ed back to time zero, one accrual period at a time. The caplet value is the averageof the discounted payoffs, Hull and White found that the values obtained in this way are notsignificantly different from those given by Black’s model. This is true even when the accrual periodsare one year in length and a very large number of trials are used.'° This suggests that the driftassumption is innocuous in most situations.
Ratchet Caps, Sticky Caps, and Flexi Caps
The LIBOR market model can be used to value some types of nonstandard caps. Consider ratchetcaps and sticky caps. These incorporate rules for determining how the cap rate for each caplet isset. In a ratchet cap, it equals the LIBOR rate at the previous reset date plus a spread. In a stickycap, it equals the previous capped rate plus a spread. Suppose that the cap rate at time 1; is K;, theLIBOR rate at time t; is R;, and the spread is s. In a ratchet cap, Kj4, = 8; +s. Ina sticky cap,Ky, = min(R;, Kj) +5.
Tables 24.2 and 24.3 provide valuations of a ratchet cap and sticky cap using the LIBOR marketmodel with one, two, and three factors. The principal is $100. The term structure is assumed to beflat at 5% per annum and the caplet volatilities are as in Table 24.1. The interest rate is resetannually. The spread is 25 basis points. Tables 24.4 and 24.5 show how the volatility was split intocomponents when two- and three-factor models were used. The results are based on 100,000Monte Carlo simulations incorporating the antithetic variable technique described in Section 18.7.The standard error of each price is about 0.001.
A third type of nonstandard cap is a flexi cap. This is like a regular cap except that there is a limiton the total number of caplets that can be exercised. Consider an annual-pay flexi cap when theprincipal is $100, the term structure is flat at 5%, and the cap volatilities are as in Tables 24.1 (p. 579),24.4 (p. 582), and 24.5 (p. 583). Suppose that all in-the-money caplets are exercised up to a maximumof five. With one, two, and three factors, the LIBOR market model gives the price of the instrumentas 3.43, 3.58, and 3.61, respectively. (See Problem 24.17 for other types of flexi caps.)
Table 24.2 Valuation of ratchet caplets
 
 
Caplet start time One Two Three(years) factor factors factors1 0.196 0.194 0.195
2 0.207 0.207 0.209
3 0.201 0.205 0.210
4 0.194 0.198 0.205
5 0.187 0.193 0.201
6 0.180 0.189 0.193
7 0.172 0.180 0.188
8 0.167 0.174 0.182
9 0.160 0.168 0.175
10 0.153 0.162 0.169
'© An exception is when the cap volatilities are very high.582 CHAPTER 24
Table 24.3 Valuation of sticky caplets
Caplet start time One Two Three(years) factor JSactors factors1 0.196 0.194 0.195
2 0.336 9.334 0.336
3 0.412 0.413 0.418
4 0.458 0.462 0.472
5 0.484 0.492 0.506
6 0.498 0.512 0.524
7 0.502 0.520 0.533
8 0.501 0.523 0.537
9 0.497 0.523 0.537
10 0.488 0.519 0.534
The pricing of a plain vanilla cap depends only on the total volatility and is independent of thenumber of factors. This is because the price of a plain vanilla caplet depends the behavior of only oneforward rate. The prices of caplets in the nonstandard instruments we have looked at are different inthat they depend on the joint probability distribution of several different forward rates. As a resultthey do depend on the number of factors.
Valuing European Swap Options
As shown by Hull and White, there is an analytic approximation for valuing European swap optionsin the LIBOR market model.!! Let 7% be the maturity of the swap option and assume that thepayment dates for the swap are 7), 7),..., Ty. Define t; = 7;,., — 7;. From equation (21.23), theswap rate at time t is given bys(t) = PU, To) P(t, Ty)
i=0 7 P(t, Tit)It is also true that
 
P(t, 1) 7 1P(t, Ty) po +E
Table 24.4 Volatility components in two-factor model
Year, k1 2 3 4 5 6 7 8 9 10Agia (%) 1410 19.52. 16.78 17.11 15.25 1406 12.65 13.06 12.36 11.63Agi (%) -6.45 -—6.70 —3.84 ~1.96 0.00 1.61 2.89 4.48 5.65 6.65
 
