1

13 Swaps, Caps, Floors, and Swaptions Swaps, caps, floors and swaptions are very useful interest rate securities. Imagine yourself the treasurer of a large corporation who has borrowed funds from a bank using a floating rate loan. A floating rate loan is a long-term debt instrument whose interest payments vary (float) with respect to the current rates for short-term borrowing. Suppose the loan was taken when interest rates were low, but now rates are high. Rates are projected to move even higher. The current interest payments on the loan are high and if they go higher, the company could face a cash flow crisis, perhaps even bankruptcy. The company’s board of directors is concerned.

2

Is there a way you can change this floating rateloan into a fixed rate loan, without retiring thedebt and incurring large transaction costs (and aloss on your balance sheet)?

The solution is to enter into a fixed for floatingrate swap or simultaneously purchase caps andfloors with predetermined strikes.

If you had thought about this earlier, you couldhave entered into a swaption at the time the loanwas made to protect the company from such acrisis.

3

A Fixed-Rate and Floating-Rate Loans

In our simple discrete time model, the short-termrate of interest corresponds to the spot rate r(t)and each period in the model requires an interestpayment.

We define a floating rate loan for L dollars (theprincipal) with maturity date T to be a debtcontract that obligates the borrower to pay thespot rate of interest times the principal L everyperiod, up to and including the maturity date,time T. At time T, the principal of L dollars isalso repaid.

4

In our frictionless and default-free setting, thisfloating rate loan is equivalent to shorting L unitsof the money market account and distributing thegains (paying out the spot rate of interest times Ldollars) every period.

Paying out the interest as a cash flow maintains thevalue of the short position in the money marketaccount at L dollars. At time T the short positionis closed out.

5

Table 13.1: Cash Flow from a Floating Rate Loan of a dollar (the Principal), with maturity date T.

0 1 2time

Borrow +1

T

P a y i n t e r e s t – [ r ( 0 ) – 1 ] – [ r ( 1 ) – 1 ] – [ r ( T – 1 ) – 1 ]

P a y p r i n c i p a l – 1

6

A s t h e f l o a t i n g - r a t e i s m a r k e t d e t e r m i n e d , i t c o s t s 0 d o l l a r s t o e n t e r i n t o a f l o a t i n g - r a t e l o a n c o n t r a c t . C o m p u t i n g t h e p r e s e n t v a l u e o f t h e c a s h f l o w s p a i d o n a f l o a t i n g - r a t e l o a n w i t h a d o l l a r p r i n c i p a l a n d m a t u r i t y d a t e T m a k e s t h i s s a m e p o i n t . U s i n g t h e r i s k - n e u t r a l v a l u a t i o n p r o c e d u r e , t h e p r e s e n t v a l u e o f t h e c a s h f l o w s t o t h e f l o a t i n g r a t e l o a n i s :

1)t(B)T(B

1tE

~)t(B

)1j(B]1)j(r[1T

tjtE~

)t(rV

( 1 3 . 1 )

E x p r e s s i o n ( 1 4 . 1 ) s h o w s t h a t t h e v a l u e o f t h e c a s h f l o w s f r o m t h e f l o a t i n g - r a t e l o a n a t t i m e t e q u a l s o n e d o l l a r , w h i c h i s t h e a m o u n t b o r r o w e d .

7

We define a fixed rate loan with interest rate c for L dollars (the principal) and with maturity date T to be a debt contract that obligates the borrower to pay (c-1) times the principal L every period, up to and including the maturity date, time T. At time T, the principal of L dollars is also repaid. A fixed-rate loan of B(0) dollars at fixed rate (C/L) and maturity T, in our frictionless and default-free setting, is equivalent to shorting the coupon bond described in Chapter 10. The (coupon) rate on the loan is defined to be (1+C/L) per period.

8

Table 13.2: Cash Flow to a Fixed Rate Loan with Coupon C, Principal L, and maturity date T.

0 1 2t i m e

B o r r o w BT…

( 0 )

P a y i n t e r e s t – C – C … – C

P a y p r i n c i p a l – L

9

C o m p u t i n g t h e p r e s e n t v a l u e o f t h e c a s h f l o w s p a i d o n t h e f i x e d - r a t e l o a n c a n m a k e t h e s a m e p o i n t . T o m a k e t h e c o u p o n r a t e c o m p a t i b l e w i t h t h e r a t e c o n v e n t i o n u s e d i n t h i s b o o k , w e d e f i n e .L/C1c U s i n g t h e r i s k - n e u t r a l v a l u a t i o n p r o c e d u r e ,

)t()T,t(LP)1j,t(CP1T

tj

)t(B)T(B

LtE

~)t(B

)1j(BC1T

tjtE~

)t(cV

B

( 1 3 . 2 )

E x p r e s s i o n ( 1 3 . 2 ) s h o w s t h a t t h e v a l u e o f t h e c a s h f l o w s t o a f i x e d - r a t e l o a n a t t i m e t e q u a l s B ( t ) , w h i c h i s t h e a m o u n t b o r r o w e d .

