Chris Dzera
5/3/2011
MAT-5900
Drs. Frey and Volpert
Swaptions are very popular but complex financial instruments that are frequently traded
over the counter on markets today. Certain kinds of swaptions can be priced analytically, but
others types of swaptions must be priced through the use of Monte Carlo simulations. These
more complex swaptions and other interest rate derivatives are priced by modeling the interest
rate, the underlying in these calculations, using one of many popular interest rate models. The
choice of the model typically depends on what interest rate derivative is being priced. Since this
paper focuses on swaptions, we will begin with an explanation of what swaptions are and how
they work to give the reader an understanding of what exactly we are attempting to price using
these models. Additionally, we will explain why interest rate derivatives and swaptions in
particular are so popular. Because the interest rate derivatives market is among the most popular
derivatives market in the world, it is important to understand how to price these derivatives and
how some of the more commonly used models work. In this paper, we will discuss various
methods of pricing interest rate derivatives. One of the first interest rate models, Vasicek’s
model, will be explained and some R-code using this model to simulate interest rate paths and
the prices of interest rate swaps and swaptions will be shown. Additionally, we will explore two
of the more popular interest rate models that can be used to price various interest rate derivatives
including swaptions, the Heath-Jarrow-Morton model and the Brace-Gatarek-Musiela model.
Included in the discussion of the BGM or LIBOR market model will be some examples of recent
research about swaption pricing that features use of the BGM model. Finally, we will explain
some of the real world factors that influence interest rates and cause them to move in a certain
1
manner. Because swaptions are so common and interest rate models are widely used in the
pricing of financial derivatives it is important to have an understanding of how they both work.
A swaption is an option on an underlying interest rate swap, and therefore any
explanation of what swaptions are and how they work must begin with a discussion of interest
rate swaps. An interest rate swap is an agreement between two parties to exchange cash flows in
the future, where one party pays a predetermined fixed rate multiplied by a notional amount and
the other party pays an agreed to floating interest rate multiplied by the same amount.1 There are
some other types of interest rate swaps that involved fixed for fixed rates, floating for floating
rates, different currencies, and different principals, but they are far less common.2 The payments
in an interest rate swap are exchanged at fixed intervals, and the payments are based on the
interest rates one interval prior. Because of this, there is no uncertainty about what the first
exchange of payments will be. Although technically there is an exchange of payments both ways,
the payments only go one way. Additionally, the interest rates are compounded based on how
many exchanges of payments happen in a given year. Interest rate swaps also have no value
when they are created, such that the payments are expected to even out over time. Because it is
unlikely that two parties will have exactly opposing needs in an interest rate swap in terms of
what rates they want to exchange and on what principal they want to exchange it, a financial
institution typically takes the opposite position from the one that another party wants to take in
the swap in exchange for a slight edge in percentage so they make money.3 Interest rate swaps
are common over the counter financial derivatives that allow organizations to change the
payments that they are making or receiving on an asset or a liability.
1
Options, Futures, and other Derivatives by John C. Hull, page 147.
http://en.wikipedia.org/wiki/Interest_rate_swap
3
Options, Futures, and other Derivatives by John C. Hull, pages 151 and 152.
2
2
As an example, let us assume that 3M is interested in exchanging fixed rate payments on
a ten year $100 million loan at 3.6% where payments are required four times per year for floating
rate payments. They contact a financial institution who agrees to a ten year interest rate swap
with them on May 2, 2013. In this case, 3M is the floating rate payer, paying LIBOR + .2%, and
the financial institution pays a fixed rate of 3.6%. Both parties pay their percentage on a $100
million notional principal, payments are exchanged every three months, and so the interest rates
are compounded four times per year. Now let us assume that the date this interest rate swap is
agreed to the LIBOR rate was 3.8%. Then on August 2, 2013, 3M will pay 3.8%, the rate three
months prior, plus .2% as agreed to. This 4% will be divided by four since it is compounded four
times annually, to give 1% on the notional amount of $100 million or a payment of $1 million.
