Swaps, interest rate derivatives, and all of their subcategories are among the most popular
financial instruments bought and sold in markets today. According to the International Swaps
and Derivatives Association, in 2009 there was $426.7 trillion in interest rate and currency swaps
outstanding.1 Additionally, the Bank for International Settlements estimates the notional amount
outstanding in June 2009 for OTC interest rate contracts was $437 trillion.2 There are many types
of swaps and interest rate derivatives. Interest rate swaps are a very important type of both swaps
and interest rate derivatives, and the Bank for International Settlements estimated the notional
amount outstanding in June 2009 for these contracts to be $342 trillion for OTC. Additionally,
swaptions are a specific kind of interest rate swap that involve an option on the contract and are
very popular as well. Each of these derivatives is very popular, and mostly because they allow
investors and organizations to hedge against specific risk factors, especially interest rate risk. In
this paper we will define and give examples swaps and interest rate derivatives. Specific kinds of
these contracts will be defined as well, with a focus on interest rate swaps and swaptions, and
examples will be given. How these financial instruments are priced will be explored as well.
Additionally, motivations for entering into these contracts will be explored, including why
interest rate risk is such an issue for companies. Because various kinds of swaps and interest rate
derivatives are so common and popular, it is important to understand what they are, how they
work, and why they are used.
A swap is defined as an agreement between parties to exchange cash flows in the future,
defining the dates they will be paid and the way the payments will be calculated.3 This contract
typically involves the future value of an interest rate, a foreign exchange rate, an equity price, or
a commodity price. The future value of some other market value may be a factor as well.4 The
two most common types of swap contracts are interest rate swaps and currency swaps. Other
kinds of swap contracts include commodity swaps, equity swaps, total return swaps, swaptions,
variance swaps, and amortizing swaps.5 What makes swap contracts different from other
financial derivatives is the fact that they almost exclusively account for more than one future
payment, whereas most kinds of financial instruments only involve one payment based on the
underlying.6 To detail this further and a more basic example, we will explore how a commodity
swap works to show how multiple future payments can be combined in one contract.
In the case of a commodity swap, a company would enter the contract if they were
interested in purchasing or selling some underlying asset in multiple years.7 We will explore the
case where the asset is exchanged once a year after the contract is agreed to and again two years
after the contract is agreed to. If the buyer wanted to pay the seller for the asset up front, a
prepaid commodity swap, they could do this by paying the present value of presently guaranteed
prices at the risk free interest rate. For this example, presume the market price of the commodity
is $100 in one year and $110 in two years. Because the riskless interest rate also must be taken
into account in this calculation, assume that this rate is 4.0% with maturity in one year and 4.5%
with maturity in two years. The price of the swap contract as outlined earlier would be 100/1.04
ASM Study Manual For Exam FM/Exam 2: Financial Mathematics & Financial Economics, Tenth Edition by Harold
Cherry, FSA, MAAA and Rick Gorvett, FCAS, MAAA, ARM, FRM, PH.D. page 643.
ASM Study Manual For Exam FM/Exam 2: Financial Mathematics & Financial Economics, Tenth Edition by Harold
Cherry, FSA, MAAA and Rick Gorvett, FCAS, MAAA, ARM, FRM, PH.D. page 643.
+ 110/(1.042), or $196.88. On the other hand, the buyer might be concerned about the default risk
of the seller and want to pay after the two year period, which would mean the parties would enter
into a postpaid commodity swap. This swap takes the same riskless interest rates into account, as
well as the $196.88 figure that we calculated earlier. In this case we determine the annual level
payment that would result in the contract being valued the same as the prepaid amount we
calculated earlier. More simply, we solve for X in the formula: X/1.04 + X/(1.042), = $196.88.
Here we get X = 104.88, which would be what the buyer pays the seller at the exchange the first
year and again at second year exchange. When the postpaid commodity swap case is looked at
further it becomes clear that there is some implicit borrowing and lending going on. The buyer
overpays the seller by $4.88 the first year, and then underpays by $5.12 the second year. This
balances itself out, because the effective interest rate between the first and second years
multiplied by that $4.88 should give us the $5.12 extra that the seller should be getting the
second time around. To calculate that rate, we do (1.045)2/1.04 – 1, which equals 5.0024%. This
checks out, because when $4.88 is multiplied by 1.050024, the result is $5.12 after rounding. The
commodity swap example shows just a few of the ways that future payments can be combined in
a swap contract.
