Interest rate Derivatives

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____________________________________________________________________________________

This note was prepared by Professor Robert M. Conroy. Copyright  2003 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to [email protected]. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of the Darden School Foundation.

Interest rate Derivatives

There is a wide variety of interest rate options available. The most widely offered are interest rate caps and floors. Increasingly we also see swaptions offered. This note will deal with the valuation of these interest rate options. However, we have included a quick review of spot and forward rates and a brief discussion of the valuation of options with delayed payments. Both of these things play a major role in the valuation of interest rate derivatives.

Spot rates and forward interest rates

It is fairly common to think of a single interest rate. However, there are many interest rates and as such, it is necessary to be specific. First, in everything that follows in this note we will be referring to the risk- free rate. In addition, for interest rates it is necessary to specify the time period covered by that rate. Spot rates are interest rates that are used to discount cash received at one point in the future back to today. For example, a six- month spot rate,i would be the appropriate rate to discount any risk- free payoff back to ₆ today. Hence, the term spot rate is the rate that begins today and ends at some future point in time.

We use the term forward rates to refer to interest rates that apply for specific time periods in the future. For example, there are interest rates that apply specifically for the time period beginning six months from now and ending at nine months from now. This would be the forward rate¹ for months 6 through 9,₆ f . ₉

The relationship between the 9- month spot, 6- month spot rate and the forward rate for months 6 through 9 is as follows:

312 612

912 ₆ ₆ ₉

9⋅ ⋅ ⋅

⋅

= ⁱ ^f

i e e

e

See Exhibit 1 for an example of forward and spot rates. We will use these rates as the market for interest rates through out the rest of this note.

1 This forward rate could also be referred to as the 3-month rate 6 months forward. This is a mouth full but it is the most common verbiage used concerning forward rates. For the purpose of this note, we will use the above descriptors to refer to forward rates for specific periods.

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Options with delayed payment

Before we actually deal with interest rate options, we need to address the issue of options with delayed payment. Consider a simple example. Suppose we have a European Call option on a share of stock (no dividends) with a maturity of 9- months and an exercise price of $50. The current share price is $52, and the volatility estimate is .30. The payoff at the maturity of this option depends on the stock price at the end of month 9,S , and the ₉ payoff would be

(

9 − ^,⁰

)

=

(

9 −⁵⁰^,⁰

)

= Max S X Max S Payoff

and we would value the option using the Black-Scholes model with the following inputs:

⋅





























=

30 .

% 24 . 1

50

$ 9

52

$ 57

. 6

$

months

R X Maturity

UAV

Call

f

σ

where the risk-free rate is the 9- month spot rate from Exhibit 1.

Now, we change the timing of the payoff on the option. Instead of the payoff being determined and paid at the end of month 9, let us set the terms of the option such that the payoff is determined at the end of month 9 but the actual payment is made at the end of month 12. If we assume that interest rates are constant, then the value of the payoff received at the end of month 12 determined at the end of month 9 is the present value discounted for the 3 months of delay or

( )

_^

 − ⋅ ⋅

= Max S₉ X e⁻⁹^f^{1 2}^⋅³¹²,0 e⁻⁹^f^{1 2}^⋅³¹²

Payoff .

We can pull the discounting factor outside of the Max function and the payoff becomes

( )

[

9 ^,⁰

]

312

1 2

9 Max S X

e

Payoff = ⁻^f ^⋅ ⋅ − .

Since the discounting factor is outside of the Max function, we can value that payoff and then discount that option value for the delay. Hence the value of the option is delayed payment is

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541 . 6

$ 57 . 6

$ 57

. 6

$ 30

.

% 24 . 1

50

$ 9

52

$

25 . 0172 12 .

12 3

3 91 2

1 2

9 = ⋅ = ⋅ =





























=

⋅

= ⁻ ^⋅ e⁻ ^⋅ e⁻ ^⋅

months

R X Maturity

UAV

e

Call ^f

f f

σ

This is a nice result. It makes it easy to value options with delayed payments. The key is to value the option assuming that payoff is realized and paid at the maturity of the option.

Then we account for the delay by discounting that option value for the delay using the forward rate applicable for the delay time period.

Options on Bonds

In principal we should not be able to value European call options on Bonds. The Black- Scholes model specifically assumes that total volatility over some time period is an increasing function of time². Bonds really have a very different price process. Interest rate changes drive changes in Bond prices. While interest rates might get more volatile as the further out in time we go, bond price sensitivity decreases the further out in time that we go. Sensitivity decreases because Duration³ decreases, as the bond gets closer to a maturity. In fact, if we ignore default risk, bond price risk goes to zero as the bond reaches maturity. The price at that point will be equal to the face value.