On the same date, the financial institution would pay 3.6% divided by four or .9% on the $100
million, or $900 thousand dollars. The net payment and only exchange of payments would be
3M paying the financial institution $100 thousand. Three months later, 3M would pay the
financial institution LIBOR on August 2, 2013 plus .2% divided by four again on the $100
million, and the financial institution would again pay $900 thousand, and the payments would
continue in this manner four times a year for a total of ten years.4
Interest rate swaps and swaptions can be used for hedging purposes to help organizations
reduce their interest rate risk. Organizations who take out a considerable amount of loans or give
them out in the form of bonds can gain or lose a lot of money depending on how interest rates
move. If an organization has to make interest rate payments and has agreed to a variable rate and
rates rise, that could cost the organization a considerable amount of money, and even more if
they have multiple variable rate loans. The same goes for if a company has multiple fixed rate
4
The basis for the example given can be found in Options, Futures, and other Derivatives, pages 148-150.
3
loans and interest rates drop. On the other hand, a party that is a bond holder or a creditor for
some institution is receiving interest rates, and could stand to lose money if they are receiving a
variable rate and rates drop or are receiving a fixed rate and rates rise. Both interest rate swaps
and swaptions allow parties to transform a fixed rate asset or liability into a floating rate asset or
liability and vice versa, by receiving inputs that essentially cancel out their payments or by
paying out what they receive and receiving a different payment instead. Ultimately, these
financial instruments allow organizations who do not have balance between fixed and floating
rate loans or bonds to become balanced and mitigate interest rate risk.5
The components of a swaption have to include the components of an interest rate swap:
the notional principal, the length of the underlying swap, the frequency of exchange of payments,
the floating rate of interest, and the fixed or strike rate. Also agreed to in a swaption contract are
the length of the option period, the date or dates when the option can be exercised, and the
premium paid for the option.6 Essentially the way a swaption works is that the option holder has
a single date or multiple dates when they can elect to enter into an interest rate swap, depending
on the type of swaption. If on any of these dates the option holder exercises their option, an
interest rate swap of the agreed to length, frequency of payments, and principal with the agreed
to rates begins. In the case that the final exercise date passes without exercise, there is no
exchange of payments beyond the premium being paid from the option holder to the seller.
Swaptions are relatively unique options, because since there are multiple payments to be
exchanged in the future the option holder is not certain whether or not they will make money
when they exercise, and the option holder in fact could lose. In the case where there are multiple
exercise dates, the holder has to weigh how much money they expect to make at that given date
5
6
The information from this paragraph was found in Options, Futures, and other Derivatives, pages 150 and 151.
http://en.wikipedia.org/wiki/Swaption
4
versus how much money they would expect to make if they waited till a later exercise date.
Swaptions are extremely difficult to price, and one reason for that is their many components.
The many types of swaptions include payer swaptions, receiver swaptions, European
swaptions, Bermudan swaptions, American swaptions with fixed tenor, and American swaptions
with fixed end date. Payer and receiver swaptions refer to the fixed and floating interest rates in a
swaption. In a payer swaption the option holder pays the fixed rate and receives the floating rate,
and in a receiver swaption the option holder pays the floating rate and receives the fixed rate.7 It
may be easier to think of a payer swaption like a simple call option, where the strike price is the
fixed or strike rate and the underlying is the floating rate. In this case, the option holder hopes
that over the course of the swap the floating rate rises above the fixed rate to the extent that it
will make them money. A receiver swaption then can be thought of like a simple put option the
same way, with the strike price being like the fixed rate and the floating rate being like the
underlying, and the option holder hoping that the floating rate dips below the fixed rate. The
European, Bermudan, and American swaptions refer to the exercise dates. For a European
swaption there is just one date where the option holder can elect to exercise and enter into an
interest rate swap.8 In a Bermudan swaption, there are multiple predetermined dates where the
option holder can elect to exercise.9 American swaptions allow the option holder to exercise at
any date within the option period. An American option with fixed tenor designates an American
swaption where, regardless of when the option holder exercises, an interest rate swap of a fixed
length begins on that date. On the other hand, an American option with fixed end date has a
predetermined end date, and if the option is exercised after a certain date the length of the
7
http://www.riskglossary.com/link/swaption.htm
http://en.wikipedia.org/wiki/Swaption
9
http://www.sdgm.com/en/Support/Glossary.aspx?term=Bermudan%20swaption
8
5
underlying interest rate swap is less than what it could have been.10 All Bermudan swaptions
have fixed end date. Because there are so many different types of swaptions and variations on
those types it is important to have an understanding of all of them.