Total Vol.(%) 15.50 20.64 17.21 17.22 15.25 14.15 12.98 13.81 13.60 13.40
" 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 583
Table 24.5 Volatility components in three-factor model
Year, kT 2 3 4 5 6 7 8 9 10Ant. (%) 13.65 19.28 16.72 16.98 1485 13.95 12.61 12.90 11.97 10.97Ag-t2 (%) —6.62 -—7.02 -4.06 —2.06 0.00 1.69 3.06 4.70 5.81 6.66Ae-t3 (%) 3.19 2.25 0.00 ~-1.98 -347 -1.63 0.00 1.51 280 3.84
 
for 1 <i < N, where G,(t) is the forward rate at time t for the period between 7; and T,,,. These twoequations together define a relationship between s(t) and the G,(t)’s. Applying Ité’s lemma (seeProblem 24.14), the variance of the swap rate, s(t), is given by
3 bial GOO
vo= yeNe 14nG()
where
foll+7G@) Disp wT + 76,0)TIN-lh4L1j=o t+
 
Ve(t) = tr Gao. SX 2h ce
VEGA 4% Lajsd SEL A faie ht 1 GVM
and A, ,(f) is the qth component of the volatility of G,(t). We approximate V(t) hy settingG,(t) = G,(0) for all j and t, The swap volatility that is substituted into the standard marketmodel for valuing a swaption is then
1 (%z| Vit) dt
0 Ji=0
1 (% we N-1 GBe gE)fel x k=0 Ee) at (24.21)
t=0
or
 
In the situation where the length of the accrual period for swap underlying the swaption is thesamé~as.the length of the accrual period for a cap, ;4(t) is the gth component of volatility of acap forward rate when the time to maturity is 7, —t. This can be looked up in Table 24.5,
The accrual periods for the swaps underlying broker quotes for European swap options do notalways match the accrual periods for the caps and floors underlying broker quotes. For example, inthe United States the benchmark caps and floors have quarterly resets while the swaps underlyingthe benchmark European swap options have semiannual resets. Fortunately, the valuation resultfor European swap options can be extended to the situation where the each swap accrual periodincludes M subperiods that could be accrual periods in a typical cap. Define t;,, as the length ofthe mth subperiod in the jth accrual period, so that
MGs » Timm=1584 CHAPTER 24
Define G,,,,(t) as the forward rate observed at time r for the z;,, accrual period. Because
M14+76,0) = [lk +4.G;,(0]
m=!
“an mo the analysis lead;
s (24.21) so that the volatility of s() ained inwe can modify the analysis leading to equation (24.21) so that the volatility of s(d) is obtained in
Aiterms of the volatilities of the G;,,,(t) rather than the volatilities of th G0). The swap volatilityto be substituted into the standard market model for valuing a swap » cation proves to be (see
Problem 24:15)1 (% vp N=1 SOM Te Bem g(t)Ge,m(OV(O)Y_ wher Bk gh IT 4,22ip | doe ( on Loam) + Th Gin (O) ) “ C4)
t=0
 
