Ch13 Swaps, Caps, Floors, And Swaptions

Whitman School of ManagementSyracuse Universityemail: [email protected]: www.som.syr.edu/facstaff/yildiray

CH 13: Swaps, Caps, Floors, and

Swaptions

Yildiray Yildirim

Agenda

• Fixed rate and floating rate loans• Interest rate swaps

• Swap valuation

• Swap rate

• Synthetic swap

• 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. Caps help limit exposure to rising interest yet allow the borrower to benefit when rates fall.

• Interest rate floors: An interest rate floor is a provision often associated with a floating‑rate loan that guarantees that a minimum interest payment

• Swaption: a call option or a put option on a swap.

2

Example: Currency risk hedging with swap

Situation

3

German bank wants to lend in U.S.

Not well-known in US$ capital market

Will fund by borrowing in EURO

Exposed to rising DM (falling US$)

Assets Liabilities

Loans (5-year, fixed rate) US$100MM @6.8%

Note (5-year, fixed-rate)EU180 MM @4.8%

4

US$ Loans

German Bank

EUInterest

EU Note

US$Interest

EU180 million

US$100 million

Dealer

EU180 million

$100 million

Initial principal exchange

6.1% (US$)

4.8% (EU)

Payments during swap

US$100 million

EU180 million

Final principal exchange

• 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. Is there a way you can change this floating rate loan into a fixed rate loan, without retiring the debt and incurring large transaction costs (and a loss on your balance sheet)?

• The solution is to enter into a fixed for floating rate swap or simultaneously purchase caps and floors with predetermined strikes.

• If you had thought about this earlier, you could have entered into a swaption at the time the loan

was made to protect the company from such a crisis.

5

(A) Fixed-Rate and Floating-Rate Loans

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

• We define a floating rate loan for L dollars (the principal) with maturity date T to be a debt contract that obligates the borrower to pay the spot rate of interest 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.

• In our frictionless and default‑free setting, this floating rate loan is equivalent to shorting L units of the money market account and distributing the gains (paying out the spot rate of interest times L dollars) every period.

• Paying out the interest as a cash flow maintains the value of the short position in the money market account at L dollars. At time T the short position is closed out.

6

7

0 1 2time

Borrow +1

T

Pay interest –[r(0)–1] –[r(1)–1] –[r(T–1)–1] Pay principal –1

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

• As the floating‑rate is market determined, it costs 0 dollars to enter into a floating‑rate loan contract. Computing the present value of the cash flows paid on a floating-rate loan with a dollar principal and maturity date T makes this same point.

• Using the risk‑neutral valuation procedure, the present value of the cash flows to the floating rate loan is:

Expression shows that the value of the cash flows from the floating‑rate loan at time t equals one dollar, which is the amount borrowed.

• 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

1 [ ( ) 1] 1( ) ( )( 1) )

)(

( 1T r jE B t E B tt tB j

tr B Tj t

(V 13.1)

9

0 1 2 time

Borrow B T …

(0)

Pay interest –C –C … –C Pay principal –L

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

• Computing the present value of the cash flows paid on the fixed-rate loan can make the same point.

• To make the coupon rate compatible with the rate convention used in this

book, we define c=(1+C/L)

• Using the risk‑neutral valuation procedure,

• Expression (13.2) shows that the value of the cash flows to a fixed‑rate loan at time t equals B(t), which is the amount borrowed.

10

1( ) ( )

( 1) ( )

1( ,

(

, )1) ( )

)

(

T C LE B t E B tt tB j B Tj t

TCP t j LP

j

t

t Tt

t

(

c

)

V

13.2B

(B) Interest Rate Swaps

An interest rate swap is a financial contract that obligates the holder to receive fixed‑rate loan payments and pay floating‑rate loan payments (or vice versa).

1 Swap Valuation• Consider an investor who has a fixed‑rate loan with interest rate c, a principal

of L dollars and a maturity date T. The cash payment at every intermediate date t is C = (c-1)L.

• The investor wants to exchange this fixed‑rate loan for a floating‑rate loan with principal L dollars, maturity date T, and floating interest payments of L(r(t‑1) ‑ 1) dollars per period.

• He does this by entering into a swap receiving fixed and paying floating.

