Chapter 14
Swap Pricing
1
© 2002 South-Western Publishing
Outline
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Swap pricing
Solving for the swap price
Valuing an off-market swap
Hedging the swap
Pricing a currency swap
Swap Pricing- theoretical
foundation
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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

You pay a fixed rate
– You receive a floating rate
– Or vice versa
…..analogous to purchasing one bond and issuing another
e.g. in a floating for fixed swap – analogous to issuing a
fixed rate bond and buying a floating rate bond
–
4
Swaps as A Pair of Bonds
(cont’d)
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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
Swap values are impacted in a similar manner to
bond values
–
5
The value changes after interest rates have changed
Swaps as A Pair of Bonds
(cont’d)
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6
If a firm is involved in a swap and pays a fixed rate
of 7% (like issuing a bond) at a time when it would
otherwise have to pay a higher rate, the swap is
saving the firm money
– swap has value
– unwinding the swap captures the value
If because of the swap you are obliged to pay more
than the current rate, the swap is beneficial to the
other party
– cost to the firm to unwind the swap
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)
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A six month swap with a single payment
date six months hence is no different than
an ordinary six-month forward contract
–
–
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Agreement to pay a fixed rate on a notional
principal amount and receive a floating rate
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
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
….the swap fixed rate is essentially the option strike price
–
10
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
The role of the forward curve for LIBOR
Implied forward rates
Initial condition pricing
Quoting the swap price
Counterparty risk implications
Swap Pricing

The swap price is determined by
fundamental arbitrage arguments
–
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13
Similar to pricing of other derivative instruments
whereby the equilibrium price essentially
eliminates arbitrage opportunities or ‘mispricing’
presents arbitrage opportunities
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
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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)
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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
–
–
–
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
…….it is these implied forward rates we will use as a
proxy for the floating rates in the swap
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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)
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The following relationship is true if both
alternatives are expected to provide the
same return:
 0 f 3  3 f 6  
0 f6 
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:
– 6f 9
–
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= 5.71%
9f12 = 5.77%
Implied Forward Rates (cont’d)
LIBOR Implied forward rate curve
%
5.77
5.71
9 x 12
6x9
5.58
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 (equilibrium)
–
The floating rate payments are uncertain
 Use
the implied forward rate curve (derived
from the spot yield curve) as a proxy for the
floating rate payments
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.
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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

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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


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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.
...this is the starting point for a swap dealer
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Quoting the Swap Price
Common practice to quote the swap price relative to the U.S.
Treasury yield curve
– interest rates are constantly changing so dealers will
quote relative to or off the U.S. treasury or Govt. Of
Canada 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
…………….The all in swap quote will then reflect swap dealer
pricing plus profit, competitive and credit considerations
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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
Valuing an Off-Market Swap

The swap value reflects the difference
between the swap price (fixed rate) and the
interest rate that would make the swap have
zero value
–
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As soon as market interest rates change after a
swap is entered, the swap has value to one party
or the other ….the initial swap fixed rate no
longer establishes a state of equilibrium
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 - the present value of the two
cash flow streams is different.
–
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Thus, the swap has value to one of the
counterparties
Valuing an Off-Market Swap
(cont’d)
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If the fixed rate in our at-the-market swap
example was 5.75% instead of 5.62%
–
–
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The value of the floating rate side would not
change
The value of the fixed rate side would be higher
than the floating rate side
The swap has value to the floating rate payer
and fixed rate receiver ...in a stand alone
situation ie if he was receiving 5.75% when
market rate is really 5.62% (arbitrage situation)
Valuing a Swap – after interest
rates change
For an existing swap with the fixed rate set at 5.62%:
 If interest rates increased to 5.75%
Value of the fixed rate side has decreased – discounting
5.62% cash flows with higher discount rates reflecting
5.75%
– Fixed rate receiver – lost value (analogy – think of owning
a fixed rate bond) – cost money to close it out
– Fixed rate payer – swap has increased in value – closing
the swap out would generate a gain
…how else could the value be realized??
–
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Hedging the Swap- by the Dealer

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
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The dealer is hurt by rising interest rates
Hedging the Swap by the Dealer
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The dealer can hedge this risk in the
futures market :
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–
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If the deal is Libor based - then use Euro-dollar
futures or more recently the new 2/5/10 year
swap futures
If the dealer faces the risk of rising rates, he
could sell Eurodollar/Swap futures and benefit
from the decline in value associated with rising
interest rates
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
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–
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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 currencies - in effect there are
two swap prices in a currency swap