Here Bjm,g(t) is the qth component of the volatility of G jn(t). It is the gth componentof thevolatility of a cap forward rate when the time to maturity is from ¢ to the beginning of the mthsubperiod in the (7;, T;,1) swap accrual period.
The expressions in : equations (24.21) and (24.22) for the swap volatility do involve the approx-imations thal Gt.) = G,(0) and G,,,(t)= G;,,(0). Hull and White compared the prices ofEuropean swap options calculated using equations (24.21) and (24.22) with the prices calculatedfrom a Monte Carlo simulation and found the two to be very close. Once the LIBOR marketmodel has been calibrated, therefore, equations (24.21) and (24.22) provide a quick way of valuingEuropean swap options. Analysts can determine whether European swap options are over- orunderpriced relative to caps. As we will see shortly, they can also use the results to calibrate themodel to the market prices of swap options.
Calibrating the Model
To calibrate the LIBOR market model we must determine the A; and how they are split into 4; ,.The first step is usually to use a principal components analysis such as that in Section 16.9 todetermine the way the A’s are split into A’s. The principal components model is
M6Fj = Yai gXyq=l
where M is the total number of factors, 3F; is the change in the forward rate for a forward contractmaturing in j accrual periods, w,, is the factor loading for the jth forward rate and qth factor, x, isthe factor score for the gth factor, and
ra a= 1 ifg=4= FUN II 0 i
if gq # a
Define s, as the standard deviation of the qth factor score. If the number of factors used in theLIBOR market model, p, is equal to the total number of factors, M, it is Correct to set
Aig = %na5qInterest Rate Derivatives: More Advanced Models 585
for 1 < j.q < M. When p < M, the 4; must be scaled so that
_ P42Ay= ve Mig
Aus.a;hig = iSq% jg (24.23)
Vege 22h
Equation (24,15) provides one way to determine the A’s so that they are consistent with capletprices. In practice it is not usually used because it often leads to wild swings in the A’s,’? Also,although the LIBOR market model is designed to be consistent with the prices of caplets, analystssometimes like to calibrate it to European swaptions. The most commonly used calibrationprocedure for the LIBOR market model is similar to that described for one-factor models inSection 23.14, Suppose that U; is the market price of the ith calibrating instrument and V; is themodel price. We choose the A’s to minimize
YU. - VP +P
i
This involves setting
where P is a penalty function chosen to ensure that the A’s are “well behaved’. Similarly toSection 23.14, P has the form
P= » wy lAjer — A + » wr i(Aigs + Aza — 2A,
When some calibrating instruments are European swaptions the formulas in equations (24.21)and (24.22) make the minimization feasible using the Levenberg—Marquardt procedure.Equation (24.23) is used to determine the A’s from the A’s.
Volatility Skews
Brokers now provide quotes on caps that are not at the money as well as on those that are at themoney. In some markets a volatility skew is observed, that is, the quoted (Black) volatility for a capor a floor is a declining function of the strike price. This can be handled using the CEV model (seeSection 20.1 for the application of the CEV model to equities). The model is
P.dF) = ++ Db. DFO" dz, (24.24)g=l
where a (0 < a < 1) is a constant . It turns out that this model can be handled in a very similar wayto the lognormal model. Caps and floors can be valued analytically using the cumulativenoncentral x distribution. There are similar analytic approximations to those given above forthe prices of European swap options.'°
'2 Sometimes there is no set of A’s consistent with a set of cap quotes.
3 For details, see L. Andersen and J. Andreasen, “Volatility Skews and Extensions of the LIBOR Market Model,”Applied Mathematical Finance, 7, no. 1 (2000), 1-32; J. C. Hull and A. White, “Forward Rate Volatilities, Swap RateVolatilities, and the Implementation of the LIBOR Market Model,” Journal of Fixed Income, \0, no. 2 (September2000), 46-62.586 CHAPTER 24
Bermudan Swap Options
A popular interest rate derivative is a Bermudan swap option. This is a swap option that can beexercised on some or all of the payment dates of the underlying swap. Bermudan swap options aredifficult to value using the LIBOR market model because the LIBOR market model relies onMonte Carlo simulation and it is difficult to evaluate early exercise decisions when Monte Carlosimulation is used. Fortunately the procedures described in Section 20.9 can be used. Longstatfand Schwartz apply the least-squares approach when there are a large number of factors. The valueof not exercising on a particular payment date is assumed to be a polynomial function of theyalues of the factors.'* Andersen shows that the optimal early exercise boundary approach can beused. He experiments with a number of ways of parametrizing the early exercise bourtdary andfinds that good results are obtained when the early exercise decision is assumed to depend only onthe intrinsic value of the option.'° Most traders value Bermudan options using one of the one-factor no-arbitrage models discussed in Chapter 23. The accuracy of one-factor models for pricingBermudan swap options has become a controversial issue.’®
24.4 MORTGAGE-BACKED SECURITIES
 
