est Rate Derivatives: The Standard Market Models 517
follows that an interest rate cap can be regarded as a portfolio of European put options on zero-coupon bonds.
Floors and Collars
Interest rate floors and interest rate collars (sometimes called floor—ceiling agreements) are definedanalogously to caps. A floor provides a payoff when the interest rate on the underlying floating-ratenote falls below a certain rate. With the notation already introduced, a floor provides a payoff attime 4) (kK =1,2,...,”) of
Ld, max(Rx — R;, 0)
Analogously to an interest rate cap, an interest rate floor is a portfolio of put options on interestrates or a portfolio of call options on zero-coupon bonds. Each of the individual optionsconstituting a floor is known as a floorlet. A collar is an instrument designed to guarantee thatthe interest rate on the underlying floating-rate note always fies between two levels. A collar is acombination of a long position in a cap and a short position in a floor. It is usually constructed sothat the price of the cap is initially equal to the price of the floor. The cost of entering into thecollar is then zero.There is a put-call parity relationship between the prices of caps and floors. This is
mas aice aararice -& value ocap price = noor price + varie oO:
In this relationship, the cap and floor have the same strike price, Rx. The swap is an agreement toreceive floating and pay a fixed rate of Rx, with no exchange of payments on the first reset date.”All three instruments have the same life and the same frequency of payments. This result can beseen to be true by noting that a long position in the cap combined with a short position in the floorprovides the same cash flows as the swap.
Valuation of Caps and Floors
As shown in equation (22.9), the caplet corresponding to the rate observed at time ¢, provides apayoff at time t,,, of
Lé, max(R; — Rx, 0)
If the rate R, is assumed to be lognormal with volatility o,, equation (22.3) gives the value of thiscaplet as
L5, PO, tey MF N(di) — Re N(A2)] (22.11)whereg, = nFed Ra) + ote?' Onv/tyIn( FRx) — opt, /2dy = (Fe Rx) — Cte! =d,-avh
Opty
? Note that swaps are usually structured so that the rate at time zero determines an exchange of payments at the firstreset date. As indicated earlier, caps and floors are usually structured so that there is no payoff at the first reset date.This difference explains why we have to exclude the first exchange of payments on the swap.518 CHAPTER 22
 
and F;, is the forward rate for the period between time ¢, and t,,1. The value of the correspondingfloorlet is, from equation (22.4),
Lb, POO, tey [Re N(—d2) — F,N(—a))I (22.12)
Note that Rx and F, are expressed with a compounding frequency equal to the frequency of resetsin these equations.
Example 22.3 Consider a contract that caps the interest rate on a $10,000 loan at 8% per annum(with quarterly compounding) for three months starting in one year. This is a caplet and could be oneelement of a cap. Suppose that the zero curve is flat at 7% per annum with quarterly compounding andthe one-year volatility for the three-month rate underlying the caplet is 20% per annum. Thecontinuously compounded zero rate for all maturities is 6.9394%. In equation (22.11), Fi, = 0.07,8, = 0.25, L = 10,000, Rx = 0.08, ty = 1.0, they = 1.25, PCO, tyyy) =e OOO! — 0.9169, ando;, = 0.20. Also,dy = L.07/0.08) +0.2° x 1/2 _= 0.20 x | ~dy = dy — 0.20 = —0.7677
—0.5677
so that the caplet price is0.25 x 10,000 x 0.9169[0.07N(—0.5677) — 0.08N(—0.7677)] = $5.162
(Note that DerivaGem gives $5.146 for the price of this caplet. This is because it assumes 365 daysper year and rounds times to the nearest whole number of days.)
Each caplet of a cap must be valued separately using equation (22.11). One approach is to use adifferent volatility for each capiet. The voiatilities are then referred to as spor volatilities. Analternative approach is to use the same volatility for all the caplets constituting any particular capbut to vary this volatility according to the life of the cap. The volatilities used are then referred toas flat volatilities.’ The volatilities quoted in the market are usually flat volatilities. However, manytraders like to work with spot volatilities because this allows them to identify underpriced andoverpriced caplets. Options on Eurodollar futures are very similar to caplets and the spot volatil-ities used for caplets on three-month LIBOR are frequently compared with those calculated fromthe prices of Eurodollar futures options.
Figure 22.3 shows a typical pattern for spot volatilities and flat volatilities as a function ofmaturity. (In the case of a spot volatility, the maturity is the maturity of a caplet; in the case of aflat volatility, it is the maturity of a cap.) The flat volatilities are akin to cumulative averages of thespot volatilities and therefore exhibit less variability. As indicated by Figure 22.3, we usuallyobserve a “hump” in the volatilities. The peak of the hump is at about the two- to three-year point.This hump is observed both when the volatilities are implied from option prices and when they arecafculated from historical data. There is no genera] agreement on the reason for the existence of thehump. One possible explanation is as follows. Rates at the short end of the zero curve arecontrolled by central banks. By contrast, two- and three-year interest rates are determined to alarge extent by the activities of traders. These traders may be overreacting to the changes theyobserve in the short rate and causing the volatility of these rates to be higher than the volatility ofshort rales. For maturities beyond two to three years, the mean reversion of interest rates, whichwill be discussed in Chapter 23, causes volatilities to decline.
3 Flat volatilities can be calculated from spot volatilities and vice versa. (See Problem 22.26.)Interest Rate Derivatives: The Standard Market Models 519
      