11

12

FIXED RATE LOAN

SWAP

pay fixedreceive fixed

pay floatinginvestor

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

13

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

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

• Let S(t) represent the value of the swap at time t .

• The value of the swap at any period t is

S(t) = B(t) ‑ L.

• Computing the present value of the cash flows from the swap can make this same point. Using the risk‑neutral valuation procedure,

• Defining c1+C/L to be one plus the coupon rate on the fixed‑rate loan, we can rewrite this as

14

1[ ( ( ) 1) ] ( )(

(1

))

T C r j LE B tt B jjS t

t

(13.3)

1[ ( )] ( ) .( 1

) ( ))

(T c r j LE B tt j t

tB

Lj

S t

(13.4)B

2 The Swap Rate

• The swap rate is defined to be that coupon rate C/L such 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 this determination of the swap rate is

under the assumption of no default risk for either counterparty to the swap contract.

EXAMPLE: SWAP VALUATION

15

16

.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

Figure 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.

• 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.

• We first compute the swap's value for an arbitrary coupon payment of C:

• Setting S(0) = 0 and solving for C yields

C = 5.7678/2.8838 = 2.

• The swap rate is C/L = 2/100 = 0.02.

17

(0) (0) 100(0,1) (0,2) ( 100) (0,3) 100

[.980392 .961169 .942322] 100(.942322) 100(2.8838) 5.7678

SCP CP C PCC

B

18

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

time 0 1 2 3

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.

(3; ) 100 100 0 (3; ) ( (2; ) 1) 2 1.60307 .39693

S uuu L Lcash flow uuu C r uu L

(2;)(2;)102(2,3;)100102(.984222)100.390667

(2;)[(1;)1]21.76056.239442

SuuuuLPuu

cashflowuuCruL

B

(1; ) (1; ) 2 (1,2; ) 102 (1,3; ) 1002(.982699) 102(.965127) 100 .408337

(1; ) [ (0) 1] 2 2 0

S u u L P u P u

cash flow u C r L

B

• We receive fixed and pay floating. The calculations are as follows.

At time 3, for each possible state:

19

(3; ) 100 100 0 (3; ) ( (2; ) 1) 2 1.60307 .39693

(3; ) 100 100 0 (3; ) 2 ( (2; ) 1)100 .39693

(3; ) 100 100 0 (3; ) 2 ( (2; ) 1)100 .039285

(3; ) 100

S uuu L Lcash flow uuu C r uu L

S uudcash flow uud r uu

S uducash flow udu r ud

S ddu

100 0

(3; ) 2 ( (2; ) 1)100 .443609(3; ) 100 100 0

(3; ) 2 ( (2; ) 1)100 .443609

cash flow ddu r ddS ddd

cash flow ddd r dd

Continuing backward through the tree, at time 2:

Finally, at time 1:

20

(2; ) (2; ) 102 (2,3; ) 100102(.984222) 100 .390667

(2; ) [ (1; ) 1] 2 1.76056 .239442(2; ) 102 (2,3; ) 100 .038500

(2; ) 2 ( (1; ) 1)100 .239442(2; ) 102 (2,3; ) 1

S uu uu L P uu

cash flow uu C r u LS ud P ud

cash flow ud r uS du P du

B

00 .079199 (2; ) 2 ( (1; ) 1)100 .240572

(2; ) 102 (2,3; ) 100 .433028 (2; ) 2 ( (1; ) 1)100 .240572.

cash flow du r dS dd P dd

cash flow dd r d

(1; ) (1; ) 2 (1,2; ) 102 (1,3; ) 1002(.982699) 102(.965127) 100 .408337

(1; ) [ (0) 1] 2 2 0(1; ) 2 (1,2; ) 102 (1,3; ) 100 .408337

(1; ) 2 ( (0) 1)100 0.

S u u L P u P u

cash flow u C r LS d P d P d

cash flow d r

B

3 Synthetic Swaps

• There are numerous ways of creating a swap synthetically.

• The first is to use a buy and hold strategy. This method is to short the money

market account (pay floating) and to synthetically create the coupon bond as a portfolio of zero-coupon bonds. This synthetic swap is independent of any particular model for the evolution of the term structure of interest rates.

