SWAPTIONS AND INTEREST RATE MODELINg

1. SWAPTIONS AND INTEREST RATE MODELINg Chris Dzera

2. Outline • Explore specific inputs into Vasicek’s model, how to find them and whether or not we can get the mean back after simulations with realistic parameters • Some discussion of interest rate models • Review some of the presentation from last time • Explanation of what swaptions are • Swaption pricing methods • How to determine whether or not to exercise • R-code for swaption pricing

3. Vasicek Inputs in Excel • Use regression parameters slope, intercept, and steyx along with stdev • Speed of mean reversion is the negative of the slope, estimate of long-term mean is the intercept divided by the speed of mean reversion, calculate volatility normally although they also indicate it can be estimated by dividing steyx by the long-run mean • Values seem to check out, quite similar to what other ways to check for these • Example

4. Popular Interest Rate Models • Heath-Jarrow-Morton model is a model where instead of modeling just the short rate (a point on the forward rate curve) we are able to capture the full dynamics of the entire forward rate curve • The Hull-White model or extended Vasicek is an extension of the HJM model where, as opposed to Vasicek’s model, has two time dependent factors that are held constant in the Vasicek model and is used to price bonds and derivatives • The Brace-Gatarek-Musiela model or LIBOR Market model is used to price exotic options, is nice because it correlates to Black’s model which is commonly used, and is essentially an industrial standard for pricing these instruments

5. Questions and Clarification • Riskless interest rate • Swaptions and interest rate swaps • Who enters? • What makes them worthwhile to enter? • Buyer vs. seller in interest rate swap • Floating vs. fixed more common • Sensitivity of swap payments to interest rate • Interest rate models • Benefits to various parties • Effectiveness & what is most common

6. Interest Rate Swaps • An interest rate swap is an agreement between two parties to exchange payments equal to an rate multiplied by a given principal • These exchanges happen at fixed intervals for a predetermined time period, and are based on rates one interval prior • At onset a swap is set up so it is worthless (or if it is done with a financial institution that institution is given a slight edge so they make money), free to set up • Interest rate swaps are mostly used for hedging

7. Interest Rate Swaps • The two rates involved are almost always a fixed interest rate and a floating interest rate • Depending on where interest rates move, the swap can have a value after onset • The first payment is certain because it is based on interest rates when the swap was agreed to • While there are technically exchanges both ways, the money only goes one way • Rates are compounded once for the amount of exchanges in a year (so if there are payments every 3 months, each rate is compounded 4 times a year)

8. Options • An option is a financial derivative where one party pays a premium to the other party in exchange for the right to determine whether or not they want to enter into a contract on specified dates (or a single date) • In this case, the option holder has a limited downside because they can see if things move in their favor or not • This is most beneficial in the case when the option holder is certain whether or not they are in the money and what will pay the most

9. Swaption Components • A swaption gives the holder the right, but not obligation, to enter into an interest rate swap • Determined when the contract is agreed to are: • Premium • Strike rate (or fixed rate in an interest rate swap) • Floating rate • Notional amount • Frequency of payments • Length of underlying swap • Length of option period (along with how many option dates there are)

10. How does it work? • Two parties agree to the aforementioned terms • On a certain date (or multiple dates) during the option period, the contract holder is given the right to choose whether or not they want to enter into the interest rate swap contract • This can happen before or after the swap would begin depending on the type of swaption • If the holder exercises then the interest rate swap begins, and if the holder does not exercise during the exercise period then the only exchange of payments is the premium • In any swap contract, unlike the options we dealt with in class, at exercise it is unknown whether or not the option holder will end up making money at the end of the underlying interest rate swap

11. Payer vs. Receiver swaption • Payer swaption – gives the owner of the swaption the right to enter an interest rate swap where they pay the fixed leg and receive the floating leg • Receiver swaption – gives the owner of the swaption the right to enter into an interest rate swap where they pay the floating leg and receive the fixedleg • Name of swaption depends on who pays/receives the fixed leg or strike rate

12. European, Bermudan, and American • There are three different ways the right to exercise the option in a swaption contract can be scheduled • European – the owner of the contract can exercise his right to enter the swap on one date, at maturity • Bermudan – the owner of the contract can exercise his right to enter the swap at certain predetermined dates between the start and end dates of the option period (multiple exercise dates) • American – the owner of the contract can exercise his right to enter the swap at any date between the start and end dates of the option period agreed to

13. American Swaptions: Fixed Tenor vs. Fixed End Date • There are two types of American swaptions: • An American swaption with fixed tenor is a when the length of the underlying swap is a fixed time length and the swap begins as soon as the option is exercised. Again, if the exercise period ends without the option holder electing to exercise it expires worthless. • An American swaption with fixed end date is when the predetermined period of time includes the length of the option period and the underlying swap, so if the first day the swap would begin passes without the swap being exercised, the length of the underlying swap decreases