Swaption pricing varies greatly depending on the type of swaption, and for some types of
swaptions there is really no standard way to price them right now. European swaptions are
typically priced using Black’s model for futures options, which is a beneficial model because the
options contract and futures contract do not have to expire at the same time as a part of Black’s
model. The model is tweaked slightly, changing the values of the underlying, the volatility, and
the discount factor. Hull and White have also shown a quick analytic way to price swaptions that
produces results similar to Monte Carlo simulations for the same parameters.11 Bermudan
swaption pricing is generally done using Monte Carlo simulations, and some examples of
research into Bermudan swaption pricing will be discussed later. Whether or not early exercise is
beneficial is typically determined by running a simulation to calibrate an absolute exercise
boundary function or a relative exercise boundary function.12 Often the Brace-Gatarek-Musiela
or LIBOR market model, which will be discussed later, is used as the interest rate model for
these simulations. The accuracy of one factor interest rate models in the pricing of Bermudan
swaptions has been questioned.13 American swaptions are difficult to price because early
exercise exists on every date. Two proposed models include 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
10
http://www.sdgm.com/en/Support/Glossary.aspx?term=American%20swaption
Much of the information in this section of the paragraph came from Options, Futures, and other Derivatives,
pages 658-661.
12
Pricing Bermudan swap options using the BGM model with arbitrage-free discretisation and boundary based
option exercise by Henrik Alpsten, pages 13 and 14
13
Options, Futures, and other Derivatives, pages 725-726.
11
6
interest rate, and another model that uses trinomial trees.14 Because there is not yet an industrial
standard like Black’s model for pricing European swaptions or the LIBOR market model for
pricing Bermudan swaptions, there is presently a lot of research being done into how American
swaptions should be priced. There are many different ways to price swaptions depending on
which type, including the use of Monte Carlo methods for Bermudan and American swaptions.
Vasicek’s model is one of the first interest rate models, introduced in 1977 by Oldrich
Vasicek. The model is a one factor short rate model, based on macroeconomic principles.15
Vasicek’s model is considered to be the first interest rate model to account for the element of
mean reversion in interest rates.16 Mean reversion is an important characteristic in interest rate
modeling, because unlike stocks interest rates typically hover around a certain value instead of
consistently rising or falling in the long run. This mean reversion element works by making a
part of the model positive if the present rate is below the long-run mean and negative if it is
below the mean, and the magnitude of the mean reversion factor is dependent on the speed of
mean reversion. The model uses geometric Brownian motion based on the volatility of the
interest rate similar to the modeling of stocks. Inputted into the model are the speed of mean
reversion and volatility as previously mentioned, along with the long run mean interest rate,
current interest rate, and a random variable as implied by the use of geometric Brownian motion.
The output of this model is an instantaneous rate, which can be added to the previous rate to give
the next rate given a time interval dt. Ultimately while the mean reversion element of Vasicek’s
model is a major benefit and was an important step in interest rate modeling, because the model
14
More can be read on the first technique in The Valuation of American-style Swaptions in a Two-factor SpotFutures Model, written by Sandra Peterson, Richard C. Stapleton, and Marti G. Subrahmanyam.
15
An Equilibrium Characterization of the Term Structure by Oldrich Vasicek
16
http://en.wikipedia.org/wiki/Vasicek_model
7
allows interest rates to become negative it is not a very commonly used one in present day
interest rate modeling.