10

B Interest Rate Swaps

An interest rate swap is a financial contract thatobligates the holder to receive fixed-rate loanpayments and pay floating-rate loan payments (orvice versa).

1 Swap Valuation

Consider an investor who has a fixed-rate loanwith interest rate c, a principal of L dollars and amaturity date T. The cash payment at everyintermediate date t is C = (c-1)L.

The investor wants to exchange this fixed-rateloan for a floating-rate loan with principal Ldollars, maturity date T, and floating interestpayments of L(r(t-1) - 1) dollars per period.

He does this by entering into a swap receivingfixed and paying floating.

11

Figure 13.1: An Illustration of a Swap Changing a Fixed Rate Loan into a Floating Rate Loan

FIXED RATE LOAN

SWAP

pay fixedreceive fixed

pay floatinginvestor

12Table 13.3: The Cash Flows and Values from a Swap Receiving Fixed and

Paying Floating

Floating Payments Fixed Payments Net Payments Swap Value

–[r(0) – 1]L –[r(1) – 1]L –[r(T–2) – 1]L –[r(T–1) – 1]L – L

0 1 2 T–1 T

+C +C +C +C+L

C – [r(0) – 1]L 0 C – [r(1) – 1]L –[r(T–2) – 1]L C–[r(T–1) – 1]L C

…

(0) – L B (1) – L B (2) – L B (T–1) – L B (T) – L B

13

L e t S ( t ) r e p r e s e n t t h e v a l u e o f t h e s w a p a t t i m e t . T h e v a l u e o f t h e s w a p a t a n y p e r i o d t i s S ( t ) = B ( t ) - L . C o m p u t i n g t h e p r e s e n t v a l u e o f t h e c a s h f l o w s f r o m t h e s w a p c a n m a k e t h i s s a m e p o i n t . U s i n g t h e r i s k - n e u t r a l v a l u a t i o n p r o c e d u r e ,

)t(B)1j(B

]L)1)j(r(C[1T

tjtE~

)t(S

. ( 1 3 . 3 )

D e f i n i n g c 1 + C / L t o b e o n e p l u s t h e c o u p o n r a t e o n t h e f i x e d - r a t e l o a n , w e c a n r e w r i t e t h i s a s

.L)t()t(B)1j(B

L)]j(rc[1T

tjtE~

)t(S

B ( 1 3 . 4 )

14

2 The Swap Rate

The swap rate is defined to be that coupon rate C/Lsuch that the swap has zero value at time 0, i.e.,such that S(0) = 0 or B(0) = L.

It is important to emphasize that thisdetermination of the swap rate is under theassumption of no default risk for eithercounterparty to the swap contract.

EXAMPLE: SWAP VALUATION

15Figure 13.2: An Example of a One-Factor Bond Price Curve Evolution. The Money Market Account Values and

Spot Rates are Included on the Tree. Pseudo-Probabilities Are Along Each Branch of the Tree.

.923845

.942322

.961169

.980392 1

.947497

.965127

.982699 1

.937148

.957211

.978085 1

1/2

1/2

1/2

1/2

1/2

1/2

.967826

.984222 1

.960529

.980015 1

.962414

.981169 1

.953877

.976147 1

.985301 1

.981381 1

.982456 1

.977778 1

.983134 1

.978637 1

.979870 1

.974502 1

1

1

1

1

1

1

1

1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

P(0,4) P(0,3) P(0,2) P(0,1) P(0,0)

=

B(0) 1

1.02

1.02

1.037958

1.037958

1.042854

1.042854

r(0) = 1.02

1.017606

1.022406

1.016031

1.020393

1.019193

1.024436

1.054597

1.054597

1.059125

1.059125

1.062869

1.062869

1.068337

1.068337

time 0 1 2 3 4

16

This evolution is arbitrage-free as it was studied in Chapter 9. Consider a swap receiving fixed and paying floating with maturity date T = 3 and principal L = 100. First, we need to determine the swap rate. To do this, we need to find the coupon payment C per period such that the value of the swap is zero, i.e., S(0) = 0.

17

W e f i r s t c o m p u t e t h e s w a p ' s v a l u e f o r a n a r b i t r a r yc o u p o n p a y m e n t o f C :

.7678.5)8838.2(C100)942322(.100]942322.961169.980392[.C

100)3,0(P)100C()2,0(CP)1,0(CP100)0()0(S

B

S e t t i n g S ( 0 ) = 0 a n d s o l v i n g f o r C y i e l d s

C = 5 . 7 6 7 8 / 2 . 8 8 3 8 = 2 .

T h e s w a p r a t e i s C / L = 2 / 1 0 0 = 0 . 0 2 .