• Unfortunately, synthetically constructing the swap via a portfolio of zero-

coupon bonds has two practical problems. One, not all zero-coupon bonds may trade. Two, the initial transaction costs will be high.

21

• The second method is the synthetic construction of swaps using forward contracts 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 date T, contract rate c (one plus a percent), and principal L to be that contract that has a certain payoff of

[r(T-1) – c]L dollars at time T.

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

• The contract rate c is set at the date the contract is initiated, say at time 0. It is set by mutual consent of the counter parties to the contract. At initiation, the contract rate need not give the FRA zero initial value (however, a typical FRA sets the rate at initiation such that the contract has zero value.

• In the case where the value of the contract at initiation is non-zero, the counter parties would sign the contract and the fair value of the FRA is exchanged in cash.

22

• Let us denote the time t value of an FRA with delivery date T and contract rate c with principal 1 dollar as Vf (t,T; c). Using the techniques of chapter 12, the time t value of this FRA is:

23

( -1)-( , ; ) ( )( )

( 1) / ( ) 1/ ( 1),

1 1( , ; ) ( )- ( )( -1) ( )

Re ( , ) (1/ ( )) ( ) :

( , -1)- ( , )

r T cV t T c E B ttf B T

But r T B T B T so

V t T c E B t cE B tt tf B T B T

calling that P t T E B T B t substitution givestP t T cP t T

At ini

,

,

, ' :(0, ; ) (0, -1)- (0, )

tiation the FRA s value would beV T c P T cP T

f

• 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.

• The value of this collection of short FRAs is:

• This is the value of the swap with maturity T and principal L receiving fixed and paying floating at time 0, as expected!

24

(0, ; ) [ (0, 1) (0, )]1 1

[ 1] (0, ) (0, )0

(0)(0)

T TV t c L L P t cP t

ft tT

L L c P t LP Tt

LS

B

• A third method for synthetically creating this swap is to use a dynamic portfolio consisting of a single zero‑coupon bond (for a one-factor model) and the money market account.

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

EXAMPLE: SYNTHETIC SWAP CONSTRUCTION

25

26

0 .396930

0 .396930

0 -.039285

0 -.039285 0 .080719

0 .080719 0 -.443609

0 -.443609


(-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

time 0 1 2 3

(2; ) 0,3

(2; ) (2; ) (2; ) (2,3; ) (2; )0 3[.390667] 1.037958 .376381

n uu and

n uu S uu n uu P uu B u

( (2; ) (2; )) ( (2; ) (2; ))(1; )3 (2,3; ) (2,3; ).630109 .200942 102..984222 .980015

S uu cash flow uu S ud cash flow udn uP uu P ud

(1; ) [ (1; ) (1; ) (1,3; )]/ B(1)0 3[.408337 102(.965127)]/1.02 96.112355.

n u S u n u P u

• We can use a dynamic self-financing trading strategy in the 3-period zero-coupon bond and the money market account using the delta approach.

At time 2, state uu the value of the swap and its cash flow are known for sure. The swap can be synthetically created by holding none of the three‑period zero‑coupon bond,

units of the money market account. The calculations for the remaining states are similar:

27

(2; ) 03(2; ) .038500/1.037958 .0370920(2; ) 03(2; ) .079199/1.042854 .0759450(2; ) 03(2; ) .433028/1.042854 .415234.0

n ud

n ud

n du

n du

n dd

n dd

(2; ) 0,3

(2; ) (2; ) (2; ) (2,3; ) (2; )0 3[.390667] 1.037958 .376381

n uu and

n uu S uu n uu P uu B u

At time 1, state u the number of three‑period zero‑coupon bonds held is

The number of units of the money market account held is

At time 1, state d the calculations are

28

( (2; ) (2; )) ( (2; ) (2; ))(1; )3 (2,3; ) (2,3; ).630109 .200942 102..984222 .980015

S uu cash flow uu S ud cash flow udn uP uu P ud

(1; ) [ (1; ) (1; ) (1,3; )]/ B(1)0 3[.408337 102(.965127)]/1.02 96.112355.

n u S u n u P u

( (2; ) (2; )) ( (2; ) (2; ))(1; )3 (2,3; ) (2,3; ).161373 ( .6736) 102,.981169 .976149

(1; ) [ (1; ) (1; ) (1,3; )]/ B(1)0 3[ .408337 102(.957211)]/1.02 96.121401.