It is also worth explaining how the value of a swap contract can change. Despite the fact
that there is implicit borrowing and lending in a swap contract, as was noted in the commodity
swap example, the contract is specifically designed so it has no value for either party. However,
the value of a commodity swap contract can change if the forward prices of the underlying assets
change. Extending the previous example further, we will consider what would happen if forward
prices of the underlying asset rise to $105 in year one and $115 in year two. Doing the same
calculations that we did earlier, the resulting prepaid swap price and annual level swap price with
these values are $206.27 and $109.88 respectively. The important aspect of this calculation is to
compare the new annual level swap price of $109.88 to the previous annual level swap price of
$104.88. Then, if the buyer were to enter a swap contract with another party for the same asset at
the prices of $105 in year one and $115 in year two, they could make a profit of $5 each year.
We now know that the new contract is worth $5 at each future date, and the old contract is worth
nothing at either date. The only remaining calculation is to find the present value of those $5
profits, which is obtained by the calculation: 5/1.04 + 5/(1.0452). This gives us a value of $9.39
for the contract. 8 These examples should help to give an understanding of how multiple
payments can be combined in a swap contract and how the value of that contract can change if
the price of the underlying asset changes. Later in this paper interest rate swaps will be covered,
but prior to discussing these financial instruments we will discuss interest rate derivatives, the
other key aspect of interest rate swaps.
An interest rate derivative is simply defined as a derivative whose payments are
dependent on future interest rates. The interest rate derivatives market is the largest derivatives
market in the world, largely because they allow for hedging against interest rate risk which will
be elaborated upon later.9 There are a tremendous amount of kinds of interest rate derivatives,
and they break down into three categories: basic, less basic, and exotic. Basic interest rate
derivatives include interest rate swaps, interest rate caps/floors, swaptions, bond options, forward
rate agreements, interest rate futures, money market interest, and cross currency swaps. Some
less basic interest rate derivatives are range accrual swaps/notes/bonds, in arrears swaps, constant
maturity or treasury swap derivatives, and some other types of interest rate swaps. Exotic interest
The examples was based off of the example was based off of the example on pages 645-648 of ASM Study
Manual For Exam FM/Exam 2: Financial Mathematics & Financial Economics, Tenth Edition by Harold Cherry, FSA,
MAAA and Rick Gorvett, FCAS, MAAA, ARM, FRM, PH.D.
rate derivatives include power reverse dual currency notes, target redemption notes, CMS
steepeners, snowballs, inverse floaters, strips of collateralized debt obligations, ratchet caps and
floors, Bermudan swaptions, and cross currency swaptions.10 Looking at the list, it becomes
evident that interest rate derivatives and swaps are often combined in various types of financial
derivatives, with interest rate swaps and swaptions being of particular interest in this paper.
Additionally, collateralized debt obligations are interesting because they played a major role in
the financial crisis a few years ago.
Because there are so many different types of interest rate derivatives, a tremendous
amount of pricing methods are necessary to be able to determine fair value for these contracts.
Although pricing models are not our focus, it is worth noting some of the more common pricing
methods so the reader has a bit of an idea of how these contracts are valued. Black created a
model that is effective in the pricing of bond options, caps, and some swaptions.11 Among the
more complex models used to prices some of the more exotic derivatives are equilibrium models
by Rendleman and Bartter; Vasicek; and Cox, Ingersoll, and Ross.12 These models utilize
assumptions about economic variables to derive a process for the short rate, and explore what the
process for this rate means about bond and option prices. Additionally, there are two factor
models created by Brennan and Schwartz and Longstaff and Schwartz that follow stochastic
processes. In the Brennan/Schwartz model the key is that the short rate reverts to the long rate,
while in the Longstaff/Schwartz model volatility is of the most importance.13 Ho and Lee also
proposed a no arbitrage model using a binomial tree of bond prices where the parameters are the
For more on Black’s model, see Options, Futures, and other Derivatives by John C. Hull, pages 648-661.
For more, see Options, Futures, and other Derivatives by John C. Hull, pages 682-685.