18

Figure 13.3: An Example of a Swap Receiving Fixed and Paying Floating with Maturity Time 3, Principal $100, and Swap Rate .02. Given first is the swap's value, then the swap's cash flow. The synthetic swap

portfolio in the money market account and three-period zero-coupon bond (n0(t; st), n3(t; st)) is given under each node.

time 0 1 2 3

0 .396930

0 .396930

0 -.039285

0 -.039285 0 .080719

0 .080719 0 -.443609

0 -.443609

S(0) = 0 Cash Flow = 0

(-97.215294, 103.165648)

.408337 0

(-96.112355, 102)

-.408337 0

(-96.121401, 102)

.390667

.239442 (.376381, 0)

-.038500 .239442

(-.037092, 0)

.079199 -.240572

(.075945, 0)

-.433028 -.240572

(-.415234, 0)

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

r(0) = 1.02

1.017606

1.016031

1.022406

1.020393

1.019193

1.024436

19

W e r e c e i v e f i x e d a n d p a y f l o a t i n g . T h ec a l c u l a t i o n s a r e a s f o l l o w s .

A t t i m e 3 , f o r e a c h p o s s i b l e s t a t e :

39693.60307.12L)1)uu;2(r(C)uuu;3(flow cash0100100LL)uuu;3(S

39693.100)1)uu;2(r(2)uud;3(flow cash0100100)uud;3(S

039285.100)1)ud;2(r(2)udu;3(flow cash0100100)udu;3(S

20

443609.100)1)dd;2(r(2)ddu;3(flow cash0100100)ddu;3(S

443609.100)1)dd;2(r(2)ddd;3(flow cash0100100)ddd;3(S

C o n t i n u i n g b a c k w a r d t h r o u g h t h e t r e e :

239442.76056.12L]1)u;1(r[C)uu;2(flow cash390667.100)984222(.102

100)uu;3,2(P102L)uu;2()uu;2(S

B

239442.100)1)u;1(r(2)ud;2(flow cash038500.100)ud;3,2(P102)ud;2(S

21

240572.100)1)d;1(r(2)du;2(flow cash079199.100)du;3,2(P102)du;2(S

.240572.100)1)d;1(r(2)dd;2(flow cash433028.100)dd;3,2(P102)dd;2(S

F i n a l l y , a t t i m e 1 :

022L]1)0(r[C)u;1(flow cash408337.100)965127(.102)982699(.2

100)u;3,1(P102)u;2,1(P2L)u;1()u;1(S

B

.0100)1)0(r(2)d;1(flow cash408337.100)d;3,1(P102)d;2,1(P2)d;1(S

F r o m F i g . 1 3 . 3 w e s e e t h a t t h e c a s h f l o w f r o m t h e s w a p c a n b e p o s i t i v e o r n e g a t i v e

22

3 Synthetic Swaps

There are numerous ways of creating a swapsynthetically.

The first is to use a buy and hold strategy. Thismethod is to short the money market account (payfloating) and to synthetically create the couponbond as a portfolio of zero-coupon bonds. Thissynthetic swap is independent of any particularmodel for the evolution of the term structure ofinterest rates.

Unfortunately, synthetically constructing the swapvia a portfolio of zero-coupon bonds has twopractical problems. One, not all zero-couponbonds may trade. Two, the initial transactioncosts will be high.

23

The second method is the syntheticconstruction of swaps using forwardcontracts written on the spot rate of interest,called Forward Rate Agreements or FRAs.

We define a forward rate agreement (FRA)on the spot rate of interest with delivery dateT, contract rate c (one plus a percent), andprincipal L to be that contract that has acertain payoff of

[r(T-1) – c]L dollars at time T.

Notice that the spot rate in this FRA’s payoffat time T is spot rate from time T-1.

24

The contract rate c is set at the date thecontract is initiated, say at time 0. It is set bymutual consent of the counter parties to thecontract. At initiation, the contract rate neednot give the FRA zero initial value (however,a typical FRA sets the rate at initiation suchthat the contract has zero value.

In the case where the value of the contract atinitiation is non-zero, the counter partieswould sign the contract and the fair value ofthe FRA is exchanged in cash.

25

L e t u s d e n o t e t h e t i m e t v a l u e o f a n F R A w i t h d e l i v e r y d a t e T a n d c o n t r a c t r a t e c w i t h p r i n c i p a l 1 d o l l a r a s V f ( t , T ; c ) . U s i n g t h e t e c h n i q u e s o f c h a p t e r 1 2 , t h e t i m e t v a l u e o f t h i s F R A i s :

B(t)B(T)

c1)r(TtEc)T;(t,fV

~ .

B u t , 1)1/B(T1)/B(T)r(T , s o

B(t)B(T)

1tEcB(t)

1)B(T1

tEc)T;(t,fV

~~ .

26

R e c a l l i n g t h a t t)(1/B(T))B(tET)P(t,~ , s u b s t i t u t i o n

g i v e s :

c)T;(t,fV = T)cP(t,1)TP(t, .

A t i n i t i a t i o n , t h e F R A ’ s v a l u e w o u l d b e :

c)T;(0,fV = T)cP(0,1)TP(0, .

