by F in the traditional risk-neutral world?d. What is the process followed by f in a world that is forward risk neutral with respect to abond maturing at time 7*, where T* 4 T? Assume that o% is the volatility of this bond.
Consider an instrument that will pay off S dollars in two years where S is the value of the Nikkeiindex. The index is currently 20,000. The dollar-yen exchange rate (yen per dollar) is 100. Thecorrelation between the exchange rate and the index is 0.3 and the dividend yield on the index is1% per annum. The volatility of the Nikkei index is 20% and the volatility of the yen-dollarexchange rate is 12%. The interest rates (assumed constant) in the U.S. and Japan are 4% and2%, respectively.
a. What is the value of the instrument?
b. Suppose that the exchange rate at some point during the life of the instrument is Q and thelevel of the index is S. Show that a U.S. investor can create a portfolio that changes in valueby approximately 5S dollars when the index changes in value by 5S yen by investingS dollars in the Nikkei and shorting SQ yen.
c. Confirm that this is correct by supposing that the index changes from 20,000 to 20,050 andthe exchange rate changes from 100 to 99.7.
d. How would you delta hedge the instrument under consideration?504 CHAPTER 21
APPENDIX 21A
Generalizations of It’s Lemma
[td’s lemma, as presented in Appendix 11A, provides the process followed by a function of a singlestochastic variable. Here we present a generalized version of It6’s lemma for the process followedby a function of several stochastic variables.
Suppose that a function f depends on the n variables x), x2,..., x, and time t. Suppose furtherthat x, follows an Ité process with instantaneous drift @; and instantaneous variance b; (1 <i <n),that is,
dx; = a; dt + b; dz; (21A.1)
where dz; (1 < i <n) is a Wiener process. Each a; and b; may be any function of all the x,’s and t.A Taylor series ‘expansion of f gives
a 4 — $t bx A.2
sf= Ya x +2 payee a5 - 8x; 6x mies i Bat bo (21A.2)
Equation (21A.1) can be discretized as5x; = a; bt + bje;V St
where ¢; is a random sample from a standardized normal distribution. The correlation p;; betweendz, and dz, is defined as the correlation between ¢; and €;. In Appendix 11A it was argued that
lim 8x7 = b? dt310Similarly,
jim dx; 5x; = b,b jp;; at
As 5t > 0, the first three terms in the expansion of 8 f in equation (21A.2) are of order ét. All otherterms are of higher order. Hence,
df= ve - a + of OF ays Dis 3 E bboudti=l
This is the generalized version of 1td’s lemma. Substituting for dx; from equation (21A.1) givesa adf— (Visas af npr ee < “bb ae yee dz, (Q1A.3)1 Xi i=] j=l
For an alternative generalization of It’s lemma, suppose that f depends on a single variable x andthat the process for x involves more than one Wiener process:
iildx =adt+ >> bj dz;
i=lMartingales and Measures 505
In this case,
of af Pf rfbx +— 8t 456 dxdtae et + 2 9x2 tbo xO +:
bx =adt+ Sobai=t
éf=
nm mlim 8x7 = SOS bb spiti=l j=lwhere as before ;; is the correlation between dz; and dz;. This leads to“Sf, mdf= Fast LSS oye, ary az; (214.4)
i=| j=l
Finally, consider the more general case where f depends on variables x; (1 <i <n) and
mdx, =a, dt +) by, dzk=1
A similar analysis shows that
n vf momdf= (Ya 1D Caan D> bubsen) at + Yo Bey hat (21.5)
Nisi f k=i i=i506 CHAPTER 21
 