We were able to implement Vasicek’s model in R to achieve many different things. The
first significant output we received from implementing the Vasicek model was a graph of
multiple interest rate random walks. Afterwards, we were able to take an average walk where
over time depending on the speed of mean reversion interest rates converged to the mean. At a
large number of runs this average walk smoothed out to give a curve that showed where interest
rates would be expected to be at a given amount of time. This average walk allowed us to price
an interest rate swap at a given point in time given those parameters based on what interest rates
were expected to be in the future. Significant manipulations of the code allowed pricing of a
European swaption based on Vasicek’s model. This pricing was based on a prior calibration
using an initial simulation in R to determine what the exercise rate should be, and a second
simulation to avoid any bias. The code remained very flexible and outside of the calibration to
determine where we should exercise could have any inputs changed very easily and go through a
new set of walks and create a good price, and the calibration process is relatively simple. While
trying various inputs in the model, we were able to see some of the downsides to the Vasicek
model, particularly the possibility for negative interest rates. At low numbers of runs, high
volatilities, and low speeds of mean reversion we were able to see negative interest rates. This
happened most of all with low speeds of mean reversion and higher volatilities. The result is not
surprising, as a low speed of mean reversion means that the mean reverting factor is somewhat
muted and the higher volatility can make more of an impact on where the interest rate moves
than the mean reversion factor in the model. Even at large numbers of runs, if speed of mean
reversion was particularly low or volatility was very high, rates seemed to be extremely
8
unpredictable. Assuming that interest rates with similar speeds of mean reversion and volatilities
exist, this would signify that the behavior of those rates would be extremely difficult to predict.
Although being able to implement one of the models we will discuss later in the paper and
pricing a Bermudan swaption as opposed to a European swaption would have been preferable,
the coding was a very interesting process that was more difficult than anticipated and pricing a
European swaption using Vasicek’s model certainly felt like an accomplishment when
completed.17
A more common model in present day interest rate modeling is the Heath-Jarrow-Morton
(HJM) model. The HJM model is a framework for interest-rate modeling that allows any specific
no arbitrage model to be fitted to it as a special case. Implementing the HJM model via Monte
Carlo simulation is relatively simple.18 Because this model naturally reproduces the market
prices of basic instruments, unlike equilibrium models, it creates a simpler and more effective
pricing method. Arbitrage pricing models like the HJM model are strongly based on the efficient
market hypothesis, a result of which is that market prices of financial instruments do not create
opportunities for arbitrage. Additionally, the HJM model implies the dynamics of the short rate.19
It is worth noting that the yield curve is an input in the HJM framework as opposed to an
output.20 Because there are many special cases of the HJM model, we will explore two of them.
One special case of the HJM model is the Hull-White or extended Vasicek model,
developed in 1989. Two features of the Hull-White model imply that the mean and variance of
the short rate are bounded, a realistic feature for the modeling of the short rate. The Hull-White
17
The code for the interest rate paths, the pricing of the interest rate swap, and the pricing of the European swaption
are included at the end of this paper, along with a graph of interest rate paths and a graph of the average interest rate
path for the inputs included in the code
18
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 82
19
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 88
20
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 81
9
model is called the extend Vasicek because the formula for the short rate in the Hull-White
model is the same as the formula for the short rate in Vasicek’s model.21 This is beneficial for the
Hull-White model, because the model is able to limit some of the downsides to the Vasicek
model by working as an arbitrage model instead of an equilibrium model, but it still takes mean
reversion into account unlike some other arbitrage models. The Hull-White model does have the
same downside as the Vasicek in that interest rates can ultimately become negative, although
with mean reversion the likelihood of obtaining a negative rate is frequently low. A benefit to the
Hull-White model is it can easily be implemented using binomial trees, meaning it can be used to
price path dependent options rather easily.22 Ultimately it is believed that the benefits of the
Hull-White model outweigh its downsides, and it is a relatively good one to use under the HJM
framework.