Capvolatility
Spot vols
Flat vols
Maturity>
 
Figure 22.3 The volatility hump
Brokers provide tables of flat implied volatilities for caps and floors. The instruments underlyingthe quotes are usually at the money. This means that the cap/floor rate equals the swap rate foraswap that has the same payment dates as the cap. Table 22.1 shows typical broker quotes for theU.S. dollar market. The tenor of the cap is three months and the cap life varies from one year toten years. The data exhibits the type of “hump” shown in Figure 22.3.
Theoretical Justification for the Model
We can show that Black’s model for a caplet is internally consistent by considering a world that isforward risk neutral with respect to a zero-coupon bond maturing at time t%,,. The analysis inSection 21.4 shows that:
1. The current value of any security is its expected value at time ¢,,, in this world multiplied bythe price of a zero-coupon bond maturing at time f,,, (see equation (21.20)).
2. The expected value of an interest rate lasting between times t, and t,,, equals the forwardinterest rate in this world (see equation (21.22)).
Table 22.1 Typical broker volatility quotes for U.S. dollar capsand floors (percent per annum)
Cap Cap Floor Floor
Life bid offer bid offer
1 year 18.00 20.00 18.00 20.002 years 23.25 24.25 23.75 24.753 years 24.00 25.00 24.50 25.504 years 23.75 24.75 24.25 25.255 years 23.50 24,50 24.00 25.007 years 21.75 22.75 22.00 23.00
10 years 20.00 21.00 20.25 21.25520 CHAPTER 22
The first of these results shows that, with the notation introduced earlier, the price of a caplet thatprovides a payoff at time é,,, is
15, PO, te. \)Eq4i[max(Ry — Rx, 9)I
where £,,, denotes the expected value in a world that is forward risk neutral with respect to azero-coupon bond maturing at time 4,1. From Appendix !2A, this becomes
Ld, POO, thy MEnsi(ReN(ds) — Re NCA)where_ MnfE ni (Ri)/Rel + oh te/2Onste
— InfE ys (Ri)/ Rul ~ ot /2
Onnlty
dq
ay = d, — oy
The second result implies thatEgy i(Ry) = Fy
Together the results lead to the cap pricing model in equation (22.11). They show that we candiscount at the t,,,-maturity interest rate observed in the market today provided that we set theexpected interest rate equal to the forward interest rate.
Use of DerivaGem
The software DerivaGem accompanying this book can be used to price interest rate caps and floorsusing Black’s model. In the Cap and Swap Option worksheet, select Cap/Floor as the Under-tying Type and Black-European as the Pricing Model. The zero curve is input using continuouslycompounded rates. The inputs include the start date and the end date of the period covered by thecap, the flat volatility, and the cap settlement frequency (i.e., the tenor). The software calculates thepayment dates by working back from the end of period covered by the cap to the beginning. Theinitial caplet/floorlet is assumed to cover a period of length between 0.5 and 1.5 times a regularperiod. Suppose, for example, that the period covered by the cap is 1.2 years to 2.8 years and thesettlement frequency is quarterly. There are six caplets covering the periods 2.55 years to 2.80years, 2.30 years to 2.55 years, 2.05 years to 2.30 years, 1.80 years to 2.05 years, 1.55 years to1.80 years, and 1.20 years to 1.55 years.
22.4 EUROPEAN SWAP OPTIONS
 