For more, see in Options, Futures, and other Derivatives by John C. Hull, pages 685-686.
short-rate standard deviation and the market price of the short rate.14 Hull and White created a
one factor model that is similar to both the Vasicek model and the Ho/Lee model. They also
proposed a similar model to the one Brennan and Schwartz created, although theirs was arbitrage
free.15 Additionally, Black and Karasinksi created a model that allows only positive interest
rates, which was an advantage over the Ho/Lee and Hull/White models although it did not have
as many analytic applications.16 There are many other ways that interest rate derivatives are
priced, but those are some of the more notable ones that could be worth researching to
Interest rate swaps are very important financial instruments that are both swaps and
interest rate derivatives. In fact, the most common type of swap contract is a “plain vanilla”
interest rate swap. Under this contract, a company agrees to pay cash flows equal to interest at a
predetermined fixed rate on a notional principal for a few years, while receiving interest at a
floating rate on the same principal for the same time period. 17 There are also interest rate swaps
that include different currencies, different principals, floating rates for floating rates, and fixed
rates for fixed rates, but these are less common and therefore our focus will remain on fixed for
floating interest rate swaps on equal principal.18 Typically in an interest rate swap the floating
rate is the London Interbank Offered Rate, the rate of interest at which a bank is prepared to
deposit money with another bank in the Eurocurrency market.19 In an interest rate swap, the
payments are exchanged at specified intervals, with the fixed rate times the notional amount
being exchanged for the floating rate one interval prior being exchanged for the fixed amount.
For more, see Options, Futures, and other Derivatives by John C. Hull, pages 686-687.
For more, see Options, Futures, and other Derivatives by John C. Hull, pages 688-690.
For more, see Options, Futures, and other Derivatives by John C. Hull, pages 689.
Options, Futures, and other Derivatives by John C. Hull, page 147.
Options, Futures, and other Derivatives by John C. Hull, page 147.
Because the first exchange happens one time interval after the contract is agreed to, there is no
uncertainty about what the first exchange of payments will be. Additionally, because interest rate
swaps have no value when they are created, the fixed rate contract is set up so that payments will
eventually even out over time.
Typically, it is assumed that the floating rates in the future will be equal to the LIBOR
rate on the LIBOR swap zero curve. This curve lasts only twelve months, but the rates can be
extrapolated over time using a formula that combines the bid rate and the offer rate.20 Because
these rates generally increase over time, typically the fixed rate payer has to pay a higher amount
initially, and it is assumed that with higher interest rates over time the floating rate payer will
have to even this out eventually. Furthermore, it is also worth noting that contrary to the
examples below, interest rate swaps typically involve three parties. Because it is unlikely that
two organizations will have exactly opposing needs in terms of wanting to exchange floating for
fixed, similar rates, and the same notional amount, a financial institution typically plays the role
of an intermediary. To take this position, they usually gain a small percentage on both sides of
the agreement, so the other parties entering the interest rate swap do receive slightly less money
than they would if they were entering an agreement with another organization that had exactly
We will now outline an example of an interest rate swap contract. This will be followed
by exploring why an organization may want to enter this type of contract. Afterwards, we will
look at valuation of an interest rate swap contract using two different methods. Let us assume
that Microsoft and Apple are going to enter into a four year interest rate swap contract agreed to
on October 22, 2010. Apple agrees to pay Microsoft .5% annually on a notional principal of
Options, Futures, and other Derivatives by John C. Hull, pages 158 and 159.
Options, Futures, and other Derivatives by John C. Hull, pages 151-152.
$400 million, and Microsoft will pay apple the six-month LIBOR rate on the same $400 million
principal in return. In this case, the rate that Apple is paying is fixed, so Apple is the fixed rate
payer in this agreement. Since the six-month LIBOR rate that Microsoft is paying can change
over time that rate is called a floating rate, and so Microsoft is the floating rate payer. The .5%
interest rate is compounded semi-annually, and payments are exchanged every six months with
the first exchange taking place April 22, 2011. On this date, Apple will pay Microsoft $1 million,
the interest on $400 million for six months at .5% annually. Microsoft would pay Apple half of a
year’s worth of the six-month LIBOR rate from six months before the payment, or on October
22, 2010, on the $400 million principal. This rate was .45%, so the payment would amount to
$900,000. As we outlined earlier, since both sides knew what the six-month LIBOR was on
October 22, 2010 there would be no uncertainty about what this payment would be. Moreover,
although we say that Apple would be paying Microsoft $1 million and Microsoft would pay
Apple $900,000, in reality the only exchange of payments on this payment date would be a
$100,000 payment going from Apple to Microsoft. The interest rate swap would have a total of
eight exchanges that would happen on April 22 and October 22 every year from 2011 through
2014. Each fixed rate payment would be $1 million from Apple to Microsoft, and the floating
rate payments from Microsoft to Apple would be calculated by taking half of the six-month
LIBOR rate from six months prior to the payment and multiplying it by the $400 million
At this point, it makes sense to wonder why Apple and Microsoft might be motivated to
enter into this contract. While it may seem to be something that does not provide much benefit or
much loss, it is also worth considering that in an actual situation a financial institution would
likely be taking a certain percentage of these payments and working on both sides. In the real
The basis for the example given can be found in Options, Futures, and other Derivatives, pages 148-150.