Another special case of the HJM model is the Ho-Lee model, developed in 1986 with the
intention of being used with binomial trees. Two basic features of the model suggest that the
short rate will vary around a quadratic function of time, and that variance increases. Because
both of these features are counter-intuitive, some have tweaked the model so that the short rate
behaves in a more realistic manner.23 The Ho-Lee model is also related the Hull-White model,
because the Hull-White model actually reduces to the Ho-Lee model when speed of mean
reversion is eliminated.24 The Ho-Lee model is probably not as appealing as the Hull-White
model, but can be used to price options on zero-coupon bonds among other applications.25
A very common model for the pricing of Bermudan swaptions is the Brace-GatarekMusiela or LIBOR market model, developed in 1997. In this model, forward rates are treated as
21
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 92
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 93
23
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 90
24
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 93
25
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 91
22
10
lognormal martingales under their respective forward measures.26 In the case of European
swaptions, our instruments of interest, the swap rate is approximately a lognormal martingale
under its own swap measure. A martingale is a stochastic process that has the property that when
we have all observations up to time s, the conditional expected value of an observation at a later
time t is equal to the observation at the earlier time s.27 This property is associated with fair
games in investments or speculations.28 Making this approximation is what allows Black’s
formula to be applied to swaption pricing. The major breakthrough for Brace, Gatarek, and
Musiela was that there was a unique and strictly positive solution to expression for the dynamics
of the forward rate with maturity at time T. This solution has interesting properties as well: it is
mean reverting, bounded from above and below by lognormal processes, and if aspects of the
expression for the rate are smooth than the solution has the same degree of smoothness.29 This
model is extremely broadly used and can be used to ratchet caps and floors, Bermudan
swaptions, target redemption notes, autocaps, zero coupon swaptions, constant maturity swaps,
and spread options among many other interest rate derivatives.30 A major appeal is its versatility
as it calibrates well to Black’s model for example which is used to price European swaptions, but
the BGM model can be implemented in Monte Carlo simulations as well.31 The BGM model is
extremely popular and is an industrial standard to price many kinds of financial derivatives,
including swaptions. We will cover some examples of recent research papers featuring use of the
BGM model to price Bermudan swaptions and other financial instruments.
26
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 182
http://en.wikipedia.org/wiki/Martingale_%28probability_theory%29
28
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 22
29
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 185
30
http://en.wikipedia.org/wiki/LIBOR_market_model
31
Interest Rate Modeling: Theory and Practice by Lixin Wu, page 200
27
11
A paper by Riccardo Rebonato of the Royal Bank of Scotland Quantitative Research
Centre focuses on the evaluation of Bermudan swaptions, which he considers to be one of the
most pressing problems in interest rate derivative pricing. Rebonato proposes a calibration
methodology of the BGM model to Monte Carlo simulations that he believes adds to one of the
two more popular recently proposed techniques. This technique is essentially to attempt to use a
calibration technique that recovers the price of European swaptions ending at the same time from
the dynamics of forward rates.32 While this is less conceptually appealing than the other
technique, it is numerically easier and Rebonato believes that he has come up with a way to do
this that recovers extremely similar prices and also calibrates well to a user-specified forwardrate covariance matrix. Rebonato believes it is significant to accomplish both of these since
almost all exotic options depend on multiple sets of variables.33 His paper is another step in
attempting to perfect the pricing of Bermudan swaptions.
Additionally, a paper by Mikkel Svenstrup of the Aarhus School of Business examines
the significance of using suboptimal exercise strategies for the pricing of Bermudan swaptions.
