© 2004 South-Western Publishing1

Chapter 14

Swap Pricing

2

Outline

Intuition into swap pricing Solving for the swap price Valuing an off-market swap Hedging the swap Pricing a currency swap

3

Intuition Into Swap Pricing

Swaps as a pair of bonds Swaps as a series of forward contracts Swaps as a pair of option contracts

4

Swaps as A Pair of Bonds

If you buy a bond, you receive interest If you issue a bond you pay interest

In a plain vanilla swap, you do both– You pay a fixed rate– You receive a floating rate– Or vice versa

5

Swaps as A Pair of Bonds (cont’d)

A bond with a fixed rate of 7% will sell at a premium if this is above the current market rate

A bond with a fixed rate of 7% will sell at a discount if this is below the current market rate

6

Swaps as A Pair of Bonds (cont’d)

If a firm is involved in a swap and pays a fixed rate of 7% at a time when it would otherwise have to pay a higher rate, the swap is saving the firm money

If because of the swap you are obliged to pay more than the current rate, the swap is beneficial to the other party

7

Swaps as A Series of Forward Contracts

A forward contract is an agreement to exchange assets at a particular date in the future, without marking-to-market

An interest rate swap has known payment dates evenly spaced throughout the tenor of the swap

8

Swaps as A Series of Forward Contracts (cont’d)

A swap with a single payment date six months hence is no different than an ordinary six-month forward contract– At that date, the party owing the greater

amount remits a difference check

9

Swaps as A Pair of Option Contracts

Assume a firm buys a cap and writes a floor, both with a 5% striking price

At the next payment date, the firm will – Receive a check if the benchmark rate is above

5%– Remit a check if the benchmark rate is below 5%

10

Swaps as A Pair of Option Contracts (cont’d)

The cash flows of the two options are identical to the cash flows associated with a 5% fixed rate swap– If the floating rate is above the fixed rate, the

party paying the fixed rate receives a check– If the floating rate is below the fixed rate, the

party paying the floating rate receives a check

11

Swaps as A Pair of Option Contracts (cont’d)

Cap-floor-swap parity

5%+ =

5% 5%

Write floor Buy cap Long swap

12

Solving for the Swap Price

Introduction The role of the forward curve for LIBOR Implied forward rates Initial condition pricing Quoting the swap price Counterparty risk implications

13

Introduction

The swap price is determined by fundamental arbitrage arguments– All swap dealers are in close agreement on what

this rate should be

14

The Role of the Forward Curve for LIBOR LIBOR depends on when you want to begin

a loan and how long it will last

Similar to forward rates:– A 3 x 6 Forward Rate Agreement (FRA) begins in

three months and lasts three months (denoted by )

– A 6 x 12 FRA begins in six months and lasts six months (denoted by )

63 f

126 f

15

The Role of the Forward Curve for LIBOR (cont’d)

Assume the following LIBOR interest rates:

Spot (0f3) 5.42%

Six Month (0f6) 5.50%

Nine Month (0f9) 5.57%

Twelve Month (0f12) 5.62%

16

The Role of the Forward Curve for LIBOR (cont’d)

LIBOR yield curve

Months0 6 9 12

5.42

5.50

5.575.62

spot

0 x 6

0 x 90 x 12

%

17

Implied Forward Rates

We can use these LIBOR rates to solve for the implied forward rates– The rate expected to prevail in three months, 3f6

– The rate expected to prevail in six months, 6f9

– The rate expected to prevail in nine months, 9f12

The technique to obtain the implied forward rates is called bootstrapping

18

Implied Forward Rates (cont’d) An investor can

– Invest in six-month LIBOR and earn 5.50%– Invest in spot, three-month LIBOR at 5.42% and

re-invest for another three months at maturity

If the market expects both choices to provide the same return, then we can solve for the implied forward rate on the 3 x 6 FRA

19

Implied Forward Rates (cont’d)

The following relationship is true if both alternatives are expected to provide the same return:

2606330

41

41

41

fff

20

Implied Forward Rates (cont’d)

Using the available data:

%58.54

0550.14

14

0542.1

63

263

f

f

21

Implied Forward Rates (cont’d)

Applying bootstrapping to obtain the other implied forward rates:– 6f9 = 5.71%– 9f12 = 5.77%

22

Implied Forward Rates (cont’d) LIBOR forward rate curve

Months0 3 6 9

5.42

5.58

5.715.77

spot

3 x 6

6 x 99 x 12

%

12

23

Initial Condition Pricing

An at-the-market swap is one in which the swap price is set such that the present value of the floating rate side of the swap equals the present value of the fixed rate side– The floating rate payments are uncertain

Use the spot rate yield curve and the implied forward rate curve

24

Initial Condition Pricing (cont’d)

At-the-Market Swap Example

A one-year, quarterly payment swap exists based on actual days in the quarter and a 360-day year on both the fixed and floating sides. Days in the next 4 quarters are 91, 90, 92, and 92, respectively. The notional principal of the swap is $1.

Convert the future values of the swap into present values by discounting at the appropriate zero coupon rate contained in the forward rate curve.