S du cash flow du S dd cash flow ddn dP du P dd

and

n d S d n d P d

At time 0,

Rather than using the 3-period zero-coupon bond, the swap could have be synthetically constructed using any other interest rate sensitive security, for example, a futures or option contract on the 3-period zero-coupon bond.

29

( (1; ) (1; )) ( (1; ) (1; ))(0)3 (1,3; ) (1,3; ).408337 ( .408337) 103.165648

.965127 .957211

(0) [ (0) (0) (0,3)]0 30 103.165648(.942322) 97.215294.

S u cash flow u S d cash flow dnP u P d

n S n P

(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 τ* on the floating‑rate

loan of Table 13.1.

• We can decompose this cap into the sum of τ * caplets.

• A caplet is defined to be an interest rate cap specific to only a single time period. Specifically, it is equivalent to a European call option on the spot interest rate with strike k and maturity the specific date of the single time period.

• For example, a caplet with maturity T and a strike k has a time T cash flow equal to:

• This cash flow is known at time T‑1 because the spot rate is known at time T‑1.

30

• The arbitrage‑free value of the T‑maturity caplet at time t is obtained using the risk‑neutral valuation procedure :

• An interest rate cap on the floating‑rate loan in Table 13.1 is then the

sum of the values of the caplets from which it is composed.

EXAMPLE: CAP VALUATION.

31

, ; max ,0 .1c t s E r kT B tt T B Tt (13.5)

32

.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

Figure 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 .

• We know that this evolution is arbitrage-free.

• Consider an interest rate cap with maturity date τ* = 3 and a strike of k = 1.02.

• This interest rate cap can be decomposed into three caplets: one at time 1,

one at time 2, and one at time 3. I(0,3)=c(0,1)+c(0,2)+c(0,3)

• We value and discuss the synthetic construction of each caplet in turn.

• The caplet at time 1, c(0,1), has zero value. Formally,

33

max (0) 1.02,0 (0)0

(1/ 2)(0) (1/ 2)(0) 1.02 0

(0,1)

.

E rc r

34

.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


(2,2; ) max( (1; ) 1.02,0) max(1.017606 1.02,0) 0c uu r u (2,2;)max((1;)1.02,0)max(1.0224061.02,0).002406 cdurd

35

.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



(1,2; ) (1/ 2)0 (1/ 2)0 (1; ) 0

(1,2; ) (1/ 2).002406 (1/ 2).002406 1.022406 .002353.

c u r u

c d

36

.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



(1/2)0(1/2).0023531.02

.001153

0,

.

(2) c

(1,2; ) (1/ 2)0 (1/ 2)0 (1; ) 0

(1,2; ) (1/ 2).002406 (1/ 2).002406 1.022406 .002353.

c u r u

c d

• Next, consider the caplet with maturity at time 2.

• By expression (13.5), at time 2 its value under each state is as follows:

• Continuing backward through the tree,

• Finally, at time 0, the caplet's value is

37

(2,2; ) max( (1; ) 1.02,0) max(1.017606 1.02,0) 0(2,2; ) max( (1; ) 1.02,0) 0(2,2; ) max( (1; ) 1.02,0) max(1.022406 1.02,0) .002406(2,2; ) max( (1; ) 1.02,0) .002406

c uu r uc ud r uc du r dc dd r d

(1,2; ) (1/ 2)0 (1/ 2)0 (1; ) 0

(1,2; ) (1/ 2).002406 (1/ 2).002406 1.022406 .002353.

c u r u

c d

(0,2) (1/ 2)0 (1/ 2).002353 1.02

.001153.

c

38

.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

time 0 1 2

Figure 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.

(2,2; ) (2,2; ) .002406 .002406(1; ) 0.3 .981169 .976147(2,3; ) (2,3; )c du c ddn dP du P dd

(1; ) (1,2; ) (1; ) (1,3; ) (1) .002353 0(.957211) 1.020 3.002307.

n d c d n d P d B

(1,2; ) (1,2; ) 0 .002353(0) .227376,3 .965127 .957211(1,3; ) (1,3; )

(0) (0,2) (0) (0,3) .001153 .227376(.942322) .211214.0 3

c u c dnP u P d

n c n P

• We can synthetically create this two‑period caplet with the money market account and a three‑period zero‑coupon bond.