world case, the floating rate payer might receive a fixed rate minus a certain percentage, or the
fixed rate payer might have to pay a certain percentage on top of the fixed rate they would pay
otherwise. The reason that the institutions engaging in the contract with the financial institution
have to give up some money is to motivate the financial institution to enter the contract, because
they have to make money somehow. Ultimately, motivations for entering into an interest rate
swap come back to the desire to hedge against interest rate risk by transforming a fixed rate asset
or liability into a floating rate asset or liability and vice versa. In this case, companies that have a
lot of debt or are creditors whose payments largely depend on how interest rates change can
hedge against that risk. All of the financial instruments discussed in this paper are used to hedge
against interest rate risk. With this knowledge, it is worth exploring how a floating rate asset or
liability can be changed into a fixed rate asset or liability when an interest rate swap contract is
Although we understand that organizations that enter into interest rate swap contracts
with financial institutions typically give up a bit of money to do so, for this portion we will
ignore that aspect of interest rate swaps and simply understand that this is the case in real life as
we explore how an asset or liability can be transformed in an interest rate swap. First let us
consider a company that has to pay fixed interest on a considerable amount of loans, and wants
to pay a floating rate on some of them instead so they can protect against dropping interest rates.
They can enter an interest rate swap as the floating rate payer on a principal equal to the loan
amount they want to exchange the fixed rates on, with the fixed rate the same as the fixed rate on
their loans and exchanges as frequently as they have to make payments on the loan. In this case,
the fixed rate that they pay on a loan is cancelled out by the fixed rate that they receive in the
interest rate swap, and the floating rate that they pay is essentially their new loan payment. The
reverse works if the company wants to exchange a floating rate on a loan for a fixed rate on a
loan. The floating rate they have coming in as a part of the interest rate swap would be the same
as the floating rate they are paying their creditors, and the fixed rate that they pay as a part of the
swap is essentially their loan payment.23
Furthermore, this can also work for an asset such as a bond where a company is getting
either a fixed or a floating rate as a return for their bond and wants to change that rate. A bond
holder with a fixed rate bond that would rather receive a floating rate would enter an interest rate
swap where they are the fixed rate payer. In this case, the coupons that they receive for the bond
would be paid out to the other party, and they would receive a floating rate as a part of the
interest rate swap which essentially becomes the new coupon. On the other hand, the holder of a
floating rate on a bond who would rather receive a fixed rate would enter an interest rate swap as
the floating rate payer. As demonstrated, interest rate swaps can be used to transform an asset to
protect a company against having to make higher payments across many of their liabilities or
receive lower payments across many of their assets due to interest rate fluctuations that are not in
their favor. 24
Earlier it was stated that interest rate swap contracts have no intrinsic worth when they
are created. However, as interest rates change the contract may have a value that is positive for
one party and negative for another. There are two different ways that we can approach the
valuation of this interest rate swap contract that will be outlined below. One consists of viewing
the interest rate swap as the difference between two bonds, while the other regards the interest
rate swap as a portfolio of forward rate agreements. We will explain how the interest rate swaps
are valued using both of these perspectives, and give an example of each valuation method.
The information from this paragraph was found in Options, Futures, and other Derivatives, page 150.
The information from this paragraph was found in Options, Futures, and other Derivatives, pages 150 and 151.
