfor forward rates in equation (24.6). BecauseF(t,t) = FO, t)+ j dF(t, t)and r(t) = F(t, t), it follows from equation (24.6) that(i) = FO.) + j v(t, t, 2,)0,(t, t, 2,) de + i v(t, t, &,) dale) (24.8)Differentiating with respect to ¢ and using v(t, t, 2,) = 0, we obtain?
dr(t) = F,(0, )dt+ ([ to. t, Q,)vy(t, t, 2,) + v(t, t, 2,)"Jdr) dt0
+ ([, Dil, ty a.) date) at + [v,(t, t, 2,)|,_,]d2() (24.9)
It is interesting to examine the terms on the right-hand side of this equation. The first and fourthterms are straightforward. The first term shows that one component of the drift in r is the slope of
> The stochastic calculus in this equation may be unfamiliar to some readers. To interpret what is going on, we canreplace integral signs with summation signs and d’s with 5’s. For example, b u(t, t, Q,)0,(t, t, Q,)dt becomesSey 0 St, t, Q;)v,(7 St, t, Q;) bt, where 6¢ = t/n.interest Rate Derivatives: More Advanced Models 577
the initial forward rate curve. The fourth term shows that the instantaneous standard deviation of ris v,(t, t, 2,)l,-,. The second and third terms are more complicated, particularly when v isstochastic. The second term depends on the history of v because it involves v(t, t,Q,) whent<t. The third term depends on the history of both v and dz,
The second and third terms terms are liable to cause the process for r to be non-Markov. Thedrift of r between time t and ¢ + 6t is liable to depend not only on the value of r at time t, but alsoon the history of r prior to time t. This means that, when we attempt to construct a tree for r, it isnonrecombining as shown in Figure 24.2. An up movement followed by a down movement doesnot lead to the same node as a down movement followed by an up movement.
This highlights the key problem in implementing a general HJM model. We have to use MonteCarlo simulation. Trees create difficulties. When we construct a tree representing term structuremovements, it is usually nonrecombining. Assuming the model has one factor and the tree isbinomial as in Figure 24.2, there are 2” nodes after n time steps. If the model has two factors, thetree must be constructed in three dimensions and there are then 4” nodes after n time steps. Forn = 30, the number of terminal nodes in a one-factor model is therefore about 10”; in a two-factormodel it is about 10/8
Extension to Several Factors
The HJM result can be extended to the situation where there are several independent factors.Suppose that
AF (t, T) = m(t, T, &)dt + Y(t, T, 2.) dzk
A similar analysis to that just given (see Problem 24.4) shows thatTm(t, T, 2) = D> (t, T, 2, s(t, t, Q,)dt (24.10)k
t
24.3 THE LIBOR MARKET MODEL
 
 
One drawback of the HJM model is that it is expressed in terms of instantaneous forward rates andthese are not directly observable in the market. Another drawback is that it is difficult to calibrate themodel to prices of actively traded instruments. This has led Brace, Gatarek, and Musiela (BGM),Jamshidian, and Miltersen, Sandmann, and Sondermann to propose an alternative.° It is known asthe LIBOR market model or the BGM model. It is expressed in terms of the forward rates that tradersare used to working with.
The Model
Define to = 0 and let t,, t),... be the reset times for caps that trade in the market today. In theUnited States the most popular caps have quarterly resets, so that it is approximately true
6 See A. Brace, D. Gatarek, and M. Musiela, “The Market Model of Interest Rate Dynamics,” MathematicalFinance, 7, no. 2 (1997), 127-55; F. Jamshidian, “LIBOR and Swap Market Models and Measures,” Finance andStochastics, 1 (1997), 293-330; K. Miltersen, K. Sandmann, and D. Sondermann, “Closed Form Solutions for TermStructure Derivatives with LogNormal Interest Rate,” Journal of Finance, 52, no. 1 (March 1997), 409-30.578 CHAPTER 24
that t; = 0.25, t) = 0.5, t; = 0.75, and so on. Define 6; = th, — ty, and:
F(t): Forward rate between times 4, and t,,, a$ seen at time t, expressed with a compoundingperiod of 3,
m(t): Index for the next reset date at time t; this means that m/(t) is the smallest integer suchthat ¢< tin(t)
¢,{t): Volatility of F(t) at time tv(t): Volatility of the zero-coupon bond price, P(t, t,), at time t
Initially, we will assume that there is only one factor. As shown in Section 21.4, in a world that isforward risk neutral with respect to P(t, t,,,), F,(f) is a martingale and follows the process
dF, (t) = (F(t) dz (24.11)where dz is a Wiener process.