As a basis for his paper, Svenstrup examines opposite claims made by Longstaff et al. (2001) and
Andersen & Andreasen (2001) about this issue. The paper by Longstaff et al. (2001) argues that
losses from the use of suboptimal exercise strategies are economically significant for swaption
holders even if they are within bid/ask spreads, because these losses could have been prevented if
better strategies were used. In fact, they make the claim that using strategies from one-factor
models that are constantly re-calibrated to market data costs Bermudan swaption holders as a
whole billions of dollars. This conclusion is based on an extensive simulation designed to
32
Accurate and Optimal Calibration to co-terminal European swaptions in a FRA-based BGM framework by
Riccardo Rebonato, page 1
33
Accurate and Optimal Calibration to co-terminal European swaptions in a FRA-based BGM framework by
Riccardo Rebonato, page 2
12
replicate present practice. 34 On the other hand, the study by Andersen & Andreasen (2001) states
that losses from using the best fit one-factor model are insignificant and the model provides a
good mark-to-market of Bermudan swaption prices. They also found that if properly calibrated to
the prices of caps and European swaptions from two-factor models, one factor models can give
Bermudan swaption prices that are slightly higher than two-factor models. Both papers use
simulation techniques, although they estimate the optimal exercise strategy differently.35
Ultimately, based on his own set of various simulations, Svenstrup ends up agreeing with
Andersen & Andreasen (2001). He also found that the strategy to estimate optimal early exercise
that Longstaff et al. (2001) used was superior to the one that Andersen & Andreasen (2001), and
that even the exercise strategy worst one factor model outperformed the one Andersen &
Andreasen (2001) used. This is an interesting result given that he ended up agreeing with
Andersen & Andreasen (2001) on the other result and one would think their strategy would have
more effective given they found that the losses from their suboptimal strategy were not
significant.36 Svenstrup’s paper had an interesting result that gave a positive outlook about the
way that Bermudan swaptions are currently priced.
Given this paper is about the simulation of interest rates and financial instruments that are
priced using interest rate models, it is worth discussing what real world factors can influence
interest rates. This is particularly interesting right now as the United States economy is in need of
a boost in some areas, and interest rates have been kept relatively low as an attempt to get out of
or continue to move away from the recession. Monetary and fiscal policy, implemented in the
case of the United States by the Federal Reserve and the United States federal government
respectively, both can influence interest rates. Typically the Federal Reserve attempts to
34
On the suboptimality of single-factor exercise strategies for Bermudan swaptions by Mikkel Svenstrup, page 2
On the suboptimality of single-factor exercise strategies for Bermudan swaptions by Mikkel Svenstrup, page 3
36
On the suboptimality of single-factor exercise strategies for Bermudan swaptions by Mikkel Svenstrup, page 27
35
13
manipulate interest rates through open market operations, or the sale or purchase of bonds.
Although it is difficult to control what happens with certain rates and how banks and individuals
react to a change in rates, the Federal Reserve can essentially put the federal funds rate exactly
where it wants to. The federal funds rate is the rate at which banks pay and receive when they
borrow money from one another. Typically a change in the money supply through an open
market sale or purchase of bonds will also change overall rates, with rates increasing if the
money supply decreases and doing the reverse in the opposite case. When the Federal Reserve
wants rates to decrease, they conduct an open market purchase of bonds. This increases bank
reserves which decreases the federal funds rate, and also increases the overall money supply
which decreases overall rates. The opposite strategy applies when the Federal Reserve wants
rates to increase.37 We have seen this be very effective recently, as the Federal Reserve has kept
interest rates very low through monetary policies including quantitative easing, and plans to
continue to do so in the near future. Fiscal policy can influence interest rates as well, although
this is probably more of an indirect result than the Federal Reserve’s intentional efforts to keep
interest rates in a certain area. Expansionary fiscal policy, increases in government spending or
tax cuts, typically cause interest rates to rise, while the reverse policies tend to keep interest rates
lower.38 This is because typically expansionary policy leads to increased output and prices and
more demand for money, while restrictive policy leads to decreased output and prices and less
demand for money. Changes in the demand for money will influence interest rates in a logical
fashion, more demand for money causes them to rise and less demand for money causes them to
fall. The fact that purchasing and selling of bonds by the Federal Reserve can impact interest
rates indicates that it is plausible that yield curves, which are impacted by these purchases and
37
38
Macroeconomics: Principles & Policy by William J. Baumol & Alan S. Blinder, Eleventh Edition, page 266-268
Macroeconomics: Principles & Policy by William J. Baumol & Alan S. Blinder, Eleventh Edition, pages 281-282
14
sales, would be helpful for the prediction of how interest rates will move when used as inputs
such as in the HJM model. However, there must be some difficulty in predicting where interest
rates move given they often depend on the behavior of unemployment and inflation, which the
Federal Reserve will manipulate interest rates to attempt to change. Presently, interest rates are
being held low but individuals must assume that they will be increased by the Federal Reserve if
inflationary pressures from rising commodity prices and quantitative easing are felt. It is
interesting to consider how this possibility affects interest rate modeling today. There are many
other factors that will influence interest rates, but from a basic macroeconomic perspective fiscal
and especially monetary policy are very important determinants of interest rates.