14. Purpose of a Swaption as opposed to an Interest Rate Swap • Because the option period can end before the swap takes place, it may be unclear why it is logical to pay a premium for the option to enter a swap contract down the road when the party could just decide whether or not they want to enter into an interest rate swap at that later date without it being at a premium • However, the option holder may be able to receive better terms on the underlying interest rate swap at exercise with a swaption than they would on a standard interest rate swap at that date • Entering into a swaption contract gives the opportunity to exercise on superior terms and already be “in the money” when the swap begins whereas an interest rate swap starts worthless

15. Example: European Receiver Swaption • A company knows that in six monthsit will enter a 10 year loan on $5 million with a 5% interest rate, and wants to reduce its interest rate risk by exchanging its payments on this contract for floating payments since it has multiple fixed rate loans out already • They enter a swaption contract with a 6 month option period, where at the end of half a yearon the expiry date of the option, if they exercise the option a 10 year swap would be initiated

16. Example: Bermudan Payer Swaption • A bond holder knows that in one year they will receive fixed rates of 3% on $2 million bond for a period of five years, and wants to receive a floating rate instead • They enter a swaption contract that expires in one year with specific exercise dates afterward that would not go too far into the swap period • If they elected to exercise after expiry, the length of the swap would be reduced by the amount of time between the end of the option period and the date the option was exercised

17. Example: American Receiver Swaption with Fixed Tenor • A company would receive a LIBOR -.02% on a 10year bond it may purchase sometime in the next two years, but would rather receive a fixed rate instead • They enter a swaption with expiry in 2 years and an underlying swap of length 10 years • If they elect to exercise the swap at any time in the option period, a 10 year swap with terms agreed to begins immediately

18. Example: American Payer Swaption with Fixed End Date • A company knows it will have to pay a floating rate of LIBOR + .03% on an 8 year loan and wants to pay a fixed rate instead • They enter a swaption contract with expiry in 2years and an end date in 30 years • If they elect to exercise the option, a swap that ends when the loan ends will begin on that date

19. European Swaption Valuation • The valuation of European swaptions can be done by tweaking Black’s model for valuing futures options • The swaption model relies on changing the value of the underlying, the volatility, and the discount factor • Black’s model benefits us in this calculation because the option contract and futures contract don’t have to mature at the same time, which is helpful because the option on a swaption and the actual swaption itself do not mature at the same time • There is also a quick way to value European swaptions that Hull and White have shown, using an analytic approach that comes up with results similar to Monte Carlo simulations for similar material • Indications are that certain exercise strategies implemented in Monte Carlo simulations may be superior to quick analytic approaches to European swaption pricing, but Black’s model gets very similar results to simulations and is the industrial standard for pricing European swaptions

20. Bermudan Swaption Valuation • Monte Carlo simulation is essentially the method that is used to price Bermudan swaptions • Valuation of Bermudan swaptions can be done by using one-factor no arbitrage models that are controversial because their accuracy has been questioned • The BGM model discussed earlier (LIBOR Market model) is the most commonly used interest rate model for swaption pricing • Early exercise methods include a least squares approach where the value of not exercising on a particular payment date is assumed to be a polynomial function of the values of the factors in the swaption, and an optimal early exercise boundary approach

21. American Swaption Valuation • Valuing American swaptions is generally considered by investors to be extremely difficult, though it is fortunate that they are not as common as European or Bermudan swaptions • There is no set way to value American swaptions, although there have been certain techniques proposed including a two factor stochastic model where the factors are the short-term interest rate and the premium of the futures rate over the short-term interest rate, and another model that uses trinomial trees

22. Determining Early Exercise • The two main early exercise strategies in Bermudan swaptions are using an absolute exercise boundary function and a relative exercise boundary function • These boundary functions are determined by running calibration simulations that select boundary functions which give the swaption its highest value, since we want to choose the best early exercise strategy • If we start from the last exercise date and assume the option has not been exercised, the boundary is found using a linear search starting at 0 and ending above the largest swap value attained during simulations, and is done until all exercise dates have a value of the boundary function • When the boundary function has been determined, a separate set of simulations are run to avoid bias

23. R-code • I used simulation to price a European Swaption in R • Issues: • How useful is this? • Why not Bermudan • Why Vasicek’s model? • How to exercise? • Accuracy • Ultimately, interesting to do, probably not very useful

24. Summing up • Swaptions are a great way to mitigate interest rate risk with a limited downside • Because they are extremely complex swaptions can be (as I found out) very difficult to price and there is a lot of demand for quick, effective ways to price swaptions since they are extremely popular • Ultimately this ended up a lot more difficult than I thought it would be, but I still learned a lot through the various bumps in the road