 
One application of the models presented in this chapter is to the mortgage-backed security (MBS)market in the United States. A mortgage-backed security is created when a financial institutiondecides to sell part of its residential mortgage portfolio to investors. The mortgages sold are putinto a pool and investors acquire a stake in the pool by buying units. The units are known asmortgage-backed securities. A secondary market is usually created for the units so that investorscan sell them to other investors as desired. An investor who owns units representing X percent of acertain pool is entitled to X percent of the principal and interest cash flows received from themortgages in the pool.
The mortgages in a pool are generally guaranteed by a government-related agency, such as theGovernment National Mortgage Association (GNMA) or the Federal National Mortgage Associ-ation (FNMA), so that investors are protected against defaults. This makes an MBS sound like aregular fixed-income security issued by the government. In fact, there is a critical differencebetween an MBS and a regular fixed-income investment. This difference is that the mortgages inan MBS poo! have prepayment privileges. These prepayment privileges can be quite valuable to thehouseholder. In the United States, mortgages typically last for 25 years and can be prepaid at anytime. This means that the householder has a 25-year American-style option to put the mortgageback to the lender at its face value.
In practice prepayments on mortgages occur for a variety of reasons. Sometimes interest rateshave fallen and the owner of the house decides to refinance at a lower rate of interest. On other
14 Sce F. A. Longstaff and E. S. Schwartz, “Valuing American Options by Simulation: A Simple Least SquaresApproach,” Review of Financial Studies, 14, no. 1 (2001), 113-47.
> L. Andersen, “A simple Approach to the Pricing of Bermudan Swaptions in the Multifactor LIBOR MarketModel.” Journal of Computational Finance, 3, no. 2 (Wintér 2000), 1-32.
© For opposing viewpoints, see ““Hactor Dependence of Bermudan Swaptions: Fact or Fiction,” by L. Andersen andJ. Andreasen, and “Throwing Away a Billion Dollars: The Cost of Suboptimal Exercise Strategies in the SwaptionMarket,” by F. A. Longstaff, P. Santa-Clara, and E. 8. Schwartz. Both are in Journal of Financial Economics, 62,no. 1 (October 2001).Interest Rate Derivatives: More Advanced Models 587
 