APPENDIX 21B
Expected Excess Return When There Are MultipleSources of Uncertainty
This appendix proves the result in equation (21.13). Suppose there are n stochastic variablesfollowing Wiener processes. Consider n+ 1! traded securities whose prices depend on some orall of the n stochastic variables. Define f; as the price of the jth (I < j <n+1) security. Weassume that no dividends or other income is paid by the 2 + 1 traded securities.® It follows fromAppendix 21A that the securities follow processes of the form
df; =m jfidt +) of; 42) (21B.1)f=1
Because there are n+1 traded securities and n Wiener processes, it is possible to form aninstantaneously riskless portfolio, TI, using the securities. Define k; as the amount of the
jth security in the portfolio, so thatntlN=) kif; (21B.2)
j=l
The k; must be chosen so that the stochastic components of the returns from the securities areeliminated. From equation (21B.1) this means that
ntl
in
i
&n
ll°otStwad&
o.i
for 1 <i <n. The return from the portfolio is then given by
ntlall = ) kyu; fiatj=lThe cost of setting up the portfolio isntl
Vaij=l
If there are no arbitrage opportunities, the portfolio must earn the risk-free interest rate, so that
atl ntlkit =r kf; (21B.4)j=l j=loratlSk Fu) —) =0 (218.5)j=l
* This is not restrictive. A non-dividend-paying security can always be obtained from a dividend-paying security byreinvesting the dividends in the security.Martingales and Measures 507
Equations (21B.3) and (21B.5) can be regarded as n + 1 homogeneous linear equations in the k,’s.The &;’s are not all zero. From a well-known theorem in linear algebra, equations (21B.3) and(21B.5) can be consistent only if, for all j, -
flej — = ei f; (21B.6)i=lornujp—r = AG; (21B.7)i=l
for some A; (1 <i < a) that are dependent only on the state variables and time. Dropping the jsubscript, we see that, for any security f dependent on the n stochastic variables,
df =pfdt+ > 0, fdz,i=1where
nwo-T= Sie;f=I
This proves the result in equation (21.13).mae,
 
CHAPTER 22
INTEREST RATE DERIVATIVES:THE STANDARD MARKET MODELS
Interest rate derivatives are instruments whose payoffs are dependent in some way on the level ofinterest rates. In the 1980s and 1990s, the volume of trading in interest rate derivatives in both theover-the-counter and exchange-traded markets increased very quickly. Many new products weredeveloped to meet particular needs of end-users. A key challenge for derivatives traders is to findgood, robust procedures for pricing and hedging these products.
Interest rate derivatives are more difficult to value than equity and foreign exchange derivatives.There are a number of reasons for this:
1, The behavior of an individual interest rate is more complicated than that of a stock price oran exchange rate.
2. For the valuation of many products, it is necessary to develop a model describing thebehavior of the entire zero-coupon yield curve.
3. The volatilities of different points on the yield curve are different.
4, Interest rates are used for discounting as well as for defining the payoff from thederivative.
Tn this chapter we look at the three most popular over-the-counter interest rate option products:bond options, interest rate caps/floors, and swap options. We explain the standard market modelsfor valuing these products and use material from Chapter 21 to show that the models are internallyconsistent. We also explain what are known as convexity and timing adjustments. These aresometimes necessary in the valuation of nonstandard interest rate derivatives.
22.1 BLACK’S MODEL
 