27

To construct a synthetic swap, note that from Table 13.3 the third row, the net payment to the swap at time T is identical to the payoff from being short a single FRA with delivery date T, contract rate c, and principal L. Hence, a synthetic swap can be constructed at time 0 by shorting a portfolio of FRAs: all with contract rate c and principal L, but with differing delivery dates. The delivery dates included in the collection of short FRAs should be times 1,2, …, T.

28

T h e v a l u e o f t h i s c o l l e c t i o n o f s h o r t F R A s i s :

T

1tL)c;t,0(

fV

T

1t)]t,0(cP)1t,0(P[L

)T,0(LPT

0t)t,0(P]1c[LL

)0(L B = S ( 0 ) .

T h i s i s t h e v a l u e o f t h e s w a p w i t h m a t u r i t y Ta n d p r i n c i p a l L r e c e i v i n g f i x e d a n d p a y i n gf l o a t i n g a t t i m e 0 , a s e x p e c t e d !

29

A third method for synthetically creating this swapis to use a dynamic portfolio consisting of a singlezero-coupon bond (for a one-factor model) and themoney market account.

This approach requires a specification of theevolution of the term structure of interest rates.

EXAMPLE: SYNTHETIC SWAPCONSTRUCTION

30

Figure 13.3: An Example of a Swap Receiving Fixed and Paying Floating with Maturity Time 3, Principal $100, and Swap Rate .02. Given first is the swap's value, then the swap's cash flow. The synthetic swap

portfolio in the money market account and three-period zero-coupon bond (n0(t; st), n3(t; st)) is given under each node.

time 0 1 2 3

0 .396930

0 .396930

0 -.039285

0 -.039285 0 .080719

0 .080719 0 -.443609

0 -.443609

S(0) = 0 Cash Flow = 0

(-97.215294, 103.165648)

.408337 0

(-96.112355, 102)

-.408337 0

(-96.121401, 102)

.390667

.239442 (.376381, 0)

-.038500 .239442

(-.037092, 0)

.079199 -.240572

(.075945, 0)

-.433028 -.240572

(-.415234, 0)

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

r(0) = 1.02

1.017606

1.016031

1.022406

1.020393

1.019193

1.024436

31

W e c a n u s e a d y n a m i c s e l f - f i n a n c i n g t r a d i n g s t r a t e g y i n t h e 3 -p e r i o d z e r o - c o u p o n b o n d a n d t h e m o n e y m a r k e t a c c o u n t u s i n gt h e d e l t a a p p r o a c h .

A t t i m e 2 , s t a t e u u t h e v a l u e o f t h e s w a p a n d i t s c a s h f l o w a r ek n o w n f o r s u r e .

T h e s w a p c a n b e s y n t h e t i c a l l y c r e a t e d b y h o l d i n g n o n e o f t h et h r e e - p e r i o d z e r o - c o u p o n b o n d ,

0)uu;2(3n , a n d

376381.037958.1]390667[.

)u;2(B)uu;3,2(P)uu;2(3n)uu;2(S)uu;2(0n

u n i t s o f t h e m o n e y m a r k e t a c c o u n t .

T h e c a l c u l a t i o n s f o r t h e r e m a i n i n g s t a t e s a r e s i m i l a r :

32

037092.037958.1/038500.)ud;2(0n

0)ud;2(3n

075945.042854.1/079199.)du;2(0n

0)du;2(3n

.415234.042854.1/433028.)dd;2(0n

0)dd;2(3n

33

A t t i m e 1 , s t a t e u t h e n u m b e r o f t h r e e - p e r i o dz e r o - c o u p o n b o n d s h e l d i s

.102980015.984222.200942.630109.

)ud;3,2(P)uu;3,2(P))ud;2(flow cash)ud;2(S())uu;2(flow cash)uu;2(S()u;1(3n

T h e n u m b e r o f u n i t s o f t h e m o n e y m a r k e t a c c o u n th e l d i s

.112355.9602.1/)]965127(.102408337[.

)1(/)]u;3,1(P)u;1(3n)u;1(S[)u;1(0n

B

34

A t t i m e 1 , s t a t e d t h e c a l c u l a t i o n s a r e

102976149.981169.

)6736.(161373.)dd;3,2(P)du;3,2(P

))dd;2(flow cash)dd;2(S())du;2(flow cash)du;2(S(

)d;1(3n

a n d

.121401.9602.1/)]957211(.102408337.[

)1(/)]d;3,1(P)d;1(3n)d;1(S[)d;1(0n

B

35

A t t i m e 0 ,

165648.103957211.965127.

)408337.(408337.)d;3,1(P)u;3,1(P

))d;1(flow cash)d;1(S())u;1(flow cash)u;1(S()0(3n

.215294.97)942322(.165648.1030

)]3,0(P)0(3n)0(S[)0(0n

R a t h e r t h a n u s i n g t h e 3 - p e r i o d z e r o - c o u p o n b o n d ,t h e s w a p c o u l d h a v e b e s y n t h e t i c a l l y c o n s t r u c t e du s i n g a n y o t h e r i n t e r e s t r a t e s e n s i t i v e s e c u r i t y , f o re x a m p l e , a f u t u r e s o r o p t i o n c o n t r a c t o n t h e 3 -p e r i o d z e r o - c o u p o n b o n d .