Swap options, or swaptions, are options on interest rate swaps and are another increasingly populartype of interest rate option. They give the holder the right to enter into a certain interest rate swapat a certain time in the future. (The holder does not, of course, have to exercise this right.) Manylarge financial institutions that offer interest rate swap contracts to their corporate clients are alsoprepared to sell them swaptions or buy swaptions from them.
To give an example of how a swaption might be used, consider a company that knows that in sixmonths it will enter into a five-year floating-rate Joan agreement and knows that it will wish toswap the floating interest payments for fixed interest payments to convert the loan into a fixed-rateloan (see Chapter 6 for a discussion of how swaps can be used in this way). Ata cost, the companyInterest Rate Derivatives: The Standard Market Models 521
 
could enter into a swaption giving it the right to receive six-month LIBOR and pay a certain fixedrate of interest, say 8% per annum, for a five-year period starting in six months. If the fixed rateexchanged for floating on a regular five-year swap in six months turns out to be fess than 8% perannum, the company will choose not to exercise the swaption and will enter into a swap agreementin the usual way. However, if it turns out to be greater than 8% per annum, the company willchoose to exercise the swaption and will obtain a swap at more favorable terms than thoseavailable in the market.
Swaptions, when used in the way just described, provide companies with a guarantee that thefixed rate of interest they will pay on a loan at some future time will not exceed some level. Theyare an alternative to forward swaps (sometimes called deferred swaps). Forward swaps involve noup-front cost but have the disadvantage of obligating the company to enter into a swap agreement.With a swaption, the company is able to benefit from favorable interest rate movements whileacquiring protection from unfavorable interest rate movements. The difference between a swaptionand a forward swap is analogous to the difference between an option on foreign exchange and aforward contract on foreign exchange.
Relation to Bond Options
It will be recalled from Chapter 6 that an interest rate swap can be regarded as an agreement toexchange a fixed-rate bond for a floating-rate bond. At the start of a swap, the vatue of the floating-rate bond always equals the notional principal of the swap. A swaption can, therefore, be regarded asan option to exchange a fixed-rate bond for the notional principal of the swap. If a swaption givesthe holder the right to pay fixed and receive floating, it is a put option on the fixed-rate bond withstrike price equal to the notional principal. If a swaption gives the holder the right to pay floatingand receive fixed, it is a call option on the fixed-rate bond with a strike price equal to the principal.
Valuation of European Swap Options
The swap rate for a particular maturity at a particular time is the fixed rate that would beexchanged for LIBOR in a newly issued swap with that maturity. The model usually used tovalue a European option on a swap assumes that the relevant swap rate at the maturity of theoption is fognormal. Consider a swaption where we have the right to pay a rate sx and receiveLIBOR on a swap that will last 1 years starting in T years. We suppose that there are m paymentsper year under the swap and that the notional principal is L.
Suppose that the swap rate for an n-year swap at the maturity of the swap option is s7. (Both s7and sx are expressed with a compounding frequency of m times per year.) By comparing the cashflows on a swap where the fixed rate is s; to the cash flows on a swap where the fixed rate is sx, wesee that the payoff from the swaption consists of a series of cash flows equal to
L
= — spr, 0
nm max(sr — Sx, 0)The cash flows are received m times per year for the n years of the life of the swap. Suppose that thepayment dates are 7), T>,..., Tmn, Measured in years from today (it is approximately true thatTJ; =T +i/m). Each cash flow is the payoff from a cafl option on s; with strike price sx.
Using equation (22.3), the value of the cash flow received at time 7; is
L5 PO: TlsoN (A) — Sx N(@2)]522 CHAPTER 22
 