 
Since the Black-Scholes model was first published in 1973, it has become a very popular tool. Asexplained in Chapter 13, the model has been extended so that it can be used to value options onforeign exchange, options on indices, and options on futures contracts. Traders have become verycomfortable with both the lognormal assumption that underlies the model and the volatilitymeasure that describes uncertainty. It is not surprising that there have been attempts to extendthe model so that it covers interest rate derivatives.
In the following few sections we will discuss three of the most popular interest ratederivatives (bond options, interest rate caps, and swap options) and describe how the log-
508Interest Rate Derivatives: The Standard Market Models 509
normal assumption underlying the Black-Scholes model can be used to value these instruments.The model we will use is usually referred to as Black’s model because the formulas are similarto those in the model suggested by Fischer Black for valuing options on commodity futures(see Section 13.8).
We start by reviewing Black’s model and showing that it provides a flexible framework for
valuine a wide range of Euronean ontions
Vauing a ce fange of Huropean options.
Using Black’s Model to Price European Options
Consider a European call option on a variable whose value is V. Define:
T: Time to maturity of the optionF: Forward price of V for a contract with maturity TF,: Value of F at time zeroK: Strike price of the optionP(t, T): Price at time t of a zero-coupon bond paying $1 at time TV;: Value of V at time To: Volatility of F
Black’s model calculates the expected payoff from the option assuming:
1. Vy has a lognormal distribution with the standard deviation of In V; equal to o JT.2. The expected value of Vr is Fo.
It then discounts the expected payoff at the 7-year risk-free rate by multiplying by P(O, 7). Thepayoff from the option is max(V; — K, 0) at time 7. As shown in Appendix 12A, the lognormalassumption implies that the expected payoff is
E(V7)N(d,) — KN(d)
where E(V;) is the expected value of V; and
4, = MED /Kl4+0°T 2= oVT2a= EWA OT gaye
onova
Because we are assuming that E(V;) = Fp and discounting at the risk-free rate, the value of theoption isce = P(O, T)LFoN(d,) — KN(d2)} (22.1)whereaa In(Fo/K) + 0° T/2oVT
_ In(Fo/K) — oT /2 _ _= OF =d, o/T510 CHAPTER 22
Similarly the value, p, of the corresponding put option is given byp = PO, T)[KN(—d) — FyN(—a,)] (22.2)
Equations (22.1) and (22.2) are equivalent to equations (13.17) and (13.18) except that Fy is heredefined as the forward price rather than the futures price.
We can extend Black’s model to allow for the situation where the payoff is calculated from thevalue of the variable V at time 7, but the payoff is actually made at some later time 7* Theexpected payoff is discounted from time T* instead of time T, so that equations (22.1) and (22.2)become:
c= P(O, T*)[FoN(d,) — KN(d))] (22.3)p= P(0,T*)[KN(—d2) — FoN(—d))] (22.4)whereLe In(Fo/K) + 0°T/2oVTgy MINOT? ggovT
An important feature of Black’s model is that we do not have to assume geometric Brownianmotion for the evolution of either V or F. All that we require is that Vr be lognormal at time T.The parameter o is usually referred to as the volatility of F or the forward volatility of V. However,its only role is to define the standard deviation of In Vp by means of the relationship
The volatility parameter does not necessarily say anything about the standard deviation of In V attimes other than time 7.
Validity of Black’s Model
It is easy to see that Black’s model is appropriate when interest rates are assumed to be eitherconstant or deterministic. In this case, as explained in Chapter 3, the forward price of V is equal toits futures price and the model is the same as the model in Section 13.8.
When interest rates are stochastic, there are two aspects of the derivation of equation (22.1) thatare open to question.
1. Why do we set E(V;) equal to the forward price, Fy, of V? This is not the same as the futuresprice.
2, Why do we ignore the fact that interest rates are stochastic when discounting?
As we apply Black’s model to bond options, caps/floors, and swap options, we will use the resultsin Section 21.4 to show that there are no approximations in equations (22.1) and (22.2) wheninterest rates are stochastic. It is correct to set E(Vr) equal to the forward price provided that wealso discount at today’s 7-year maturity zero rate. Black’s model therefore has a soundertheoretical basis and wider applicability than is sometimes supposed.Interest Rate Derivatives: The Standard Market Models 511
22.2. BOND OPTIONS
 