When determining at the value of the interest rate swap in terms of bond prices, the
perspective of the floating rate payer can be regarded as a long position on a fixed rate bond and
a short position on a floating rate bond. The perspective of the fixed rate payer, then, is a long
position on a floating rate bond and a short position on a fixed rate bond. In this case, the value
of the interest rate swap from the perspective of the floating rate payer is Vswap = Bfloating – Bfixed,
and the value from the perspective of the fixed rate payer is Vswap = Bfixed – Bfloating.25
For example, consider a financial institution that agreed to pay six-month LIBOR and
receive a fixed rate of .5% per year compounded semi-annually on a principal of $400 million as
a part of an interest rate swap contract that has one and a quarter years remaining. We also
assume that LIBOR rates at three, six, and nine month maturities are .5%, .52%, and .54% and
when extrapolated to fifteen months the LIBOR rate is .58%. At each upcoming payment date
the fixed rate payer will be paying $1 million, $1 million, and $401 million.26 As a part of this
calculation we must also be aware that the discount factors for each of these cash flows are e-
, e-.0054*.75, and e-.0058*1.25. Moreover, there is a principal of $400 million, interest due of
.5*.0052*$400,000,000, and a time of .25. Therefore, the floating rate bond can be valued like it
produces a cash flow of $401.04 million in three months, and using the first discount factor it has
a present value of $400.539 million. The discount factors applied to all of the fixed rate cash
flows give a value of $400.098 million, so the value of the interest rate swap is -$441,000 for the
institution paying the floating rate of six-month LIBOR and $441,000 for the fixed rate payer.27
An interest rate swap can also be valued by considering it to be a portfolio of forward rate
agreements. Recall the interest rate swap between Apple and Microsoft from earlier. This interest
The information from this paragraph was found in Options, Futures, and other Derivatives, pages 159 and 160.
The $401 million payment includes the $400 million notional principal, which both parties technically exchange
in terms of the numbers, although these payments cancel each other out.
The basis for the example given can be found in Options, Futures, and other Derivatives, pages 159-161.
rate swap was a four year contract with semiannual payments, and both sides were aware of what
the first payment would be at the time the interest rate swap was negotiated. Each of the other
seven exchanges can be regarded as a forward rate agreement. For example, the exchange on
October 22, 2011 can be characterized as a forward rate agreement where interest at .5% is
exchanged for interest at six-month LIBOR observed April 22, 2011 on the principal of $400
million. The same works for the remaining six exchanges, each is valued the same way with the
only change being the six-month LIBOR rate observed six months prior to the exchange. We can
value a forward rate agreement by assuming that forward interest rates are realized, so again the
LIBOR swap zero curve will be used to determine forward rates for each LIBOR rate to
determine interest rate swap cash flows. After this, cash flows will be calculated by assuming
that LIBOR rates will equal forward rates, and these interest rate swap cash flows will be
discounted using the rates found on the LIBOR swap zero curve to obtain present value.28
Considering the example of the swap contract we valued earlier by considering it to be
the difference between two bonds, we have a swap with semiannual exchanges that has a year
and a quarter remaining on $400 million notional principal. The fixed rate payer pays .5% on the
principal, or $1 million each payment date, and the floating rate payer pays six-month LIBOR.
The LIBOR rates were found to be .5% for three months, .52% for six months, .54% for nine
months, and .58% for fifteen months. Based on this, we know that the first floating rate cash
flow is $1.04 million. To find the next cash flow, we must calculate a forward rate corresponding
to the difference between three and nine months. To do that, we calculate: (.0054)(.75) –
(.005)(.25), and divide this result by two to find the average. This gives a rate of .0056, and so
we use the following formula to determine the rate with semiannual compounding, and arrive at
The information from this paragraph was found in Options, Futures, and other Derivatives, page 161.
a nearly identical answer: 2(e.(0056/2) – 1) = .005607843729. The cash outflow for the floating rate
payer is therefore $1.12 million at the second date, and using the same calculation for the third
date we get $1.28 million. Using the same discount factors as in the calculation using the
difference of bond prices, the present value for the exchange in three months is $39,950.03, the
present value for the exchange in nine months is $121,078.10, and the present value for the
exchange in fifteen months is $280,012.72. All of these values are in favor of the fixed rate
payer, and so the present net cash flow is the sum of all of these values, $441,010.85 in favor of
the fixed rate payer. At the outset of the interest rate swap the sum of the values of the forward
rate agreements would be zero since the swap has no worth technically, though the values of the
forward rate agreements themselves would not be zero.30
Now, the focus of the paper will turn from swaps, interest rate derivatives, and interest
rate swaps to swaptions. Swaptions are options on swap contracts, specifically on interest rate
swaps. An option refers to the fact that one party pays a premium for the right to whether or not
they want to enter that contract at a certain specified date or perhaps multiple dates.31 This limits
the downside that party in the contract, since between entering the swaption and deciding
whether or not they would like to exercise, the underlying may have changed in a way that does
not favor the contract holder. The first swaption contract was created by William Lawton in
1983.32 Swaptions are very popular, like the swaps, interest rate derivatives, and interest rate
swaps that help compose what swaptions are, because they help to hedge against interest rate
risk. In the following paragraphs, we will explore the components of a swaption contract, explain
This equation can be found on page 77 of Options, Futures, and other Derivatives.