25

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)First obtain the discount factors:

013701.10542.3609111 3

R

027653.10550.360

909111 6

R

26

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)First obtain the discount factors:

042239.10557.360

92909111 9

R

056981.10562.360

9292909111 12

R

27

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)Next, apply the discount factors to both the fixed and floating rate sides of the swap to solve for the swap fixed rate that will equate the two sides:

055042.0013951.014001.013575.013515.

056981.136092%77.5

042239.136092%71.5

027653.136090%58.5

013701.136091%42.5

floating

PV

28

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)Apply the discount factors to both the fixed and floating rate sides of the swap to solve for the swap fixed rate that will equate the two sides:

XXXXX

XXXXPV

979612.0241779.245199.243273.249361.056981.1

36092%

042239.136092%

027653.136090%

013701.136091%

fixed

29

Initial Condition Pricing (cont’d)

At-the-Market Swap Example (cont’d)Solving the two equations simultaneously for X gives X = 5.62%. This is the equilibrium swap fixed rate, or swap price.

30

Quoting the Swap Price

Common practice to quote the swap price relative to the U.S. Treasury yield curve– Maturity should match the tenor of the swap

There is both a bid and an ask associated with the swap price– The dealer adds a swap spread to the

appropriate Treasury yield

31

Counterparty Risk Implications From the perspective of the party paying

the fixed rate– Higher when the floating rate is above the fixed

rate

From the perspective of the party paying the floating rate– Higher when the fixed rate is above the floating

rate

32

Valuing an Off-Market Swap

The swap value reflects the difference between the swap price and the interest rate that would make the swap have zero value– As soon as market interest rates change after a

swap is entered, the swap has value

33

Valuing an Off-Market Swap (cont’d)

An off-market swap is one in which the fixed rate is such that the fixed rate and floating rate sides of the swap do not have equal value– Thus, the swap has value to one of the

counterparties

34

Valuing an Off-Market Swap (cont’d)

If the fixed rate in our at-the-market swap example was 5.75% instead of 5.62%– The value of the floating rate side would not

change– The value of the fixed rate side would be lower

than the floating rate side– The swap has value to the floating rate payer

35

Hedging the Swap

Introduction Hedging against a parallel shift in the yield

curve Hedging against any shift in the yield curve Tailing the hedge

36

Introduction

If interest is predominantly in one direction (e.g., everyone wants to pay a fixed rate), then the dealer stands to suffer a considerable loss– E.g., the dealer is a counterparty to a one-year,

$10 million swap with quarterly payments and pays floating

The dealer is hurt by rising interest rates

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Introduction (cont’d)

The dealer can hedge this risk in the eurodollar futures market– Based on LIBOR– If the dealer faces the risk of rising rates, he

could sell eurodollar futures and benefit from the decline in value associated with rising interest rates

38

Hedging Against A Parallel Shift in the Yield Curve

Assume the yield curve shifts upward by one basis point– The present value of the fixed payments

decreases– The present value of the floating payments also

decreases, but by a smaller amount– The net effect hurts the floating rate payer

39

Hedging Against A Parallel Shift in the Yield Curve (cont’d)

The dealer could sell eurodollar (ED) futures to hedge– Need one ED futures contract for every $25

change in value of the swap– Need to choose between the various ED futures

contracts available

40

Hedging Against A Parallel Shift in the Yield Curve (cont’d)

How to choose between the ED futures contracts available?– With a stack hedge, the hedger places all the

futures contracts at a single point on the yield curve, usually using a nearby delivery date

– With a strip hedge, the hedger distributes the futures contracts along the relevant portion of the yield curve depending on the tenor of the swap

41

Hedging Against Any Shift in the Yield Curve

The yield curve seldom undergoes a parallel shift

To hedge against any change, determine how the swap value changes with changes at each point along the yield curve

42

Hedging Against Any Shift in the Yield Curve (cont’d)

Steps involved in hedging:– Convert the annual LIBOR rate into effective

rates :

numberpayment yearper payments swap ofnumber

swap theof tenor over the LIBORTpayment for rateinterest effective

where

11

TNRZ

TN

NRZ

T

T

T

43

Hedging Against Any Shift in the Yield Curve (cont’d)

Steps involved in hedging (cont’d):– Next, determine the number of futures needed at

each payment date:

NT

FTTZ1

$1,000,000principal notional Swap

44

Tailing the Hedge

Futures contracts are marked to market daily

Forward contracts are not marked to market

This introduces a time value of money differential for long-tenor swaps– Hedging equations would overhedge

45

Tailing the Hedge (cont’d)

To remedy the situation, simply reduce the size of the hedge by the appropriate time value of money adjustment (tail the hedge):

TR)1(HedgeHedge untailed

tailed

46

Tailing the Hedge (cont’d)

Tailing the Hedge ExampleAssume we have determined that we need 100 ED futures contracts for delivery two years from now. The two-year interest rate is 6.00%. How many ED futures do you need if you tail the hedge?

47

Tailing the Hedge (cont’d)

Tailing the Hedge Example (cont’d)You need 89 ED futures contracts:

89)06.1(

100Hedge 2tailed

48

Pricing A Currency Swap To value a currency swap:

– Solve for the equilibrium fixed rate on a plain vanilla interest rate swap for each of the two countries

Determine the relevant spot rates over the tenor of the swap

Determine the relevant implied forward rates

– Find the equilibrium swap price for an interest rate swap in both countries