A bond option is an option to buy or sell a particular bond by a certain date for a particular price.In addition to trading in the over-the-counter market, bond options are frequently embedded inbonds when they are issued to make them more attractive to either the issuer or potentialpurchasers.
Embedded Bond Options
One example of a bond with an embedded bond option is a callable bond. This is a bond thatcontains provisions allowing the issuing firm to buy back the bond at a predetermined price atcertain times in the future. The holder of such a bond has sold a call option to the issuer. The strikeprice or call price in the option is the predetermined price that must be paid by the issuer to theholder. Callable bonds cannot usually be called for the first few years of their life. (This is knownas the lock out period.) After that the call price is usually a decreasing function of time. Forexample, in a 10-year callable bond, there might be no call privileges for the first two years. Afterthat, the issuer might have the right to buy the bond back at a price of 110 in years 3 and 4 of itslife, at a price of 107.5 in years 5 and 6, at a price of 106 in years 7 and 8, and at a price of 103 inyears 9 and 10. The value of the call option is reflected in the quoted yields on bonds. Bonds withcall features generally offer higher yields than bonds with no call features.
Another type of bond with an embedded option is a puttable bond. This contains provisions thatallow the holder to demand early redemption at a predetermined price at certain times in thefoture. The holder of such a hond has purchased a put aption on the bond as well as the bonditself. Because the put option increases the value of the bond to the holder, bonds with put featuresprovide lower yields than bonds with no put features. A simple example of a puttable bond is a10-year bond where the holder has the right to be repaid at the end of five years. (This is sometimesreferred to as a retractable bond.)
Loan and deposit instruments also often contain embedded bond options. For example, a five-year fixed-rate deposit with a financial institution that can be redeemed without penalty at any timecontains an American put option on a bond. (The deposit instrument is a bond that the investorhas the right to put back to the financial institution at any time.) Prepayment privileges on loansand mortgages are similarly call options on bonds.
Finally, we note that a loan commitment made by a bank or other financial institution is a putoption on a bond. Consider, for example, the situation where a bank quotes a five-year interest rateof 12% per annum to a potential borrower and states that the rate is good for the next two months.The client has, in effect, obtained the right to sell a five-year bond with a 12% coupon to thefinancial institution for its face value any time within the next two months.
European Bond Options
Many over-the-counter bond options and some embedded bond options are European. We nowconsider the standard market modcls used to value European options.
The assumption usually made is that the bond price at the maturity of the option is lognormal.Equations (22.1) and (22.2) can be used to price the option with Fy equal to the forward bondprice. The variable o is defined so that o/T is the standard deviation of the logarithm of the bondprice at the maturity of the option.512 CHAPTER 22
A bond is a security that provides a known cash income. From Section 3.3, Fy can be calculatedusing the formulaBy-1P(O, T)
 
y= (22.5)
where By is the bond price at time zero and J is the present value of the coupons that will be paidduring the life of the option. In this formula, both the spot bond price and the forward bond priceare cash prices rather than quoted prices. The relationship between cash and quoted bond prices isexplained in Section 5.9.
The strike price, K, in equations (22.1) and (22.2) should be the cash strike price. In choosing thecorrect value for K, the precise terms of the option are therefore important. If the strike price isdefined as the cash amount that is exchanged for the bond when the option is exercised, K shouldbe put equal to this strike price. If, as is more common, the strike price is the quoted priceapplicable when the option is exercised, K should be set equal to the strike price plus accruedinterest at the expiration date of the option. Traders refer to the quoted price of a bond as the“clean price” and the cash price as the ‘‘dirty price.”
Example 22.1 Consider a 10-month European call option on a 9.75-year bond with a face value of$1,000. (When the option matures the bond will have 8 years and 11 months remaining.) Supposethat the current cash bond price is $960, the strike price is $1,000, the 10-month risk-free interest rateis 10% per annum, and the volatility of the forward bond price in 10 months is 9% per annum. The
bond pays a semiannual coupon of 10% and coupon payments of $50 are expected in three monthsmonths. (This means that the accrued interest is $25 and the quoted bond price is $935.) We
une montns. (i he accrued v 18 $2) anc the quotec oond price 1s $953.)
suppose that the three-month and nine-month risk-free interest rates are 9.0% and 9.5% per annum,respectively. The present value of the coupon payments is therefore
  
 
50e 025%009 4 5Q9°-0.75%0.095 _ 95 45or $95.45, From equation (22.5), the bond forward price is given by= (960 — 95.45)e"!*9**33 — 939.68
(a) Ifthe strike price is the cash price that would be paid for the bond on exercise, the parameters forequation (22.1) are Fy = 939.68, K = 1000, P(0, 7) =e!) — 0.9200, « = 0.09, andT = 10/12. The price of the call option is $9.49.
(b) If the strike price is the quoted price that would be paid for the bond on exercise, one month’saccrued interest must be added to K because the maturity of the option is one month after acoupon date. This produces a value for K of
1,000 + 50 x 0.16667 = 1,008.33
The values for the other parameters in equation (22.1) are unchanged (ie., Fy = 939.68,P(O, T) = 0.9200, a = 0.09, and T = 0.8333). The price of the option is $7.97.
Figure 22.1 shows how the standard deviation of the logarithm of a bond’s price changes withtime. The standard deviation is zero today because there is no uncertainty about the bond’s pricetoday. It is also zero at the bond’s maturity because we know that the bond’s price will equal itsface value at maturity. Between today and the maturity of the bond, the standard deviation firstincreases and then decreases. The volatility, o, that should be used when a European option on theInterest Rate Derivatives: The Standard Market Models 513
 