In practice, it often most convenient to value interest rate derivatives by working in a world thatis always forward risk neutral with respect to a bond maturing at the next reset date. We refer tothis as a rolling forward risk-neutral world.” In this world we can discount from time tea, to time tusing the zero rate observed at time t, for a maturity %,;. We do not have to worry about whathappens to interest rates between times t, and f,41.
The rolling forward risk-neutral world is a world that is at time t forward risk neutral withrespect to the bond price, P(t, ty). Equation (24.11) gives the process followed by F;,(t) in a worldthat is forward risk neutral with respect to P(t, t;4.;). From Section 21.7, it follows that the processfollowed by F;,(t) in the rolling forward risk-neutral world is
AAO = & Omg — Peri OVO dt + EOF. dz (24.12)The relationship between forward rates and bond prices is
P(t, ti) 14+3,Fi(0)Be yah + bi FiltP(t, ti41)orIn P(t, t;) — In PC, 444) = laf] + FO)
Using It6’s lemma we can calculate the process followed the left-hand side and the right-hand sideof this equation. Equating the coefficients of dz, we obtain
5 FOG)
1+5 FO Cn)
0,(t) — vj4i(t) =
so that from equation (24.12) the process followed by F;,(t) in the rolling forward risk-neutralworld is
dF) SG FHOGMEM
- 24.14FQ) Sty E+ 8 FO dt + y(t) dz (24.14)
7 In the terminology of Section 21.4, this world corresponds to-using a “rolling CD” as the numeraire. A rolling CDis one where we start with $1, buy a bond maturing at time ¢,, reinvest the proceeds at time ¢, in a bond maturing attime ¢, reinvest the proceeds at time tz in a bond maturing at time £3, and so on. Strictly speaking, the interest ratetrees we constructed in Chapter 23 are in a rolling forward risk-neutral world rather than the traditional risk-neutralworld. The numeraire is a CD rolled over at the end of each lime step.Interest Rate Derivatives: More Advanced Models 579
The HJM result in equation (24.6) is the limiting case of this as the 4; tend to zero. (SeeProblem 24.9).
Forward Rate Volatilities
We now simplify the modc! by assuming that “4h ra Tan on of the anberwe HOW Simply wie MoOGci CY a55UITINE wat ¢ptiy is a function only or the nhumoer
accrual periods between the next reset date and time t,. Define A; as the value of ¢,(t) when thereare i such accrual periods. This means that ¢,(t) = Ay_9 iS a step function.
The A, can (at least in theory) be estimated from the volatilities used to value caplets in Black’smodel (i.e., from the spot volatilities in Figure 22.3). Suppose that o is the Black volatility for thecaplet that corresponds to the period between times ¢, and t,,;. Equating variances we must have
f a hoalei Wide
koR, =D AZ_i8}-1 (24.15)
i=]This equation can be used to obtain the A’s iteratively.
