Chris Dzera
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Explore specific inputs into Vasicek’s model,
how to find them and whether or not we can
get the mean back after simulations with
realistic parameters
Some discussion of interest rate models
Review some of the presentation from last time
Explanation of what swaptions are
Swaption pricing methods
How to determine whether or not to exercise
R-code for swaption pricing
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Use regression parameters slope, intercept, and
steyx along with stdev
Speed of mean reversion is the negative of the
slope, estimate of long-term mean is the
intercept divided by the speed of mean
reversion, calculate volatility normally
although they also indicate it can be estimated
by dividing steyx by the long-run mean
Values seem to check out, quite similar to what
other ways to check for these
Example
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Heath-Jarrow-Morton model is a model where
instead of modeling just the short rate (a point on
the forward rate curve) we are able to capture the
full dynamics of the entire forward rate curve
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The Hull-White model or extended Vasicek is an extension of
the HJM model where, as opposed to Vasicek’s model, has two
time dependent factors that are held constant in the Vasicek
model and is used to price bonds and derivatives
The Brace-Gatarek-Musiela model or LIBOR
Market model is used to price exotic options, is
nice because it correlates to Black’s model which is
commonly used, and is essentially an industrial
standard for pricing these instruments
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Riskless interest rate
Swaptions and interest rate swaps
Who enters?
 What makes them worthwhile to enter?
 Buyer vs. seller in interest rate swap
 Floating vs. fixed more common
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Sensitivity of swap payments to interest rate
Interest rate models
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Benefits to various parties
Effectiveness & what is most common
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An interest rate swap is an agreement between
two parties to exchange payments equal to an
rate multiplied by a given principal
These exchanges happen at fixed intervals for a
predetermined time period, and are based on
rates one interval prior
At onset a swap is set up so it is worthless (or if
it is done with a financial institution that
institution is given a slight edge so they make
money), free to set up
Interest rate swaps are mostly used for hedging
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The two rates involved are almost always a fixed
interest rate and a floating interest rate
Depending on where interest rates move, the swap
can have a value after onset
The first payment is certain because it is based on
interest rates when the swap was agreed to
While there are technically exchanges both ways,
the money only goes one way
Rates are compounded once for the amount of
exchanges in a year (so if there are payments every
3 months, each rate is compounded 4 times a year)
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An option is a financial derivative where one
party pays a premium to the other party in
exchange for the right to determine whether or
not they want to enter into a contract on
specified dates (or a single date)
In this case, the option holder has a limited
downside because they can see if things move
in their favor or not
This is most beneficial in the case when the
option holder is certain whether or not they are
in the money and what will pay the most
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A swaption gives the holder the right, but not
obligation, to enter into an interest rate swap
Determined when the contract is agreed to are:
Premium
Strike rate (or fixed rate in an interest rate swap)
Floating rate
Notional amount
Frequency of payments
Length of underlying swap
Length of option period (along with how many
option dates there are)
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Two parties agree to the aforementioned terms
On a certain date (or multiple dates) during the option
period, the contract holder is given the right to choose
whether or not they want to enter into the interest rate
swap contract
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This can happen before or after the swap would begin
depending on the type of swaption
If the holder exercises then the interest rate swap
begins, and if the holder does not exercise during the
exercise period then the only exchange of payments is
the premium
In any swap contract, unlike the options we dealt with
in class, at exercise it is unknown whether or not the
option holder will end up making money at the end of
the underlying interest rate swap
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Payer swaption – gives the owner of the
swaption the right to enter an interest rate
swap where they pay the fixed leg and receive
the floating leg
Receiver swaption – gives the owner of the
swaption the right to enter into an interest rate
swap where they pay the floating leg and
receive the fixed leg
Name of swaption depends on who
pays/receives the fixed leg or strike rate
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There are three different ways the right to exercise
the option in a swaption contract can be scheduled
European – the owner of the contract can exercise
his right to enter the swap on one date, at maturity
Bermudan – the owner of the contract can exercise
his right to enter the swap at certain
predetermined dates between the start and end
dates of the option period (multiple exercise dates)
American – the owner of the contract can exercise
his right to enter the swap at any date between the
start and end dates of the option period agreed to
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There are two types of American swaptions:
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An American swaption with fixed tenor is a when
the length of the underlying swap is a fixed time
length and the swap begins as soon as the option is
exercised. Again, if the exercise period ends without
the option holder electing to exercise it expires
worthless.