At time 1, state u no position is required. At time 1, state d the number of three‑period zero‑coupon bonds is

The number of units of the money market account held is:

At time 0:

39

(2,2; ) (2,2; ) .002406 .002406(1; ) 0.3 .981169 .976147(2,3; ) (2,3; )c du c ddn dP du P dd

(1; ) (1,2; ) (1; ) (1,3; ) (1) .002353 0(.957211) 1.020 3.002307.

n d c d n d P d B

(1,2; ) (1,2; ) 0 .002353(0) .227376,3 .965127 .957211(1,3; ) (1,3; )

(0) (0,2) (0) (0,3) .001153 .227376(.942322) .211214.0 3

c u c dnP u P d

n c n P

40

.001131 (.173297,–.186358)

.000189 (.049195, -.052760)

.002118 (.468115, -.507241)

1/2

1/2

r(0) = 1.02

1/2

1/2

r(1;u) = 1.017606

1/2

1/2

r(1;d) = 1.022406

0 (0, 0)

.000385 (.000371, 0)

0 (0, 0)

.004330 (.004152, 0)

max[1.016031-1.02, 0] = 0

max[1.016031-1.02, 0] = 0

r(2;uu) = 1.016031

1/2

1/2

max[1.020393-1.02, 0] = .000393

max[1.020393-1.02, 0] = .000393

max[1.019193-1.02, 0] = 0

max[1.019193-1.02, 0] = 0

max[1.024436-1.02, 0] = .004436

max[1.024436-1.02, 0] = .004436

r(2;ud) = 1.020393

1/2

1/2

r(2;du) = 1.019193

1/2

1/2

r(2;dd) = 1.024436

1/2

1/2

time 0 1 2 3

Figure 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.

• The interest rate cap's value is the sum of the three separate caplets' values, i.e.,

41

(0,1) (0,2) (0,3)0 .001153 .001131 .

(0

0,3022

)84 .

c c cdolla s

Ir

(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 τ* on the floating-rate loan of Table 13.1.

• This interest rate floor can be decomposed into the sum of τ * floorlets.

• A floorlet is an interest rate floor specific to only a single time period. The floorlet is a European put on the spot interest rate with strike price k and maturity the date of the single time period.

• For example, a floorlet with maturity T and strike k has a time T cash flow of

• This cash flow is known at time T‑1 because the spot rate is known at time T‑1.

42

max 1 ,0 .k r T

• The arbitrage‑free value of the T‑maturity floorlet at time t is obtained using the risk‑neutral valuation procedure:

• An interest rate floor on the floating‑rate loan in Table 13.1 equals the sum of

the values of the τ* floorlets of which it is composed.

43

1, ; m .0ax ,d t T s E B T Bk r T tt t (13.7)

EXAMPLE: FLOOR VALUATION.

• Again, consider Figure 13.2. As before, we know that this evolution is arbitrage-free.

• Consider an interest rate floor with maturity date τ* = 3 and strike k = 1.0175.

• This interest rate floor can be decomposed into three floorlets: one at time 1, one at time

2, and one at time 3.

• We value and discuss the synthetic construction of each floorlet in turn.

• The floorlet at time 1, d(0, 1), has zero value. Formally,

44

max 1.0175 (0),0 (0)01/ 2(0) (1/ 2)0 1.

(

0

1

2 0.

0, )d E r r

• Next, consider the floorlet with maturity at time 2. By expression (13.7) its value at time 2 is zero under all states; i.e.,

• Hence, at time 1 and time 0 its value is also zero.

• The calculations for the remaining three‑period floorlet are contained in Fig.

13.6.

45

2,2; max 1.0175 (1; ),0 max 1.0175 1.017606,0 0

2,2; max(1.0175 (1; ),0) 0

(2,2; ) max(1.0175 (1; ),0) max(1.0175 1.022406,0) 0(2,2; ) max(1.0175 (1; ),0) 0.

d uu r u

d ud r u

d du r dd dd r d

46

.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

Figure 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.