occasions, a mortgage is prepaid simply because the house is being sold. A critical element invaluing an MBS is the determination of what is known as the prepayment function. This is afunction describing expected prepayments on the underlying pool of mortgages at a time ¢ in termsof the yield curve at time t and other relevant variables.
A prepayment function is very unreliable as a predictor of actual prepayment experience for an
individual mortease. When many similar morteage loans are combined in the same pool, there is a
individual Mortgage. When many simuar mortgage ioans are Comoined in the same pooi, there
“law of large numbers” effect at work and prepayments can be predicted more accurately from ananalysis of historical data. As mentioned, prepayments are not always motivated by pure interestrate considerations. Nevertheless, there is a tendency for prepayments to be more likely wheninterest rates are low than when they are high. This means that investors require a higher rate ofinterest on an MBS than on other fixed-income securities to compensate for the prepaymentoptions they have written.
Collateralized Mortgage Obligations
The MBSs we have described so far are sometimes referred to as pass-throughs. All investors receivethe same return and bear the same prepayment risk. Not all mortgage-backed securities work inthis way. In a collateralized mortgage obligation (CMO) the investors are divided into a number ofclasses and rules are developed for determining how principal repayments are channeled todifferent classes.
As an example of a CMO, consider an MBS where investors are divided into three classes:class A, class B, and class C. All the principal repayments (both those that are scheduled and thosethat are prepayments) are channeled to class A investors until investors in this class have beencompletely paid off. Principal repayments are then channeled to class B investors until theseinvestors have been completely paid off. Finally, principal repayments are channeled to class Cinvestors. In this situation, class A investors bear the most prepayment risk. The class A securitiescan be expected to last for a shorter time than the class B securities, and these, in turn, can beexpected to last less long than the class C securities.
The objective of this type of structure is to create classes of securities that are more attractive toinstitutional investors than those created by the simpler pass-through MBS. The prepayment risksassumed by the different classes depend on the par value in each class. For example, class C bearsvery little prepayment risk if the par values in classes A, B, and C are 400, 300, and 100,respectively. Class C bears rather more prepayment risk in the situation where the par values inthe classes are 100, 200, and 500.
Os and POs
In what is known as a stripped MBS, principal paymenis are separated from interest payments. Allprincipal payments are channeled to one class of security, known as a principal only (PO). Allinterest payments are channeled to another class of security khown as an interest only (IO). BothTOs and POs are risky investments. As prepayment rates incyéase, a PO becomes more valuable andan IO becomes less valuable. As prepayment rates decrease, the reverse happens. In a PO, a fixedamount of principal is returned to the investor, but the timing is uncertain. A high rate ofprepayments on the underlying pool leads to the principal being received early (which is, ofcourse, good news for the holder of the P 0}. A low rate of prepayments on the underlying pooldelays the return of the principal and reduces the yield provided by the PO. In the case of an IO,the total of the cash flows received by the investor is uncertain. The higher the rate of prepayments,the lower the total cash flows received by the investor, and vice versa.588 CHAPTER 24
Valuing Mortgage-Backed Securities
Mortgage-backed securities are usually valued using Monte Carlo simulation. Either the HJM orLIBOR market models can be used to simulate the behavior of interest rates month by monththroughout the life of an MBS. Consider what happens on one simulation trial. Each monthexpected prepayments are calculated from the current yield curve and the history of yield curvemovements. These prepayments determine the expected cash flows to the holder of the MBS andthe cash flows are discounted to time zero to obtain a sample value for the MBS. An estimate ofthe value of the MBS is the average of the sample vakies over many simulation trials.
Option-Adjusted Spread
A critical input to any term structure model is the initial zero-coupon yield curve. This is a curve.generated in the way described in Chapter 5, providing the relationship between yield and maturityfor zero-coupon bonds that have no embedded options.
In addition to calculating theoretical prices for mortgage-backed securities and other bonds withembedded options, traders also like to compute what is known as the option-adjusted spread (OAS).This is a measure of the spread over the yields on government Treasury bonds provided by theinstrument when all options have been taken into account.
To calculate an OAS for an instrument, it is first priced using the zero-coupon governmentTreasury curve as the input to the pricing model. The price of the instrument given by the model iscompared with the price in the market. A series of iterations is then used to determine the parallelshift to the input Treasury curve that causes the model price to be equal to the market price. Thisparallel shift is the OAS.
To illustrate the nature of the calculations, suppose that the market price is $102.00 and that theprice calculated using the Treasury curve is $103.27. As a first trial we might choose to try a60-basis-point parallel shift to the Treasury zero curve. Suppose that this gives a price of $101.20for the instrument. This is less than the market price of $102.00 and means that a parallel shiftsomewhere between 0 and 60 basis points will lead to the model price being equal to the marketprice. We could use linear interpolation to calculate
103.27 — 102.0000 x 93.97 - 10120 08!
or 36.81 basis points as the next trial shift. Suppose that this gives a price of $101.95. This indicatesthat the OAS is slightly less than 36.81 basis points. Linear interpolation suggests that the next trial
shift be
103.27 — 102.0036.81 x T5555 — 191-65 = 35-4
or 35.41 basis points, and so on.
SUMMARY
 