36

C Interest Rate Caps A simple interest rate cap is a provision often attached to a floating-rate loan that limits the interest paid per period to a maximum amount, k-1, where k is 1 plus a percentage. Interest rate caps trade separately. Consider an interest rate cap with cap rate k and maturity date * on the floating-rate loan of Table 13.1. We can decompose this cap into the sum of * caplets.

37

A c a p l e t i s d e f i n e d t o b e a n i n t e r e s t r a t e c a ps p e c i f i c t o o n l y a s i n g l e t i m e p e r i o d . S p e c i f i c a l l y ,i t i s e q u i v a l e n t t o a E u r o p e a n c a l l o p t i o n o n t h es p o t i n t e r e s t r a t e w i t h s t r i k e k a n d m a t u r i t y t h es p e c i f i c d a t e o f t h e s i n g l e t i m e p e r i o d .

F o r e x a m p l e , a c a p l e t w i t h m a t u r i t y T a n d a s t r i k ek h a s a t i m e T c a s h f l o w e q u a l t o : .0,k1Trmax

T h i s c a s h f l o w i s k n o w n a t t i m e T - 1 b e c a u s e t h es p o t r a t e i s k n o w n a t t i m e T - 1 .

38

T h e a r b i t r a g e - f r e e v a l u e o f t h e T - m a t u r i t y c a p l e t a t t i m e t i s o b t a i n e d u s i n g t h e r i s k - n e u t r a l v a l u a t i o n p r o c e d u r e :

.tBTB0,k1TrmaxtE~

ts;T,tc ( 1 3 . 5 )

A n i n t e r e s t r a t e c a p o n t h e f l o a t i n g - r a t e l o a n i n T a b l e 1 3 . 1 i s t h e n t h e s u m o f t h e v a l u e s o f t h e c a p l e t s f r o m w h i c h i t i s c o m p o s e d . E X A M P L E : C A P V A L U A T I O N .

39Figure 13.2: An Example of a One-Factor Bond Price Curve Evolution. The Money Market Account Values and Spot Rates are Included on the Tree. Pseudo-Probabilities Are Along Each Branch of the Tree.

.923845

.942322

.961169

.980392 1

.947497

.965127

.982699 1

.937148

.957211

.978085 1

1/2

1/2

1/2

1/2

1/2

1/2

.967826

.984222 1

.960529

.980015 1

.962414

.981169 1

.953877

.976147 1

.985301 1

.981381 1

.982456 1

.977778 1

.983134 1

.978637 1

.979870 1

.974502 1

1

1

1

1

1

1

1

1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

P(0,4) P(0,3) P(0,2) P(0,1) P(0,0)

=

B(0) 1

1.02

1.02

1.037958

1.037958

1.042854

1.042854

r(0) = 1.02

1.017606

1.022406

1.016031

1.020393

1.019193

1.024436

1.054597

1.054597

1.059125

1.059125

1.062869

1.062869

1.068337

1.068337

time 0 1 2 3 4

40

W e k n o w t h a t t h i s e v o l u t i o n i s a r b i t r a g e - f r e e .

C o n s i d e r a n i n t e r e s t r a t e c a p w i t h m a t u r i t y d a t e * = 3 a n d a s t r i k e o f k = 1 . 0 2 .

T h i s i n t e r e s t r a t e c a p c a n b e d e c o m p o s e d i n t ot h r e e c a p l e t s : o n e a t t i m e 1 , o n e a t t i m e 2 , a n d o n ea t t i m e 3 .

W e v a l u e a n d d i s c u s s t h e s y n t h e t i c c o n s t r u c t i o n o fe a c h c a p l e t i n t u r n .

T h e c a p l e t a t t i m e 1 , c ( 0 , 1 ) , h a s z e r o v a l u e . F o r m a l l y ,

.002.1)0)(2/1()0)(2/1(

)0(r0,02.1)0(rmax0E~

)1,0(c

41

N e x t , c o n s i d e r t h e c a p l e t w i t h m a t u r i t y a t t i m e 2 . B y e x p r e s s i o n ( 1 3 . 5 ) , a t t i m e 2 i t s v a l u e u n d e r e a c h s t a t e i s a s f o l l o w s :

002406.)0,02.1)d;1(rmax()dd;2,2(c002406.)0,02.1022406.1max()0,02.1)d;1(rmax()du;2,2(c

0)0,02.1)u;1(rmax()ud;2,2(c0)0,02.1017606.1max()0,02.1)u;1(rmax()uu;2,2(c

C o n t i n u i n g b a c k w a r d t h r o u g h t h e t r e e ,

.002353.022406.1002406).2/1(002406).2/1()d;2,1(c

0)u;1(r0)2/1(0)2/1()u;2,1(c

F i n a l l y , a t t i m e 0 , t h e c a p l e t ' s v a l u e i s

.001153.