whereq __ Inls0/8q) + 0° T /21= UF VTIn(so/sx) — 0° T/2dy = Ed ~ VT2 OVE 17
and 59 is the forward swap rate.The total value of the swaption is
mn L
Dey PO. TION (dr) — sx N(da)]
i=}
Defining A as the value of a contract that pays 1/m at times 7; (1 <i < mn), the value of the
swaption becomesLA[soN(d1) — 8xN(@)] (22.13)where
mn
1A= my PO, T;)
If the swaption gives the holder the right to receive a fixed rate of sx instead of paying it, the payofffrom the swaption is
L—max(sx — Sr, 0)m
This is a put option on sy. As before, the payoffs are received at times 7; (1 <i < mn). Equa-tion (22.4) gives the value of the swaption as
LA[sx N(—d)) — 8) N(—d,)] (22.14)
Example 22.4 Suppose that the LIBOR yield curve is flat at 6% per annum with continuouscompounding. Consider a swaption that gives the holder the right to pay 6.2% in a three-year swapstarting in five years. The volatility for the swap rate is 20%. Payments are made semiannually andthe principal is $100. In this case,
A= Ae 0.06x9.5 Ee 006x6 1 4-0.066.5 9006x7497 0.06%7.5 $e 0. 06%8) = 2.0035=35 =24.
A rate of 6% per annum with continuous compounding translates into 6.09% with semiannualcompounding. It follows that, in this example, sp = 0.0609, sx = 0.062, T = 5, o = 0.2, so that
_ In(0.0609/0.062) +0.2? x 5/2~ 0.2/5
dy = d, — 0.205 = —0.2636
a = 0.1836
From equation (22.13), the value of the swaption is100 x 2.0035{0.0609 x N(0.1836) — 0.062 x N(—0.2636)] = 2.07
or $2.07. (This is in agreement with the price given by DerivaGem.)}
Brokers provide tables of implied volatilities for European swap options. The instrumentsInterest Rate Derivatives: The Standard Market Models 523
Table 22.2 Typical broker quotes for U.S. European swap options (mid-marketvolatilities percent per annum)
Swap length
Expiration Llyear 2years 3 years 4 years Syears 7 years 10 years
 
1 month 17.75 17.75 17.75 17.50 17.00 17.00 16.003 months 19.50 19.00 19.00 18.00 17.50 17.00 16.006 months 20.00 20.00 19.25 18.50 18.75 17.75 16.75
1 year 2250 21.75 20.50 20.00 1950 18.25 16.782 years 22.00 22.00 20.75 19.50 19.75 18.25 16.753 years 21.50 21.00 20.00 = 19.25 19.00 «17.75 16.504 years 20.75 20.25 19.25 18.50 «18.25 17.50 16.00
5 years 20.00 19.50 18.50 17.75 17.50 17.00 {5.50
underlying the quotes are usually at the money. This means that the strike swap rate equals theforward swap rate. Table 22.2 shows typical broker quotes provided for the U.S. dollar market.The tenor of the underlying swaps (i.e., the frequency of resets on the floating rate) is six months.The life of the option is shown on the vertical scale. This varies from one month to five years.The life of the underlying swap at the maturity of the option is shown on the horizontal scale.This varies from one year to ten years. The volatilities in the one-year column of the tablecorrespond to instruments that are similar to caps. They exhibit the hump discussed earlier. Aswe move to the columns corresponding to options on longer-lived swaps, the hump persists butbecomes fess pronounced.
Theoretical Justification for the Swap Option Model
We can show that Black’s model for swap options is internally consistent by considering a worldthat is forward risk neutral with respect to the annuity A. The analysis in Section 21.4 shows that:
1. The current value of any security is the current value of the annuity multiplied by the
expected value ofsecurity price at time 7”
value of the annuity at time T
in this world (see equation (21.25)).
2. The expected value of the swap rate at time T in this world equals the forward swap rate (seeequation (21.24)).
The payoff at time T from a swap option where we have the right to pay sx and receive floating isthe value of an annuity times
L—max(sp ~ Sx, 0)m
The first result shows that the value of the swaption is
LAE ,{max(sy — Sx, 0)]524 CHAPTER 22
From Appendix 12A, this isLA[E,(s7)N(dy) — 5¢N(d2)}
where4, = BlEalsr)/sel + 0°T/21 — oF!2b= In[E4(s7)/sx] — 0° T/2 =d,—-ovF
oVT
The second result shows that £'4(s;) equals so. Taken together, the results lead to the swap optionpricing formula in equation (22.13). They show that we are entitled to treat interest rates asconstant for the purposes of discounting provided that we also set the expected swap rate equal tothe forward swap rate.
22.56 GENERALIZATIONS
 