Swaptions are very complex financial instruments that are very popular because they
allow for hedging against interest rate risk. Given that swaptions are options on interest rate
swaps, it is critical to have some understanding of what the underlying interest rate swap is so we
understand what the option is on. Additionally, it is important to understand the many different
types of swaptions. A swaption can either be a payer or a receiver swaption, which refers to the
fixed and floating interest rates in the underlying interest rate swap. Additionally, a swaption can
be a European swaption, a Bermudan swaption, or an American swaption. These differences
refer to the amount of exercise dates the option holder has during the option period. Additionally,
an American swaption can have fixed tenor or fixed end date which refers to how the length of
the underlying interest rate swap can be changed depending on how long the option holder waits
to exercise their option and enter into the interest rate swap, assuming they do. The pricing of
swaptions requires that some form of interest rate model is used, as the underlying in a swaption
contract is an interest rate. Bermudan swaptions are priced using Monte Carlo simulations,
typically using the LIBOR market or BGM model as the interest rate model in these simulations.
15
Our attempt to price swaptions in R started with trying to determine which interest rate model
should be used, and led to some extensive research on the subject of interest rate modeling.
Although some of the literature was difficult to work through, much was learned about interest
rate modeling in general, especially the ones that were discussed in this paper. The end goal of
this project was to price a Bermudan swaption accurately using an appropriate model. However,
struggles with determining how to implement the BGM or HJM model in R and determining
how to calibrate an early exercise strategy led to the inability to accomplish the goal. Despite this
failure, a considerable amount was learned about interest rate modeling and swaptions over the
course of this project. While the level of success that was hoped for was unfortunately not
achieved, what was ultimately able to be accomplished felt significant.
16
R-code
##Vasicek interest rate paths
r<-.12 #The initial riskless interest rate
mean<-.1 #Long run mean interest rate
smr<-.3 #Speed of reversion of the interest rate to the mean
sigma<-.03 #Volatility of the interest rate
n<-200 #Number of intervals in the simulation
runs<-50000 #Number of simulations of interest rate paths
t<-10 #Length of the interest rate simulation
dt<-t/n #Time step
walks<-matrix(double(runs*(n+1)),n+1,runs) #Creating the matrix of random walks
walks[1,]<-r #Setting the first step of each walk in the matrix equal to the current rate
##Simulating the random walks implementing Vasicek’s model using a matrix of random
variables
for (j in 2:(n+1)){
walks[j,]<-walks[j-1,]+smr*(mean-walks[j-1,])*dt+sigma*rnorm(runs,0,sqrt(dt))}
##plotting the walks
x<-seq(0,t,dt)
matplot(x,walks[,1:10], type="l", lty=1)
##Finding the average at each iteration
average<-double(n+1) #Creating the vector of average walks
for(i in 0:n+1){
average[i]<-mean(walks[i,])} #Designating spot i in the average vector as the mean of the ith
iteration
##Plotting the average walk
matplot(x,average,type="l",lty=1)
##Pricing an interest rate swap
rk<-.005 #Riskless interest rate
freq<-1/2 #Frequency of exchange of payments in years, so ½ would be twice a year
principal<-1000000 #The notional principal in the interest rate swap
k<-.105 #The fixed rate in the interest rate swap
T<-seq(freq,t/freq,by=freq) #A vector of times by frequency