02.1002353).2/1(0)2/1()2,0(c

42Figure 13.4: An Example of a Two-Period Caplet with a 1.02 Strike. The synthetic caplet portfolio in the

money market account and three-period zero-coupon bond (n0(t;st), n3(t;st)) is given under each node.

time 0 1 2

.001153 (.211214, -.227376)

0 (0, 0)

.002353 (.002307, 0)

1/2

1/2

r(0) = 1.02

0

1/2

1/2

r(1;u) = 1.017606

0

.002406

1/2

1/2

r(1;d) = 1.022406

.002406

43

W e c a n s y n t h e t i c a l l y c r e a t e t h i s t w o - p e r i o d c a p l e tw i t h t h e m o n e y m a r k e t a c c o u n t a n d at h r e e - p e r i o d z e r o - c o u p o n b o n d .

A t t i m e 1 , s t a t e u n o p o s i t i o n i s r e q u i r e d . A t t i m e1 , s t a t e d t h e n u m b e r o f t h r e e - p e r i o d z e r o - c o u p o nb o n d s i s

.0976147.981169.002406.002406.

)dd;3,2(P)du;3,2(P)dd;2,2(c)du;2,2(c)d;1(3n

T h e n u m b e r o f u n i t s o f t h e m o n e y m a r k e t a c c o u n th e l d i s :

.002307.

02.1)957211(.0002353.

)1(B)d;3,1(P)d;1(3n)d;2,1(c)d;1(0n

44

A t t im e 0 , t h e c a lc u la t io n s a r e

,227376.957211.965127.

002353.0)d;3,1(P)u;3,1(P

)d;2,1(c)u;2,1(c)0(3n

a n d

.211214.)942322(.227376.001153.

)3,0(P)0(3n)2,0(c)0(0n

45Figure 13.5: An Example of a Three-Period Caplet with a 1.02 Strike. The Synthetic Caplet Portfolio in the

Money Market Account and Four-Period Zero-Coupon Bond (n0(t;st), n4(t;st)) is given under each node.

time 0 1 2 3

46

T h e i n t e r e s t r a t e c a p ' s v a l u e i s t h e s u m o f t h et h r e e s e p a r a t e c a p l e t s ' v a l u e s , i . e . ,

.dollars 002284.001131.001153.0)3,0(c)2,0(c)1,0(c)3,0(I

47

D Interest Rate Floors An interest rate floor is a provision often associated with a floating-rate loan that guarantees that a minimum interest payment of k-1 is made, where k is 1 plus a percentage. Interest rate floors trade separately. Consider an interest rate floor with floor rate k and maturity date * on the floating-rate loan of Table 13.1. This interest rate floor can be decomposed into the sum of * floorlets.

48

A f l o o r l e t i s a n i n t e r e s t r a t e f l o o r s p e c i f i c t o o n l y as i n g l e t i m e p e r i o d .

T h e f l o o r l e t i s a E u r o p e a n p u t o n t h e s p o t i n t e r e s tr a t e w i t h s t r i k e p r i c e k a n d m a t u r i t y t h e d a t e o ft h e s i n g l e t i m e p e r i o d .

F o r e x a m p l e , a f l o o r l e t w i t h m a t u r i t y T a n d s t r i k ek h a s a t i m e T c a s h f l o w o f .0,1Trkmax

T h i s c a s h f l o w i s k n o w n a t t i m e T - 1 b e c a u s e t h es p o t r a t e i s k n o w n a t t i m e T - 1 .

49

T h e a r b i t r a g e - f r e e v a l u e o f t h e T - m a t u r i t y f l o o r l e t a t t i m e t i s o b t a i n e d u s i n g t h e r i s k - n e u t r a l v a l u a t i o n p r o c e d u r e :

.tBTB0,1TrkmaxtE~

ts;T,td ( 1 3 . 7 )

A n i n t e r e s t r a t e f l o o r o n t h e f l o a t i n g - r a t e l o a n i n T a b l e 1 3 . 1 e q u a l s t h e s u m o f t h e v a l u e s o f t h e * f l o o r l e t s o f w h i c h i t i s c o m p o s e d .

50

E X A M P L E : F L O O R V A L U A T I O N . A g a i n , c o n s i d e r F i g u r e 1 3 . 2 . A s b e f o r e , w e k n o w t h a t t h i s e v o l u t i o n i s a r b i t r a g e - f r e e . C o n s i d e r a n i n t e r e s t r a t e f l o o r w i t h m a t u r i t y d a t e * = 3 a n d s t r i k e k = 1 . 0 1 7 5 . T h i s i n t e r e s t r a t e f l o o r c a n b e d e c o m p o s e d i n t o t h r e e f l o o r l e t s : o n e a t t i m e 1 , o n e a t t i m e 2 , a n d o n e a t t i m e 3 . W e v a l u e a n d d i s c u s s t h e s y n t h e t i c c o n s t r u c t i o n o f e a c h f l o o r l e t i n t u r n . T h e f l o o r l e t a t t i m e 1 , d ( 0 , 1 ) , h a s z e r o v a l u e . F o r m a l l y ,

