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Chris Dzera 12/14/2010 MAT-5900 Dr. Volpert 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. 1 http://en.wikipedia.org/wiki/Swap_%28finance%29 2 http://en.wikipedia.org/wiki/Interest_rate_derivative 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 3 http://en.wikipedia.org/wiki/Swap_%28finance%29 4 http://en.wikipedia.org/wiki/Swap_%28finance%29 5 http://en.wikipedia.org/wiki/Swap_%28finance%29 6 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. 7 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 8 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. 9 http://en.wikipedia.org/wiki/Interest_rate_derivative 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 10 http://en.wikipedia.org/wiki/Interest_rate_derivative 11 For more on Black’s model, see Options, Futures, and other Derivatives by John C. Hull, pages 648-661. 12 For more, see Options, Futures, and other Derivatives by John C. Hull, pages 682-685. 13 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 understand. 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. 14 For more, see Options, Futures, and other Derivatives by John C. Hull, pages 686-687. 15 For more, see Options, Futures, and other Derivatives by John C. Hull, pages 688-690. 16 For more, see Options, Futures, and other Derivatives by John C. Hull, pages 689. 17 Options, Futures, and other Derivatives by John C. Hull, page 147. 18 http://en.wikipedia.org/wiki/Interest_rate_swap 19 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 opposing needs.21 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 20 Options, Futures, and other Derivatives by John C. Hull, pages 158 and 159. 21 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 principal.22 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 22 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 entered into. 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. 23 The information from this paragraph was found in Options, Futures, and other Derivatives, page 150. 24 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- .005*.25 , 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 25 The information from this paragraph was found in Options, Futures, and other Derivatives, pages 159 and 160. 26 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. 27 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 28 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 29 This equation can be found on page 77 of Options, Futures, and other Derivatives. 30 The basis for the example given can be found in Options, Futures, and other Derivatives, pages 161 and 162. 31 http://en.wikipedia.org/wiki/Swaption 32 http://en.wikipedia.org/wiki/Swaption 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 33 http://en.wikipedia.org/wiki/Swaption 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 34 http://www.riskglossary.com/link/swaption.htm 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 swaptions. 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 35 http://en.wikipedia.org/wiki/Swaption 36 http://www.sdgm.com/en/Support/Glossary.aspx?term=Bermudan%20swaption 37 http://www.sdgm.com/en/Support/Glossary.aspx?term=American%20swaption 38 http://www.sdgm.com/en/Support/Glossary.aspx?term=American%20swaption 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 understand them. 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. 39 http://www.sdgm.com/en/Support/Glossary.aspx?term=American%20swaption 40 http://en.wikipedia.org/wiki/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 41 Each of the following four examples was created by the author, and they are not based on any other specific examples. 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 42 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. 43 Much of the information in this paragraph came from Options, Futures, and other Derivatives, pages 725-726. 44 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.