© 2004 south-western publishing 1 chapter 14 swap pricing
Post on 18-Dec-2015
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TRANSCRIPT
2
Outline
Intuition into swap pricing Solving for the swap price Valuing an off-market swap Hedging the swap Pricing a currency swap
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Intuition Into Swap Pricing
Swaps as a pair of bonds Swaps as a series of forward contracts Swaps as a pair of option contracts
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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
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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
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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
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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
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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
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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%
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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
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Swaps as A Pair of Option Contracts (cont’d)
Cap-floor-swap parity
5%+ =
5% 5%
Write floor Buy cap Long swap
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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
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Introduction
The swap price is determined by fundamental arbitrage arguments– All swap dealers are in close agreement on what
this rate should be
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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
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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%
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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 9
0 x 12
%
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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
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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
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Implied Forward Rates (cont’d)
The following relationship is true if both alternatives are expected to provide the same return:
2
606330
41
41
41
fff
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Implied Forward Rates (cont’d)
Applying bootstrapping to obtain the other implied forward rates:– 6f9 = 5.71%
– 9f12 = 5.77%
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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 9
9 x 12
%
12
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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
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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:
013701.10542.360
9111 3
R
027653.10550.360
909111 6
R
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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
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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:
055042.0
013951.014001.013575.013515.056981.1
36092
%77.5
042239.136092
%71.5
027653.136090
%58.5
013701.136091
%42.5
floating
PV
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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:
X
XXXX
XXXXPV
979612.0
241779.245199.243273.249361.056981.1
36092
%
042239.136092
%
027653.136090
%
013701.136091
%
fixed
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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
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
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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
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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
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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
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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
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Hedging the Swap
Introduction Hedging against a parallel shift in the yield
curve Hedging against any shift in the yield curve Tailing the hedge
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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
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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
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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
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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
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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
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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 LIBOR
Tpayment for rateinterest effective
where
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T
N
R
Z
T
N
N
RZ
T
T
T
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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
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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
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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
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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?
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Tailing the Hedge (cont’d)
Tailing the Hedge Example (cont’d)
You need 89 ED futures contracts:
89)06.1(
100Hedge
2tailed
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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