Standard deviation oflogarithm of bond price
 
 
Bond Timematurity
Figure 22.1 Standard deviation of logarithm of bond price as a function of time
bond is valued is
standard deviation of logarithm of bond price at maturity of option
time to maturity of option
Figure 22.2 shows a typical pattern for o as a function of the life of the option. In general,declines as the life of the option increases.
 
Yield Volatilities
The volatilities that are quoted for bond options are often yield volatilities rather than pricevolatilities. The duration concept, introduced in Chapter 5, is used by the market to convert aquoted yield volatility into a price volatility. Suppose that D is the modified duration of the bondunderlying the option at the option maturity, as defined in Chapter 5. The relationship between the
 
Life ofoption>Bondmaturity
Figure 22.2 Variation of volatility, 0, for bond with life of option514 CHAPTER 22
change in the forward bond price, F, and its forward yield, y,, is
oFFD oveor= ~ —py, XEJF
Volatility is a measure of the standard deviation of percentage changes in the value of a variable.This equation therefore suggests that the volatility, o, of the forward bond price used in Black’smodel can be approximately related to the volatility, o,, of the forward bond yield by
ao = Dyyo, (22.6)
where yo is the initial value of y-. When a yield volatility is quoted for a bond option, the implicitassumption is usually that it will be converted to a price volatility using equation (22.6), and thatthis volatility will then be used in conjunction with equation (22.1) or (22.2) to obtain a price.Suppose that the bond underlying a call option will have a modified duration of five years at optionmaturity, the forward yield is 8%, and the forward yield volatility quoted by a broker is 20%. Thismeans that the market price of the option corresponding to the broker quote is the price given byequation (22.1) when the volatility variable, o, is
5 x 0.08 x 0.2 = 0.08or 8% per annum.
The Bond_Options worksheet of the DerivaGem software accompanying this book can be usedto price European bond options using Black’s model by selecting Black-European as the PricingModel. The user inputs a yield volatility, which is handled in the way just described. The strikeprice can be the cash or quoted sirike price.
Example 22.2 Consider a European put option on a ten-year bond with a principal of 100. Thecoupon is 8% per year payable semiannually. The life of the option is 2.25 years and the strike priceof the option is 115. The forward yield volatility is 20%. The zero curve is flat at 5% with continuouscompounding. DerivaGem shows that the quoted price of the bond is 122.84. The price of the optionwhen the strike price is a quoted price is 2.37. When the strike price is a cash price, the price of theoption is $1.74. (Note that DerivaGem’s prices may not exactly agree with manually calculated pricesbecause DerivaGem assumes 365 days per year and rounds times to the nearest whole number ofdays. See Problem 22.22 for the manual calculation.)
Theoretical Justification for the Model
In Section 21.4, we explored alternatives to the usual risk-neutral valuation assumption for valuingderivatives. One alternative was a world that is forward risk neutral with respect to a zero-couponbond maturing at time 7. We showed that:
1. The current value of any security is its expected value at time T in this world multiplied bythe price of a zero-coupon bond maturing at time 7 (see equation (21.20)).
2. The expected value of any traded security at time 7 in this world equals its forward price (seeequation (21.21)).Interest Rate Derivatives: The Standard Market Models 515
The first of these results shows that the price of a call option with maturity T years on a bond isc = P(0, T)E7[max(B; — K, 0)] (22.7)
where B; is the bond price at time T and E; denotes expected value in a world that is forwardrisk neutral with respect to a zero-coupon bond maturing at time 7. The second result impliesthat
E,(By) = Fo (22.8)
Assuming the bond price is lognormal! with the standard deviation of the logarithm of the bondprice equal to o./T, Appendix 12A shows that equation (22.7) becomes
c= PO, TEr(Br)N(d,) — KN(d3)]
wherea= in[E(B,)/K] +.0°T/2_ = BLE r(Br)/K + a T/2oVT2he eo Te = 4d ~oVToO
Using equation (22.8), we find that this reduces to Black’s model formula in equation (22.1). Wehave shown that we can use today’s 7-year maturity interest rate for discounting provided that wealso set the expected bond price equal to the forward bond price.
22.3 INTEREST RATE CAPS
 