We have presented three different versions of Black’s model: one for bond options, one for caps, and
one for swap options. Each of the models is internally consistent, but they are not consistent with
each other. For example, when future bond prices are lognormal, future zero rates and swap rates are
not; and when future zero rates are lognormal, future bond prices and swap rates are not.These results we have produced can be generalized as follows:
1. Consider any instrument that provides a payoff at time T dependent on the price of a securityobserved at time J. Its current value is P(0, T) times the expected payoff provided thatexpectations are calculated in a world where the expected value of the underlying securityequals its forward price.
2. Consider any instrument that provides a payoff at time 7, dependent on the 7>-maturityinterest rate observed at time 7;. Its current value is P(0,7,) times the expected payoffprovided that expectations are calculated in a world where the expected value of the
underlying interest rate equals the forward interest rate.
3. Consider any instrument that provides a payoff in the form of an annuity. We suppose thatthe size of the annuity is determined at time T as a function of the swap rate for an n-yearswap starting at time T. We also suppose that annuity lasts for n years and payment dates forthe annuity are the same as those for the swap. The value of the instrument is A times theexpected payoff per year, where (a) A is current value of the annuity when payments are atthe rate $1 per year and (b) expectations are taken in a world where the expected future swaprate equals the forward swap rate.
The first of these results is a generalization of the European bond option model; the second is ageneralization of the cap/floor model; the third is a generalization of the swap option model.
22.6 CONVEXITY ADJUSTMENTS
 
 
This section discusses what happens when an instrument provides a payoff at time T dependent ona bond yield calculated at time T.Interest Rate Derivatives: The Standard Market Models 525
The forward yield on a bond is defined as the yield implied by the forward bond price. Supposethat By is the price of a bond at time T, yz is its yield, and the relationship between B; and yr is
Br = G(yr)
Define Fy as the forward bond price at time zero for a contract maturing at time T and yp as theforward bond yield at time zero. ‘Ihe definition of a forward bond yield means that
Fy = G(yp)
The function G is nonlinear. This means that, when the expected future bond price equals theforward bond price (so that we are a world that is forward risk neutral with respect to a zero-couponbond maturing at time 7), the expected future bond yield does not equal the forward bond yield.
This is illustrated in Figure 22.4, which shows the relationship between bond prices and bondyields at time 7. For simplicity, we suppose that there are only three possible bond prices, B,, B2,and B;, and that they are equally likely in a world that is forward risk neutral with respect to P(t, T).We assume that the bond prices are equally spaced, so that B; ~ B, = B, — B. The expected bondprice is B, and this is also the forward bond price. The bond prices translate into three equally likelybond yields, Y;, ¥2, and ¥;, which are not equally spaced. The variable Y; is the forward bond yieldbecause it is the yield corresponding to the forward bond price. The expected bond yield is theaverage of Y;,, ¥2, and Y; and is clearly greater than ¥».
Consider now a derivative that provides a payoff dependent on the bond yield at time T. We knowfrom equation (21.20) that it can be valued by (a) calculating the expected payoff in a world that isforward risk neutral with respect to a zero-coupon bond maturing at time T and (b) discounting atthe current risk-free rate for maturity T. We know that the expected bond price equals the forwardprice in the world being considered. We therefore need to know the value of the expected bond yieldwhen the expected bond price equals the forward bond price. The analysis in Appendix 22A showsthat an approximate expression for the required expected bond yield is
1 G"(y0)
E7(yr) = yo > syjo3T
 