• The time 3, 2,1 and 0 payoffs to the floorlet, using expression (13.7), are

47

(3,3; ) max(1.0175 (1; ),0) max(1.0175 1.016031,0) .001469(3,3; ) max(1.0175 (1; ),0) .001469(3,3; ) max(1.0175 (1; ),0) max(1.0175 1.020393,0) 0(3,3; ) max(1.0175 (1; ),0) 0(3,3;

d uuu r uud uud r uud udu r udd udd r udd

) max(1.0175 (1; ),0) max(1.0175 1.019193,0) 0(3,3; ) max(1.0175 (1; ),0) 0(3,3; ) max(1.0175 (1; ),0) max(1.0175 1.024436,0) 0(3,3; ) max(1.0175 (1; ),0) 0

duu r dud dud r dud ddu r ddd ddd r dd

(2,3; ) (1/ 2).001469 (1/ 2).001469 1.016031 .001446.

(1,3; ) (1/ 2).001446 (1/ 2)0 1.017606 .000711.

(0,3) (1/ 2).000711 (1/ 2)0 1.02 .000348.

d uu

d u

d

• To synthetically construct the floorlet, we use the four‑period zero‑coupon bond and the money market account. The calculations are as follows: At time 2 state uu; at time 1 state u; at time 0; floor value:

48

(3,3; ) (3,3; ) .001469 .001469(2; ) 04 .985301 .981381(3,4; ) (3,4; )

(2; ) (2,3; ) (2; ) (2,4; ) (2; )0 4.001446 0(.984222) 1.037958 .001393.

(2,3; ) (2,3; )(1; )4

d uuu d uudn uuP uuu P uud

n uu d uu n uu P uu B uu

d uu d udn u

.001446 0 .198175.967826 .960529(2,4; ) (2,4; )

(1; ) (1,3; ) (1; ) (1,4; ) (1; ) .000711 (.198175).947497 1.02 .183392.0 4(1,3; ) (1,3; ) .000711 0(0)4 .947497 .93(1,4; ) (1,4; )

P uu P ud

n u d u n u P u B u

d u d dnP u P d

.0686627148

(0) (0,3) (0) (0,4)0 4.000348 .068662(.923845) .063085.

(0,1) (0,2) (0,3(0,3 )0 0 .000348 .00034

)8 .

J

n d n P

d d ddollars

(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.

49

• A European call option on the swap S(t) with an expiration date T* T and a strike price of K dollars is defined by its payoff at time T*, which is equal to max [S(T*)‑K,0].

• The arbitrage‑free value of the swaption is obtained using the risk‑neutral valuation procedure; i.e.,

• A simple manipulation of expression (13.9) generates an important insight.

• Recall that a swap can be viewed as a long position in a coupon bearing bond and a short position in the money market account.

• Substituting this insight gives:

50

max[ ( *) ,0]/ ( *) )( ) (E S T K B TO ttt B (13.9)

max[ ( *) ( ),0]/ ( *( ) ) ( )E T L K B T B tO t t (13.10)B

• 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

51

52

0 .396930

0 .396930

0 -.039285

0 -.039285 0 .080719

0 .080719 0 -.443609

0 -.443609


(-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

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.

53

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:

54

Time 1, state u:

O(1;u) = max[S(1;u)-K,0] = max[.408337, 0] = .408337

Time 1, state d:

O(1;d) = max[S(1;d)-K,0] = max[-.408337, 0] = 0

Time 0:

• We can synthetically create this swaption with the money market account and

a three‑period zero‑coupon bond. At time 0 the calculations are as follows:

(0) [(1/ 2) (1; ) (1/ 2) (1; )]/ (0)[(1/ 2)(.408337) (1/ 2)0]/1.02 .200165.

O O u O d r

(1; ) (1; ) .408337 0(0) 51.58383 .965127 .957211(1,3; ) (1,3; )(0) [ (0) (0) (0,3)]/ (0)0 3

.200165 (51.5838).942322 48.4083.

O u O dnP u P d

n O n P B

Ch13 Swaps, Caps, Floors, And Swaptions

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