A popular interest rate option offered by financial institutions in the over-the-counter market isan interest rate cap. Interest rate caps can best be understood by first considering a floating ratenote where the interest rate is periodically reset equal to LIBOR. The time between resets isknown as the tenor. Suppose that the tenor is three months. The interest rate on the note for thefirst three months is the initial three-month LIBOR rate; the interest rate for the next threemonths is set equal to the three-month LIBOR rate prevailing in the market at the three-monthpoint; and so on.
An interest rate cap is designed to provide insurance against the rate of interest on the floating-rate note rising above a certain level. This level is known as the cap rate. Suppose that the principalamount is $10 million, the tenor is three months, the life of the cap is five years, and the cap rateis 8%. (Because the payments are made quarterly, this cap rate is expressed with quarterlycompounding.) The cap provides insurance against the interest on the floating rate note risingabove 8%. Suppose that on a particular reset date the three-month LIBOR interest rate is 9%. Thefloating rate note would require
0.25 x 0.09 x $10,000,000 = $225,000
of interest to be paid three months later. With a three-month LIBOR rate of 8% the interestpayment would be0.25 x 0.08 x $10,000,000 = $200,000516 CHAPTER 22
The cap therefore provides a payoff of $25,000.' Note that the payoff does not occur on the resetdate when the 9% is observed. It occurs three months later. This reflects the usual time lag betweenan interest rate being observed and the corresponding payment being required.
At each reset date during the life of the cap, we observe LIBOR. If LIBOR is less than 8%, thereis no payoff from the cap three months later. If LIBOR is greater than 8%, the payoff is onequarter of the excess applied to the principal of $10 million. Note that caps are usually defined sothat the initial LIBOR rate, even if it is greater than the cap rate, does not lead to a payoff on thefirst reset date. In our example the cap lasts for five years. There are, therefore, a total of 19 resetdates (at times 0.25, 0.5, 0.75,..., 4.75 years) and 19 potential payoffs from the caps (at times 0.50,0.75, 1.00,..., 5.00 years).
The Cap as a Portfolio of Interest Rate Options
Consider a cap with a total life of T, a principal of ZL, and a cap rate of Rx. Suppose that the resetdates are ty, f,..., f, and define ¢,,, = T. Define R, as the interest rate for the period between timet, and t,4, observed at time t; (1 < & <n). The cap leads to a payoff at time t,,, (k = 1, 2,..., 2) of
Ld, max(R; — Rx, 0) (22.9)
where 6; = ty41 — t. (Both R, and Rx are expressed with a compounding frequency equal to thefrequency of resets.)
Equation (22. 9) is a call option on the LIBOR rate observed at time fk with the payoff occurringat time te. The cap is a portfolio of x such options. LIBOR rates are observed at times t), to,..., thand the corresponding payoffs occur at times ft, t3,..., fn41;. The n call options underlying the cap
are known as caplets.
A Cap as a Portfolio of Bond Options
An interest rate cap can also be characterized as a portfolio of put options on zero-coupon bondswith payoffs on the puts occurring at the time they are calculated. The payoff in equation (22.9) attime ¢;.41 is equivalent to
k_ R,-—RTe Reh, MCR — Ree 9)
at time ¢,. A few lines of algebra show that this reduces to
LL + Rgd,) 0)
TTR, (22.10)
max (u -
The expressionL(+ Rxdy)1+8,R,
is the value at time t, of a zero-coupon bond that pays off L(] + Rxd,) at time t,,,. The expressionin equation (22.10) is therefore the payoff froma put option with maturity 4 on a zero-couponbond with maturity ¢,,; when the face value of the bond is L(1 + Rxd,) and the strike price is L. It
' This calculation assumes exactly one quarter of a year between reset dates. In practice the calculation takes accountof the exact number of days between reset dates using a specified day count convention.inter