(22.15)
G0)
Bondprice
  
 
>
Yield
Figure 22.4 Relationship between bond prices and bond yields526 CHAPTER 22
where G’ and G” denote the first and second partial derivatives of G, Ey denotes expectations inaworld that is forward risk neutral with respect to P(t, T), and o, is the forward yield volatility. Itfollows that we can discount expected payoffs at the current risk-free rate for maturity T providedthat we assume that the expected bond yield is
G"(y0)G'(yo)
yo — 496057
 
rather than yo. The difference between the expected bond yield and the forward bond yield,
Gyo)G'(y0)
 
—tyjoyT
is known as a convexity adjustment. It corresponds to the difference between Y, and the expectedyield in Figure 22.4. (The convexity adjustment is positive because G’(yg) < 0 and G’(yp) > 02)
Application 1: Interest Rates
As our first application of equation (22.15), we consider an instrument that provides a cash flow attime T equal to the interest rate between times T and 7* applied to a principal of L. (This examplewill be useful when we consider LIBOR-in-arrears swaps in Chapter 25.) Note that the interest rateapplicable to the time period between times T and 7* is normally paid T*; here we are assumingthat it is paid early, at time T.
The cash flow at time 7 in the instrument we are considering is LR;t, where t = T* — T and R;is the zero-coupon interest rate applicable to the period between T and T* (expressed with acompounding period of 1). The variable Ry can be viewed as the yield on a zero-coupon bondmaturing at time T*. The relationship between the price of this bond and its yield is
 
 
G() = 1+ ytFrom equation (22.15),G’(R )E7(Rr) = Ro — Ron SRorRioktT= 22.E7(Rr) = Ro +7 + Rot (22.16)
where Ry is the forward rate applicable to the period between T and T* and cp is the volatility ofthe forward rate.The value of the instrument is therefore
 
RoogtTPO, TyLx( Ry + o7RT )
1+ RotExample 22.5 Consider a derivative that provides a payoff in three years equal to the one-year
zero-coupon rate (annually compounded) at that time multiplied by $1000. Suppose that the zero ratefor all maturities is 10% per annum with annual compounding and the volatility of the forward rateInterest Rate Derivatives: The Standard Market Models 527
applicable to the time period between year 3 and year 4 is 20%. In this case, Ry = 0.10, og = 0.20,T =3, t= 1, and P(0, 3) = 1/1.10° = 0.7513. The value of the derivative is
0.10? x 0.20? x 1 x *)
0.7313 x 1000 x 1(0.10 + Tro10x1
or $75.95. (This compares with a price of $75.13 when no convexity adjustment is made.)
Application 2: Swap Rates
Consider next a derivative providing a payoff at time T equal to a swap rate observed at that time.A swap rate is a par yield. For the purposes of calculating a convexity adjustment, we make anapproximation and assume that the N-year swap rate at time T equals the yield at that time on anN-year bond with a coupon equal to today’s forward swap rate. This enables equation (22.15) tobe used.
Example 22.6 Consider an instrument that provides a payoff in three years equal to the three-yearswap rate at that time multiplied by $100. Suppose that payments are made annually on the swap, thezero rate for all maturities is 12% per annum with annual compounding, the volatility for the three-year forward swap rate in three years (implied from swap option prices) is 22%. We approximate theswap rate as the yield on a 12% bond, so that the relevant function G(y) isGu) = + oN wty dtr A+y)
, 0.12 0.24 3.36GO) =F aay aaG+tye Utyl (+y)
” 0.24 0.72 13.44G"y) = ——4 + >tGQ+xy (+ U+y)
In this case the forward yield, yo, is 0.12, so that G’(yo) = —2.4018 and G” (yp) = 8.2546. Fromequation (22.15),
 
8.2546= 1 2 2 _Er(yr) = 0.12 + 5 x 0.12" x 0.22 x 3x FGp7g = 01236
We should therefore assume a forward swap rate of 0.1236 (= 12.36%) rather than 0.