Chapter 14
Swap Pricing
1
© 2002 South-Western Publishing
Outline





2
Intuition into swap pricing
Solving for the swap price
Valuing an off-market swap
Hedging the swap
Pricing a currency swap
Intuition Into Swap Pricing



3
Swaps as a pair of bonds
Swaps as a series of forward contracts
Swaps as a pair of option contracts
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

–
–
–
4
You pay a fixed rate
You receive a floating rate
Or vice versa
Swaps as A Pair of Bonds
(cont’d)
5

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
Swaps as A Pair of Bonds
(cont’d)
6

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
Swaps as A Series of Forward
Contracts
7

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
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
–
8
At that date, the party owing the greater
amount remits a difference check
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
–
–
9
Receive a check if the benchmark rate is above
5%
Remit a check if the benchmark rate is below 5%
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
–
–
10
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
Swaps as A Pair of Option
Contracts (cont’d)

Cap-floor-swap parity
Write floor
+
5%
11
Long swap
Buy cap
=
5%
5%
Solving for the Swap Price






12
Introduction
The role of the forward curve for LIBOR
Implied forward rates
Initial condition pricing
Quoting the swap price
Counterparty risk implications
Introduction

The swap price is determined by
fundamental arbitrage arguments
–
13
All swap dealers are in close agreement on what
this rate should be
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:
–
–
14
A 3 x 6 Forward Rate Agreement (FRA) begins in
three months and lasts three months (denoted
by 3 f 6 )
A 6 x 12 FRA begins in six months and lasts six
months (denoted by 6 f12 )
The Role of the Forward Curve
for LIBOR (cont’d)

15
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%
The Role of the Forward Curve
for LIBOR (cont’d)
LIBOR yield curve
%
5.62
5.57
0 x 12
0x9
5.50
0x6
5.42
spot
0
16
3
6
9
Months
Implied Forward Rates

We can use these LIBOR rates to solve for
the implied forward rates
–
–
–

17
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
Implied Forward Rates (cont’d)

An investor can
–
–

18
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
Implied Forward Rates (cont’d)

The following relationship is true if both
alternatives are expected to provide the
same return:
 0 f 3  3 f 6   0 f 6 
1 
1 
  1 

4 
4  
4 

19
2
Implied Forward Rates (cont’d)

Using the available data:
 .0542  3 f 6   .0550 
  1 
1 
1 

4 
4  
4 

3 f 6  5.56%
20
2
Implied Forward Rates (cont’d)

Applying bootstrapping to obtain the other
implied forward rates:
– 6f 9
–
21
= 5.71%
9f12 = 5.75%
Implied Forward Rates (cont’d)
LIBOR forward rate curve
%
5.75
5.71
9 x 12
6x9
5.56
3x6
5.42
spot
0
22
3
6
9
Months
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
23
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.
24
Initial Condition Pricing (cont’d)
At-the-Market Swap Example (cont’d)
First obtain the discount factors:
 91

1  R3  1  
 .0542   1.013701
 360

 91  90

1  R6  1  
 .0550   1.027653
 360

25
Initial Condition Pricing (cont’d)
At-the-Market Swap Example (cont’d)
First obtain the discount factors:
 91  90  92

1  R9  1  
 .0557   1.042239
360


 91  90  92  92

1  R12  1  
 .0562   1.056981
360


26
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:
91
90
92
92
5.56%
5.71%
5.75%
360 
360 
360 
360
PVfloating 
1.013701
1.027653
1.042239
1.056981
 .013515  .013526  .014001  .013902
5.42%
 0.054944
27
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:
91
90
92
92
X%
X%
X%
360 
360 
360 
360

1.013701 1.027653 1.042239 1.056981
 .249361X  .243273 X  .245199 X  .241779 X
X%
PVfixed
 0.979612 X
28
Initial Condition Pricing (cont’d)
At-the-Market Swap Example (cont’d)
Solving the two equations simultaneously for X gives X =
5.61%. This is the equilibrium swap fixed rate, or swap price.
29
Quoting the Swap Price

Common practice to quote the swap price
relative to the U.S. Treasury yield curve
–

There is both a bid and an ask associated
with the swap price
–
30
Maturity should match the tenor of the swap
The dealer adds a swap spread to the
appropriate Treasury yield
Counterparty Risk Implications

From the perspective of the party paying
the fixed rate
–

From the perspective of the party paying
the floating rate
–
31
Higher when the floating rate is above the fixed
rate
Higher when the fixed rate is above the floating
rate
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
–
32
As soon as market interest rates change after a
swap is entered, the swap has value
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
–
33
Thus, the swap has value to one of the
counterparties
Valuing an Off-Market Swap
(cont’d)

If the fixed rate in our at-the-market swap
example was 5.75% instead of 5.61%
–
–
–
34
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
Hedging the Swap




35
Introduction
Hedging against a parallel shift in the yield
curve
Hedging against any shift in the yield curve
Tailing the hedge
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

36
The dealer is hurt by rising interest rates
Introduction (cont’d)

The dealer can hedge this risk in the
Eurodollar futures market
–
–
37
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
Hedging Against A Parallel Shift
in the Yield Curve

Assume the yield curve shifts upward by
one basis point
–
–
–
38
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
Hedging Against A Parallel Shift
in the Yield Curve (cont’d)

The dealer could sell Eurodollar (ED)
futures to hedge
–
–
39
Need one ED futures contract for every $25
change in value of the swap
Need to choose between the various ED futures
contracts available
Hedging Against A Parallel Shift
in the Yield Curve (cont’d)

How to choose between the ED futures
contracts available?
–
–
40
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
Hedging Against Any Shift in
the Yield Curve
41

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
Hedging Against Any Shift in
the Yield Curve (cont’d)

Steps involved in hedging:
–
Convert the annual LIBOR rate into effective
rates :
T

 N
R
Z T  1    1 
 N 
 T
where
Z T  effective interest rate for payment T
R  LIBOR over the tenor of the swap
N  number of swap payments per year
T  payment number
42
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:
swap notional principal
$1,000,000
FT 
T

1   ZT  
N

43
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
–
44
Hedging equations would overhedge
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):
Hedge untailed
Hedge tailed 
(1  R)T
45
Tailing the Hedge (cont’d)
Tailing the Hedge Example
Assume 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?
46
Tailing the Hedge (cont’d)
Tailing the Hedge Example (cont’d)
You need 89 ED futures contracts:
100
Hedge tailed 
 89
2
(1.06)
47
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


–
48
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