The basis for the example given can be found in Options, Futures, and other Derivatives, pages 161 and 162.
how the contract actually works, break down the different kinds of swaptions, give examples of
each type, and briefly touch on valuation techniques.
A swaption contract has many components, some of which are the same as the
components of an interest rate swap and a few that are different. Like an interest rate swap, parts
of swaptions include the length of the underlying swap, the notional amount, and the frequency
of settlement of payments. Additionally, the floating and fixed rates are predetermined in both of
these contract types, although in swaptions the fixed rate is occasionally called the strike rate
which will be discussed later. There are two components of a swaption contract that are not parts
of a typical interest rate swap, the length of the option period and the premium.33 How these, and
the other components of a swaption contract work will be outlined in the following paragraph.
The ways swaption contracts work can be somewhat confusing and are worth outlining.
Initially, two parties agree to the components listed in the previous paragraph. During the option
period there will be one or multiple dates when the contract holder can choose whether or not
they would like to exercise the contract and enter into an interest rate swap as outlined into the
contract. Later, the different kinds of exercise dates and how they work will be covered in detail.
If the contract holder does not elect to enter into the contract, the only exchange of payments is
the premium previously agreed to. Why the contract holder would enter into this contract at a
premium instead of just deciding whether or not they want to participate in an interest rate swap
on their own will be delineated as well. An interesting aspect of swaptions that is different from a
basic call or put option is that the option in a swaption contract happens before the contract
holder knows whether or not they will make money. This is part of why swaptions are very
difficult to price, because since there is uncertainty about whether or not the contract holder will
make money on the contract at the exercise date it can be unclear whether or not they will
exercise the contract and when they will do so if they have more than one exercise date.
One of the most important things to distinguish in a swaption contract is whether that
swaption is a payer swaption or a receiver swaption. In a payer swaption, the contract holder has
the right to enter into an interest rate swap where they pay the fixed leg and receive the floating
leg. On the other hand, in a receiver swaption the contract holder has the right to enter into an
interest rate swap where they receive the fixed leg and pay the floating leg.34 It may be easier to
understand what payer and receiver swaptions are if they are thought of in terms of call and put
options. The floating interest rate in the contract is similar to the underlying in a call or put
option, and the fixed interest rate is similar to the strike price which is why it is sometimes called
the strike rate. For a payer swaption, the contract holder will make money at payment dates if the
floating rate is above the fixed rate, similar to how a call option holder will make money if the
price of the underlying is above the strike price. A receiver swaption is then similar to a put
option, and the contract holder will make money if the floating rate is below the fixed rate.
Again, swaptions are more complex to exercise because, for example, a payer swaption contract
holder may elect to exercise the contract even if the floating leg is below the fixed leg if the
contract has many payment dates and the contract holder anticipates the floating leg rising above
the fixed leg in the future. However, these types of swaptions are two of the more important ones
and are very important to understand. Later, examples of different types of payer and receiver
swaptions will be given.