payment<-double(t/freq) #A vector of the length of the total number of payments
17
##Designating each location in the payment vector to be the average at the previous location
times the ##principal minus the principal times the strike rate, all discounted to present day
for (h in 1:(t/freq)){
payment[h]<-(average[((h-1)*t)+1]*principal-principal*k)*exp(-rk*T[h])}
sum(payment) #The present value of the interest rate swap from the perspective of the fixed rate
payer
##Swaption Interest rate paths
r<-.11 #The initial riskless interest rate
mean<-.1 #Long run mean interest rate
smr<-.3 #Speed of reversion of the interest rate to the mean
sigma<-.03 #Volatility of the interest rate
n<-200 #Number of intervals in the simulation
runs<-50400 #Number of simulations of interest rate paths
exercise<-.5 #Time to exercise
length<-4.5 #Length of the underlying interest rate swap
t<-exercise+length #Length of the interest rate simulation
dt<-t/n #Time step
walks<-matrix(double(runs*(n+1)),n+1,runs) #Creating the matrix of random walks
walks[1,]<-r #Setting the first step of each walk in the matrix equal to the current rate
##Simulating the random walks implementing Vasicek’s model using a matrix of random
variables
for (j in 2:(n+1)){
walks[j,]<-walks[j-1,]+smr*(mean-walks[j-1,])*dt+sigma*rnorm(runs,0,sqrt(dt))}
##Finding the average at each iteration
average<-double(n+1) #Creating the vector of average walks
for(i in 0:n+1){
average[i]<-mean(walks[i,])} #Designating spot i in the average vector as the mean of the ith
iteration
##pricing a swaption
rk<-.005#Riskless interest rate
freq<-1/2 #Frequency of exchange of payments in years, so ½ would be twice a year
principal<-10000 #Notional principal of the interest rate swap
k<-.1075 #Strike rate
exrate<-average[(exercise*n)/t] #The exercise rate calibrated from a prior simulation for
maximal payoff
T<-seq(freq+exercise,t/freq,by=freq) #A vector of times by frequencies
payment<-double(length/freq) #A vector of the length of the total number of payments
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##Swaption payoff that creates a total payment for every given random walk, and exercises
based on a ##previously calibrated exercise rate, then places each payoff in a vector of length
runs
SwaptionPayoff<-double(runs)
for (g in 1:runs){
for (h in 1:(length/freq)){
payment[h]<-(walks[(n/t*freq)*(h-1)+1*(n/(t/exercise))+1,g]*principal-principal*k)*exp(rk*T[h])}
if(walks[((exercise*n)/t),g]>exrate){SwaptionPayoff[g]<-sum(payment)}
else if (walks[((exercise*n)/t),g]<exrate){SwaptionPayoff[g]<-0}}
mean(SwaptionPayoff) #The swaption price
sd(SwaptionPayoff)/sqrt(runs) #Standard error of the swaption price
##Creating an optimal payoff function where the payoff at any given point is the value of the
interest rate ##swap if it is positive, and zero if it is negative – this is what would happen if we
could perfectly predict ##where interest rates would move which we cannot – thus values in the
SwaptionPayoff vector will be ##negative but none in this vector will
OptimalPayoff<-double(runs)
for (q in 1:runs){
for (w in 1:(length/freq)){
payment[w]<-(walks[(n/t*freq)*(w-1)+1*(n/(t/exercise))+1,q]*principal-principal*k)*exp(rk*T[w])}
OptimalPayoff[q]<-pmax(sum(payment),0)}
mean(OptimalPayoff) #Price of a swaption if we could use an optimal payoff strategy
sd(OptimalPayoff)/sqrt(runs) #Standard error of the optimal payoff price
average
0.110
0.105
0.10
0.05
walks[, 1:10]
0.15
0.115
0.120
0.20
##Below on the left is a plot of random interest rate walks with the same inputs as in the code
above, to ##the right is a plot of the average random walk with the same parameters and 200,000
runs
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