Example 24.1 Assume that the 6; are all equal and the Black caplet spot volatilities for the firstthree caplets are 24%, 22%, and 20%. This means that Ap = 24%. Because
Ag+ A} =2 x 0.227A; is 19.80%. Also, becauseAO + AT + AR =3 x 0.20"A; is 15.23%.Example 24.2 Consider the data in Table 24.1 on the caplet volatilities. These exhibit the hump
discussed in Section 22.3. The A’s are shown in the second row. Notice that the hump in the A’s ismore pronounced than the hump in the o’s.
implementation of the Model
The LIBOR market model can be implemented using Monte Carlo simulation. Expressed in termsof the A,’s equation (24.14) is
dF, (1) EL BFAD Ak—m(0)PEMD Sy RDA mom yb a ey dz 24.16Fu) 24, IRR mo (24.16)
Table 24.1 Volatility data; accrual period = 1 year
Year, kI 2 3 4 5 6 7 8 9 10or (%) 15.50 18.25 17.91 17.74 17.27 16.79 16.30 = 16.01 15.76 15.54
Agi (%) 15.50 20.64 17.21 17.22 15.25 14.15 12.98 13.81 13.60 13.40
5 In practice the A’s are determined using a least-squares calibration that we will discuss later.580 CHAPTER 24
or
 
~ SF (DA me Army _ Apn)”2
din F(t) = ( EEO
i=m(t)
) dt + Micmy 4 (24.17)
If, as an approximation, we assume in the calculation of the drift of In F(t) that F(t) = Fi(t,)for t; <t < t,,,, then
 
k 8: F(t) Ay jp An) ARFi(tias) = F(tyexo| ( > ERG) - S ‘a + Asie | (24.18)i=j+l ie;
where ¢ is a random sample from a normal distribution with mean equal to zero~and standarddeviation equal to one.
Extension to Several Factors
The LIBOR market model can be extended to incorporate several independent factors. Supposethat there are p factors and ¢,, is the component of the volatility of F,(t) attributable to the gthfactor. Equation (24.14) becomes (see Problem 24.13)
dF,(t) _ k. 8:F(t) we Eig DSeg(t)
P~ dt t+ ) bight dz 24.19FQ) iEaty 14640 X iglt) deg (24.19)
Define 4;,, as the gth component of the volatility when there are i accrual periods between the nextreset date and the maturity of the forward contract. Equation (24.18) then becomes
 
8 F(t) OP Aim jtg he jng py Mj P_ EME gai hi-j-l.qAk-j-L¢ g=l *k-j-lqFdtia) = rttyexo| (> 145 F) 2 Ja + DA ita6av |LNSIFL TOPRIM iF / q=l 4
(24.20)
where the €, are random samples from a normal distribution with mean equal to zero and standarddeviation equal to one.
The approximation that the drift of a forward rate remains constant within each accrual periodallows us to jump from one reset date to the next in the simulation. This is convenient because, asalready mentioned, the rolling forward risk-neutral world allows us to discount from one resetdate to the next. Suppose that we wish to simulate a zero curve for N accrual periods. On eachtrial we start with the forward rates at time zero. These are Fo(0), F\(0),..., Fy_)(0) calculatedfrom the initial zero curve; we then use equation (24.20) to calculate Fy(t;), Fo(t)),..., Py—(t))swe then use equation (24.20) again to calculate Fy(t)), F3(t),..., Fy—(t2); and so on untilFy_\(ty_1) is obtained. Note that as we move through time the zero curve gets shorter andshorter. For example, suppose each accrual period is three months and N = 40. We start with aten-year zero curve. At the six-year point (at time t,) the simulation gives us information on afour-year zero curve.
Hull and White tested the drift approximation by valuing caplets using equation (24.20) andcomparing the prices to those given by Black’s model.” The value of F;,(t,) is the realized rate for
° See J. C. Hull and A. White, “Forward Rate Volatilities, Swap Rate Volatilities, and the Implementation of theLIBOR Market Model,” Journal of Fixed Income, 10, no. 2 (September 2000), 46-62.interest Rate Derivatives: More Advanced Models 581
the time period between 1, and t,,; and enables the caplet payoff at time t,,, to be calculated. Thispayoff is discount