The other main way swaption contracts can break down is by exercise date. The three
different kinds of swaptions in this category are European swaptions, Bermudan swaptions, and
American swaptions. European swaptions are the simplest kind of swaption in terms of exercise
date and the easiest to price, in a European swaption the contract holder has the right to exercise
the contract at maturity of the option period.35 In other words, there is only one exercise date in a
European swaption contract. Bermudan swaptions give the owner of the contract the right to
enter a swaption at multiple predetermined exercise dates.36 If the contract holder elects to enter
into the contract at one of those predetermined dates after the swap could have taken place, the
swap begins at that date, does not get any longer, and none of the prior payments happen. For
example, if a contract holder has a three year option period on a 15 year swaption, and the first
date they can exercise their option is one year into the option period, if they exercise two years
into the option period then the swaption is just 14 years long. These swaptions are more
complicated to price than European swaptions, but considerably easier to price than American
American swaptions are more difficult to price than European and Bermudan swaptions,
and subsequently are not as common. In an American swaption contract, the contract holder can
enter into an interest rate swap at any date during the option period agreed to.37 These contracts
are so difficult to price because a considerable amount of dates need to be taken into account to
figure out when the option might be exercised and what might happen at the multiple payment
dates if it does. American swaptions also have two different subcategories, American swaptions
with fixed tenor or with fixed end date. An American swaption with fixed tenor is an American
swaption where the length of the underlying swap does not change based on when it is exercised,
so as long as the option is exercised an interest rate swap with a fixed predetermined length
begins.38 On the other hand, an American swaption with fixed end date is similar to a Bermudan
swaption, so if the contract holder waits to exercise the swaption its end date will remain the
same and the length of the underlying swap will decrease.39 Although these types of swaptions
are uncommon, they do occur and are relatively complex, therefore it is worth taking the time to
Why it makes sense to enter into a swaption contract at a premium as opposed to simply
waiting to enter into an interest rate swap at an important date without paying a premium for it
may be somewhat unclear. The explanation is that entering a swaption contract allows for the
potential to enter into an interest rate swap at a later date with terms that are more in the contract
holder’s favor than they would be if they waited.40 Because there is uncertainty about what will
happen with the floating interest rate compared to the fixed rate during the option period, the
way the fixed rate is set to ensure the interest rate swap has no value cannot assume any dramatic
increases or decreases. Therefore, the terms of the interest rate swap that the contract holder
could enter may, at the time the contract would be entered into, presume that the contract holder
will be making money on the interest rate swap. Consequently, a contract holder in a payer
swaption would want to exercise their option on the contract if the floating rate rose during the
option period, and would probably decline to exercise if the floating rate went down. The reverse
applies for a receiver swaption, they would probably only exercise their option if the floating
interest rate dropped during the option period. Swaptions do make sense to enter into, because
the premium that the contract holder pays can get them into a more favorable contract at some
time in the future than they would have been able to if they did not enter the swaption.
Following are a few examples of different kinds of swaption contracts that cover each of
the individual types of swaptions that have been discussed in the paper.41 First, let us consider a
small business that knows it will be entering into a five year loan at some date two years into the
future as a part of expansion efforts. The principal of this loan will be $5 million, and the
business will be paying 2% interest on said loan. Because the company already has a similar
fixed rate on some other loans, they would like to reduce their interest rate risk by entering into a
swaption contract. As a part of the contract, they have the right to enter into a five year interest
rate swap with a two year option period and a fixed rate of 2%, and because they know when
they will be taking out the loan they only require one exercise date. In this interest rate swap they
would want to pay the floating leg and receive the fixed leg, therefore this is an example of a
European receiver swaption.
Now, consider a bond holder who knows they will be receiving 3% interest on a $2
million for a period of five years, but if interest rates rise they would like to be able to receive a
floating rate instead. In this case, the bond holder might be inclined to enter into a swaption
contract with multiple exercise dates, so that if interest rates rise above 3% they can exchange his
fixed interest rate for a higher floating rate. A likely swaption they might enter into would have
five exercise dates spread out over a two year period that begins when the bond begins. If
exercised, the bond holder would have the right to enter into a swaption that ends at maturity of
the bond. The fixed interest rate would be 3%, the notional amount $2 million, and the bond
holder would be paying the fixed rate he receives on the bond in exchange for a floating rate.
Consequently, this is an example of a Bermudan payer swaption given the multiple but not
Each of the following four examples was created by the author, and they are not based on any other specific
continuous exercise date and the fact that the bond holder has the right to enter an interest rate
swap pay the fixed leg and receive the floating leg.