.002.10)2/1()0(2/1

)0(r0),0(r0175.1max0E~

)1,0(d

51

N e x t , c o n s i d e r t h e f l o o r l e t w i t h m a t u r i t y a t t i m e 2 . B y e x p r e s s i o n ( 1 3 . 7 ) i t s v a l u e a t t i m e 2 i s z e r o u n d e r a l l s t a t e s ; i . e . ,

.0)0),d;1(r0175.1max()dd;2,2(d0)0,022406.10175.1max()0),d;1(r0175.1max()du;2,2(d

0)0),u;1(r0175.1max(ud;2,2d00,017606.10175.1max0),u;1(r0175.1maxuu;2,2d

H e n c e , a t t i m e 1 a n d t i m e 0 i t s v a l u e i s a l s o z e r o . T h e c a l c u l a t i o n s f o r t h e r e m a i n i n g t h r e e - p e r i o d f l o o r l e t a r e c o n t a i n e d i n F i g . 1 3 . 6 .

52Figure 13.6: An Example of a Three-Period Floorlet with a 1.0175 Strike. The Synthetic Floorlet Portfolio in the Money Market Account and Four-Period Zero-Coupon Bond (n0(t;st), n4(t;st)) is given under each node.

time 0 1 2 3

.000348 (-.063085, .068662)

.000711 (-.183392, .198175)

0 (0, 0)

r(1;u) = 1.017606

r(1;d) = 1.022406

0 (0, 0)

0 (0, 0)

.001446 (.001393, 0)

0 (0, 0)

max(1.0175-1.016031, 0) = .001469

max(1.0175-1.016031, 0) = .001469

r(2;uu) = 1.016031

max(1.0175-1.020397, 0) = 0

max(1.0175-1.020393, 0) = 0

max(1.0175-1.019193, 0) = 0

max(1.0175-1.019193, 0) = 0

max(1.0175-1.024436, 0) = 0

max(1.0175-1.024436, 0) = 0

r(2;ud) = 1.020393

r(2;du) = 1.019193

r(2;dd) = 1.024436

r(0) = 1.02

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

53

T h e t i m e 3 p a y o f f s t o t h e f l o o r l e t , u s i n g e x p r e s s i o n ( 1 3 . 7 ) , a r e

0)0),dd;1(r0175.1max()ddd;3,3(d0)0,024436.10175.1max()0),dd;1(r0175.1max()ddu;3,3(d

0)0),du;1(r0175.1max()dud;3,3(d0)0,019193.10175.1max()0),du;1(r0175.1max()duu;3,3(d

0)0),ud;1(r0175.1max()udd;3,3(d0)0,020393.10175.1max()0),ud;1(r0175.1max()udu;3,3(d

001469.)0),uu;1(r0175.1max()uud;3,3(d001469.)0,016031.10175.1max()0),uu;1(r0175.1max()uuu;3,3(d

54

T h e f l o o r l e t h a s a p o s i t i v e v a l u e o n l y a t t i m e 2 ,s t a t e u u . I t s v a l u e i s

.001446.016031.1001469).2/1(001469).2/1()uu;3,2(d

C o n t i n u i n g b a c k w a r d i n t h e t r e e , t h e f l o o r l e t h a s ap o s i t i v e v a l u e o n l y a t t i m e 1 , s t a t e u :

.100071.017606.10)2/1(001446).2/1()u;3,1(d

F i n a l l y , i t s t i m e 0 v a l u e i s

.000348.02.10)2/1(000711).2/1()3,0(d

55

T o s y n t h e t i c a l l y c o n s t r u c t t h e f l o o r l e t , w e u s e t h ef o u r - p e r i o d z e r o - c o u p o n b o n d a n d t h e m o n e ym a r k e t a c c o u n t . T h e c a l c u l a t i o n s a r e a s f o l l o w s :

A t t i m e 2 , s t a t e u u ,

.001393.037958.1)984222(.0001446.

)uu;2(B)uu;4,2(P)uu;2(4n)uu;3,2(d)uu;2(0n

0981381.985301.001469.001469.

)uud;4,3(P)uuu;4,3(P)uud;3,3(d)uuu;3,3(d)uu;2(4n

A t t i m e 1 , s t a t e u ,

.183392.

02.1947497).198175(.000711.

)u;1(B)u;4,1(P)u;1(4n)u;3,1(d)u;1(0n

198175.960529.967826.

0001446.)ud;4,2(P)uu;4,2(P

)ud;3,2(d)uu;3,2(d)u;1(4n

56

F i n a l l y , a t t i m e 0 ,

.063085.)923845(.068662.000348.

)4,0(P)0(4n)3,0(d)0(0n

068662.937148.947497.