An American swaption with fixed end date is when
the predetermined period of time includes the length
of the option period and the underlying swap, so if
the first day the swap would begin passes without
the swap being exercised, the length of the
underlying swap decreases
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Because the option period can end before the swap
takes place, it may be unclear why it is logical to pay a
premium for the option to enter a swap contract down
the road when the party could just decide whether or
not they want to enter into an interest rate swap at that
later date without it being at a premium
However, the option holder may be able to receive
better terms on the underlying interest rate swap at
exercise with a swaption than they would on a
standard interest rate swap at that date
Entering into a swaption contract gives the opportunity
to exercise on superior terms and already be “in the
money” when the swap begins whereas an interest rate
swap starts worthless
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A company knows that in six months it will
enter a 10 year loan on $5 million with a 5%
interest rate, and wants to reduce its interest
rate risk by exchanging its payments on this
contract for floating payments since it has
multiple fixed rate loans out already
They enter a swaption contract with a 6 month
option period, where at the end of half a year
on the expiry date of the option, if they exercise
the option a 10 year swap would be initiated
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A bond holder knows that in one year they will
receive fixed rates of 3% on $2 million bond for
a period of five years, and wants to receive a
floating rate instead
They enter a swaption contract that expires in
one year with specific exercise dates afterward
that would not go too far into the swap period
If they elected to exercise after expiry, the
length of the swap would be reduced by the
amount of time between the end of the option
period and the date the option was exercised
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A company would receive a LIBOR -.02% on a
10 year bond it may purchase sometime in the
next two years, but would rather receive a
fixed rate instead
They enter a swaption with expiry in 2 years
and an underlying swap of length 10 years
If they elect to exercise the swap at any time in
the option period, a 10 year swap with terms
agreed to begins immediately
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A company knows it will have to pay a floating
rate of LIBOR + .03% on an 8 year loan and
wants to pay a fixed rate instead
They enter a swaption contract with expiry in 2
years and an end date in 30 years
If they elect to exercise the option, a swap that
ends when the loan ends will begin on that
date
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The valuation of European swaptions can be done by tweaking
Black’s model for valuing futures options
The swaption model relies on changing the value of the
underlying, the volatility, and the discount factor
Black’s model benefits us in this calculation because the option
contract and futures contract don’t have to mature at the same
time, which is helpful because the option on a swaption and the
actual swaption itself do not mature at the same time
There is also a quick way to value European swaptions that Hull
and White have shown, using an analytic approach that comes up
with results similar to Monte Carlo simulations for similar
material
Indications are that certain exercise strategies implemented in
Monte Carlo simulations may be superior to quick analytic
approaches to European swaption pricing, but Black’s model gets
very similar results to simulations and is the industrial standard
for pricing European swaptions
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Monte Carlo simulation is essentially the method that
is used to price Bermudan swaptions
Valuation of Bermudan swaptions can be done by
using one-factor no arbitrage models that are
controversial because their accuracy has been
questioned
The BGM model discussed earlier (LIBOR Market
model) is the most commonly used interest rate model
for swaption pricing
Early exercise methods include a least squares
approach where the value of not exercising on a
particular payment date is assumed to be a polynomial
function of the values of the factors in the swaption,
and an optimal early exercise boundary approach
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Valuing American swaptions is generally
considered by investors to be extremely difficult,
though it is fortunate that they are not as common
as European or Bermudan swaptions
There is no set way to value American swaptions,
although there have been certain techniques
proposed including a two factor stochastic model
where the factors are the short-term interest rate
and the premium of the futures rate over the shortterm interest rate, and another model that uses
trinomial trees
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The two main early exercise strategies in Bermudan
swaptions are using an absolute exercise boundary
function and a relative exercise boundary function
These boundary functions are determined by running
calibration simulations that select boundary functions
which give the swaption its highest value, since we
want to choose the best early exercise strategy
If we start from the last exercise date and assume the
option has not been exercised, the boundary is found
using a linear search starting at 0 and ending above the
largest swap value attained during simulations, and is
done until all exercise dates have a value of the
boundary function
When the boundary function has been determined, a
separate set of simulations are run to avoid bias
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I used simulation to price a European Swaption
in R
Issues:
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How useful is this?
 Why not Bermudan
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Why Vasicek’s model?
How to exercise?
Accuracy
Ultimately, interesting to do, probably not very
useful
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Swaptions are a great way to mitigate interest
rate risk with a limited downside
Because they are extremely complex swaptions
can be (as I found out) very difficult to price
and there is a lot of demand for quick, effective
ways to price swaptions since they are
extremely popular
Ultimately this ended up a lot more difficult
than I thought it would be, but I still learned a
lot through the various bumps in the road