 
We have covered a number of advanced models for valuing interest rate derivatives, In the two-factor Hull-White model, the short rate reverts to a level dependent on another stochastic variable.This stochastic variable, in turn, reverts to zero. Unlike the models considered in Chapter 23, thismodel can match the volatility structure observed today in the market, not just today but at allfuture times.Interest Rate Derivatives: More Advanced Models 589
The HJM and LIBOR market models provide approaches that give the user complete freedom inchoosing the volatility term structure. The LIBOR market model has two key advantages over theHJM model. First, it is developed in terms of the forward rates that determine the pricing of caps,rather than in terms of instantaneous forward rates. Second, it is relatively easy to calibrate to theprice of caps or European swap options. The HJM and LIBOR market models both have theserious disadvantage that they cannot be represented as recombining trees. In practice, this meansthat they must be implemented using Monte Carlo simulation.
The mortgage-backed security market in the United States has given birth to many exoticinterest rate derivatives: CMOs, IOs, POs, and so on. These instruments provide cash flows to theholder that depend on the prepayments on a pool of mortgages. These prepayments depend on,among other things, the level of interest rates. Because they are heavily path dependent, mortgage-backed securities usually have to be valued using Monte Carlo simulation. These are thereforeideal candidates for applications of the HIM model and LIBOR market models.
SUGGESTIONS FOR FURTHER READING
 
Amin, K., and A. Morton, “Implied Volatility Functions in Arbitrage-Free Term Structure Models,”Journal of Financial Economics, 35 (1994), 141-80.
Andersen, L., “A Simple Approach to the Pricing of Bermudan Swaption in the Multi-Factor LIBORMarket Model,” Applied Mathematical Finance, 7, no. 1 (2000), 1-32.
Brace, A., D. Gatarek, and M. Musiela, “The Market Model of Interest Rate Dynamics,”Mathematical Finance, 7, no. 2 (1997), 127-55.
Buhler, W., M. Ulrig-Homberg, U. Walter, and T. Weber, “An Empirical Comparison of Forward andSpot-Rate Models for Valuing Interest Rate Options,” Journal of Finance, 54, no. 1 (February 1999),269-305.
Carverhill, A., “When Is the Short Rate Markovian,” Mathematical Finance, 4 (1994), 305-12.
Cheyette, O., “Term Structure Dynamics and Mortgage Valuation,” Journal of Fixed Income, March1992, 28-41.
Duffie, D., and R. Kan, “A Yield-Factor Model of Interest Rates,” Mathematical Finance, 6, no. 4(1996), 379-406.
Heath, D., R. Jarrow, and A. Morton, “Bond Pricing and the Term Structure of Interest Rates:A Discrete Time Approximation,” Journal of Financial and Quantitative Analysis, 25, no. 4 (December1990), 419-40,
Heath, D., R. Jarrow, and A. Morton, “Bond Pricing and the Term Structure of the Interest Rates:A New Methodology,” Econometrica, 60, no. 1 (1992), 77-105.
Heath, D., R. Jarrow, A. Morton, and M. Spindel, “Easier Done Than Said,” RISK, May 1993,pp. 77-80.
Hull, J., and A. White, “Bond Option Pricing Based on a Model for the Evolution of Bond Prices,”
Advances in Futures and ae 6 (1993), 1-13.Hull, I., and A. White, “Numerical Procedures for Implementing Term Structure Models Tl: Two-
Factor Models,” Journal of Derivatives, 2, no. 2 (Winter 1994), 37-48.
Hull, J.. and A. White, “Forward Rate Volatilities, Swap Rate Volatilities, and the Implementation ofthe LIBOR Market Model,” Journal of Fixed Income, 10, no. 2 (September 2000), 46-62.590 CHAPTER 24
Inui, K., and M. Kijima, ‘““A Markovian Framework in Multifactor Heath, Jarrow, and MortonModels,” Journal of Financial and Quantitative Analysis, 33, no. 3 (September 1998), 423-40.
Jamshidian, F., ‘‘LIBOR and Swap Market Models and Measures,” Finance and Stochastics, 1 (1977),293-330.
Jarrow, R. A., Modeling Fixed Income Securities and Interest Rate Options, McGraw-Hill, New York,1995,
Jarrow, R. A., and S. M. Turnbull, “Delta, Gamma, and Bucket Hedging of Interest Rate Derivatives,”Applied Mathematical Finance, \ (1994), 21-48.
Jeffrey,