Chapter 14
Swap Pricing
1
© 2004 South-Western Publishing
Outline
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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
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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
–
–
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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
–
–
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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
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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:
–
–
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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
6
9
12
Months
Implied Forward Rates

We can use these LIBOR rates to solve for
the implied forward rates
–
–
–
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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)
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An investor can
–
–
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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  5.58%
20
2
Implied Forward Rates (cont’d)

Applying bootstrapping to obtain the other
implied forward rates:
– 6f9
–
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= 5.71%
9f12 = 5.77%
Implied Forward Rates (cont’d)
LIBOR forward rate curve
%
5.77
5.71
9 x 12
6x9
5.58
3x6
5.42
spot
0
22
3
6
9
12
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
5.58%
5.71%
5.77%
360 
360 
360 
360
PVfloating 
1.013701
1.027653
1.042239
1.056981
 .013515  .013575  .014001  .013951
5.42%
 0.055042
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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
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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.
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Quoting the Swap Price
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Common practice to quote the swap price
relative to the U.S. Treasury yield curve
–
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There is both a bid and an ask associated
with the swap price
–
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Maturity should match the tenor of the swap
The dealer adds a swap spread to the
appropriate Treasury yield
Counterparty Risk Implications
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From the perspective of the party paying
the fixed rate
–
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From the perspective of the party paying
the floating rate
–
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Higher when the floating rate is above the fixed
rate
Higher when the fixed rate is above the floating
rate