Let us now consider a company that would receive LIBOR -.02% on a 15 year bond it is
considering buying sometime in the next year, but would like to fixed rate instead because the
company has another floating rate asset and wants to protect itself from a drop in interest rates. It
would then make sense for this company to enter into a swaption contract where they could enter
into the interest rate swap whenever they elect to purchase the bond. To protect against interest
rates dropping between now and the purchase, they would like to align an interest rate swap that
has terms similar to those that the company could get on the market right now, thus the decision
to purchase a swaption. Therefore, they would enter into a swaption with expiry in one year that
could be exercised at any date in that one year period, since they do not know when they might
purchase the bond. Additionally, the company would like the interest rate swap to last exactly 15
years since that would be the length of the bond whenever they purchase it. Since they would
like to exchange floating rates for fixed rates on an asset, they would be paying those floating
rates and receiving fixed rates in return. To achieve this, they would purchase an American
receiver swaption with a fixed tenor of 15 years and one year to expiry.
In our final example, consider a company that just took out 30 year loan paying interest
of LIBOR + .03%, but wants to protect itself against rising interest rates because they are paying
floating rates on other liabilities. They might enter into a swaption contract where they have a
few years, say three, to decide whether or not they want to exchange those floating rates for fixed
rates. Since they already have taken out the loan, they would like the interest rate swap to end
when the loan ends. Additionally, since they want to exchange floating payments for fixed
payments on a loan, they would want to be the fixed rate payer in the interest rate swap.
Therefore they would purchase an American payer swaption with fixed tenor and an option
period that expires in three years.
Since we have discussed the basic kinds of swaption contracts, at this point it is
worthwhile to discuss pricing methods of swaptions so the reader has an understanding of what
techniques can be used to determine the premiums for these complex financial instruments.
European swaptions are complicated to price because, although there is only one exercise date,
whether or not the option will be exercised and what the multiple payments in the interest rate
swap could be must be taken into account. One way to value European swaptions is by tweaking
Black’s model for valuing futures options. Essentially, the tweaks that need to be made for
Black’s model to be able to price European swaptions are changing the value of the underlying,
the volatility, and the discount factor. In Black’s model, the options contract and the futures
contract do not have to expire at the same time. This helps us, because in swaptions the option
period ends before the interest rate swap would end if the option was exercised. Hull and White
have also shown a quick way to value European swaptions using an analytic approach that gives
us results similar to what Monte Carlo simulations for the same or similar values would give.42
Because European swaptions are the simplest kind of swaption there are good ways to price
them. We will not entirely find the same thing for Bermudan and American swaptions.
Because pricing of Bermudan and American swaptions has to take into account when the
option will be exercised, if it will be exercised, and what the various payments will be if it is
exercised, they are very difficult to price. Typically, Bermudan swaptions are valued using a one
factor no arbitrage model. However, the accuracy of this model has been called into question and
thus the model is somewhat controversial. Another method to price Bermudan swaptions
involves a least squares approach where the value of not exercising on a particular exercise date
Much of the information in this paragraph came from Options, Futures, and other Derivatives, pages 658-661.
is assumed to be a polynomial function of the components of the swaption. An optimal early
exercise boundary approach can also be used. Additionally, Monte Carlo simulation is an
important technique that is used to value Bermudan swaptions.43 Because American swaptions
can be exercised on any date during the option period in addition to having to determine what
happens after that date and combining future exchanges of money, they are very tricky to value.
One proposed technique to value American swaptions is a two factor stochastic model where the
factors are the short-term interest rate and the premium of the futures rate over the short-term
interest rate.44 Better techniques to value both Bermudan and American swaptions are in high
demand. Due to the multitude of factors involved and uncertainty, Bermudan and American
swaptions are very difficult to price and there are arguably no good techniques for the pricing of
these exotic options right now.
Swaps, interest rate derivatives, interest rate swaps, and swaptions are all very popular
financial derivatives. They help individuals and business to hedge against changes in the market
that are not in their favor, especially interest rate risk in interest rate derivatives, interest rate
swaps, and swaptions. The ability to transform an asset or a liability to fit the needs of a person
or organization is very important, but occasionally people who do not understand these
derivatives can be taken advantage of. Because these derivatives are so popular and incredibly
common, it is important for individuals to understand how they work, what goes into their
pricing, and how their values can change. The useful qualities of swaps, interest rate derivatives,
interest rate swaps, and swaptions make them very important and popular financial instruments
for the market today, and thus being knowledgeable about how they work is important for all
people who are influenced by financial markets.
Much of the information in this paragraph came from Options, Futures, and other Derivatives, pages 725-726.
More can be read on this technique in The Valuation of American-style Swaptions in a Two-factor Spot-Futures
Model, written by Sandra Peterson, Richard C. Stapleton, and Marti G. Subrahmanyam.