0000711.)d;4,1(P)u;4,1(P

)d;3,1(d)u;3,1(d)0(4n

T h e i n t e r e s t r a t e f l o o r ' s v a l u e i s t h e s u m o f t h et h r e e s e p a r a t e f l o o r l e t s ' v a l u e s ; i . e . ,

.dollars 000348.000348.00)3,0(d)2,0(d)1,0(d)3,0(J

57

E Swaptions This section values swaptions, which are options issued on interest rate swaps. An interest rate swap changes floating to fixed rate loans or vice-versa. Swaptions, then, are “insurance contracts” issued on the decision to enter into a fixed rate or floating rate loan in the future. Consider the swap receiving fixed and paying floating discussed earlier in this chapter. This swap has a swap rate C/L, a maturity date T, and a principal equal to L dollars. Its time t value is denoted by S(t) and is given in expression (13.4). This simplest type of swaption is a European call option on this swap.

58

A E u r o p e a n c a l l o p t i o n o n t h e s w a p S ( t ) w i t h a n e x p i r a t i o n d a t e T * T a n d a s t r i k e p r i c e o f K d o l l a r s i s d e f i n e d b y i t s p a y o f f a t t i m e T * , w h i c h i s e q u a l t o m a x [ S ( T * ) - K , 0 ] . T h e a r b i t r a g e - f r e e v a l u e o f t h e s w a p t i o n i s o b t a i n e d u s i n g t h e r i s k - n e u t r a l v a l u a t i o n p r o c e d u r e ; i . e . ,

)t(B* )T(B/]0,K* )T(Sma x [tE~

)t(O ( 1 3 . 9 )

A s i m p l e m a n i p u l a t i o n o f e x p r e s s i o n ( 1 3 . 9 ) g e n e r a t e s a n i m p o r t a n t i n s i g h t .

59

R e c a l l t h a t a s w a p c a n b e v i e w e d a s a l o n g p o s i t i o n i n a c o u p o n b e a r i n g b o n d a n d a s h o r t p o s i t i o n i n t h e m o n e y m a r k e t a c c o u n t . S u b s t i t u t i n g t h i s i n s i g h t g i v e s :

)t(B* )T(B/]0),KL(* )T(ma x [tE~

)t(O B

( 1 3 . 1 0 )

60

This shows that :

a European call option with strike K and expiration T* on a swap receiving fixed and paying floating with maturity T, principal L, and swap rate C/L

is equivalent to

a European call option with a strike L+K and an expiration date of T* on a (noncallable) coupon bond B(t;st) with maturity T, coupon C, and principal L.

The pricing and synthetic construction of these bond options was discussed in Chapter 11. Thus, we have already studied the pricing and synthetic construction of swaptions

61

Figure 13.3: An Example of a Swap Receiving Fixed and Paying Floating with Maturity Time 3, Principal $100, and Swap Rate .02. Given first is the swap's value, then the swap's cash flow. The synthetic swap

portfolio in the money market account and three-period zero-coupon bond (n0(t; st), n3(t; st)) is given under each node.

time 0 1 2 3

0 .396930

0 .396930

0 -.039285

0 -.039285 0 .080719

0 .080719 0 -.443609

0 -.443609

S(0) = 0 Cash Flow = 0

(-97.215294, 103.165648)

.408337 0

(-96.112355, 102)

-.408337 0

(-96.121401, 102)

.390667

.239442 (.376381, 0)

-.038500 .239442

(-.037092, 0)

.079199 -.240572

(.075945, 0)

-.433028 -.240572

(-.415234, 0)

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

r(0) = 1.02

1.017606

1.016031

1.022406

1.020393

1.019193

1.024436

62

EXAMPLE: EUROPEAN CALL OPTION ON A SWAP. Recall that the swap in this example is receiving fixed and paying floating. It has a swap rate C/L = 0.02, a maturity date T = 3, and a principal L = 100. The evolution of the zero-coupon bond price curve is as given in Figure 13.2. Consider a European call option on this swap. Let the maturity date of the option be T* = 1, and let the strike price be K = 0. Using the risk-neutral valuation procedure, the value of the swaption is as follows:

63

T i m e 1 , s t a t e u :O ( 1 ; u ) = m a x [ S ( 1 ; u ) - K , 0 ] = m a x [ . 4 0 8 3 3 7 , 0 ] =. 4 0 8 3 3 7

T i m e 1 , s t a t e d :O ( 1 ; d ) = m a x [ S ( 1 ; d ) - K , 0 ] = m a x [ - . 4 0 8 3 3 7 , 0 ] = 0

T i m e 0 :

.200165.02.1/]0)2/1()408337)(.2/1[()0(r/)]d;1(O)2/1()u;1(O)2/1[()0(O

W e c a n s y n t h e t i c a l l y c r e a t e t h i s s w a p t i o n w i t h t h em o n e y m a r k e t a c c o u n t a n d a t h r e e - p e r i o dz e r o - c o u p o n b o n d . A t t i m e 0 t h e c a l c u l a t i o n s a r ea s f o l l o w s :

5838.51957211.965127.

0408337.)d;3,1(P)u;3,1(P

)d;1(O)u;1(O)0(3n

.4083.48942322).5838.51(200165.

)0(B/)]3,0(P)0(3n)0(O[)0(0n