One solution is to think differently about bond price volatility. If we are willing to assume that at any point in time that bond prices are log normally distributed, then we can assume that there is some total volatility of the bond price at the option maturity time T, σ . Given this we can estimate a Black-Scholes model volatility,_T σ , as bs

T

T bs

σ = σ

Let us look at a specific example. Suppose we had a 5-year, 4% coupon bond with semi- annual coupon payments. The current cash bond price⁴ is 103.539 per 100. Consider a call option on this bond with an exercise price⁵ of $100 and 21 months to maturity. We estimate that the volatility of the bond price 21 months forward is σ = .08. Note that _T this is an estimate of the price volatility of a bond that has a maturity of not 5 years but

2 This means that the longer the time period the more variable the possible outcomes.

3 Modified duration is a measure of bond price sensitivity. The basic relationship is

(

^yield

)

Duration

ice = − ⋅ ∆

∆Pr

% . As duration decreases, the %?price decreases for any given change in yield.

4 The cash price does include accrued interest. The quoted price excludes accrued interest. However, in this case, where it is exactly 6 months to the next dividend the accrued interest is 0 and the cash and quoted prices are the same.

5 In this simple case, the exercise price is the quoted price, which does not include accrued interest. If the exercise price were the cash price, the exercise price would be $100 plus 3 months of accrued interest, $1 ($100x.04x3/12=$1.00).

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3.25 years. As such, the volatility for the maturity of the option (21 months or 1.75 years) that we would use in the Black-Scholes model is

0605 75 .

. 1

08

. =

=

= T

T bs

σ σ

The last necessary piece is the Underlying Asset Value or UAV. In order to deal with this we appeal to our dividend analogy once again. If I hold the bond, I get the coupon payments. The option holder does not get the coupon payments. Hence, the UAV for the option is the bond price less the present value of the coupon payments or using the rates in Exhibit 1,

625 . 97

$ 2

$ 539 . 103

$

2

$ 2

$ 539 . 103

$

5 . 1 0160 . 1

0136 . 5

. 0112 .

5 . 1 1

5 .

.₆ _{1 2} _{1 8}

=

⋅

−

⋅

−

⋅

−

=

⋅

−

⋅

−

⋅

−

=

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

e e

e UAV

e e

e

UAV ⁱ ⁱ ⁱ

( )

^⋅





























=

0605 .

% 72 . 1

100

$ 21

625 . 97

$ 41

. 3

$

21

months

i r

X Maturity

UAV

Call

bs f

σ

If the exercise price had been in terms of the cash price ($101=$100 + accrued interest⁶), the option value would have been

( )

^⋅





























=

0605 .

% 72 . 1

101

$ 21

625 . 97

$ 936

. 2

$

21

months

i r

X Maturity

UAV

Call

bs f

σ Bond Price Volatility and Yield volatility

While it seems easy to say that we need the bond price volatility, σ , it is much more _T difficult to actually estimate it. Since each bond is unique⁷ and the maturity of a particular bond decreases over time, we cannot look at a series of bond prices and calculate a volatility of the bond price with a specific maturity. One way around this problem is to use Modified Duration⁸, which is bond specific and the volatility of yields, which are not necessarily bond specific.

6 See footnote 4.

7 Each bond is unique in that the coupon ra tes are usually bond specific as is the maturity.

8 See Darden technical note, Duration and Convexity (UVA -F-1238) for a fuller discussion of Modified Duration.

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The basic relationship between bond prices and Modified Duration is Yield

ModDur ice=− ⋅∆

∆Pr

%

If we modify this slightly we can get the following:

Yield Yield

ModDur ice

Yield Yield Yield

ModDur ice

∆

⋅

−

=

∆

⋅∆

⋅

−

=

∆

% Pr

%

If we calculate the standard deviation of the %? of the above⁹, we get the bond price volatility as a function of yield volatility.

(

%∆Pr^ice

)

=^ModDur⋅^Yield⋅σ

(

%∆^Yield

)

σ .

Yield volatilities can be calculated directly from yields. Exhibit 2 lists the yields on US Treasury bonds with maturities of 3 years, 5 years, 7 years and 10 years. Volatility for each of these yields is also shown. Not surprisingly, shorter-term yields are more volatile than long-term yields. Note that the way we calculated the yield volatility is the same as we would have calculated the volatility on stock. As such, we can use this as a Black-Scholes volatility.

Let us use the same bond listed above to estimate the price volatility based on the forward and spot rates in Exhibit 1 and the yield volatilities in Exhibit 2. For both the Modified Duration and the Yield, we need to calculate these for a bond 1.75 years from today with a maturity of 3.25 years. We need the duration and the yield for the underlying bond as of the maturity of the option. Exhibit 3 shows the calculation of the bond price, yield to maturity and modified duration for the bond as of today given the forward and spot rates in Exhibit 1. Exhibit 4 shows the forward bond price, forward yield to maturity and forward modified duration for the same bond forward 1.75 years to coincide with the maturity of the option. This is the relevant data for calculating the Black-Scholes bond price volatility. Using the data from Exhibit 4 and the yield volatility of the 3-year maturity¹⁰, we get a volatility of

(

^%^∆^Pr^ice

)

⁼^ModDur^⋅^Yield^⋅^σ

(

^%^∆^Yield

)

⁼²^.⁹⁹^⋅^.⁰⁴¹³^⋅^.⁴⁸⁰⁹⁼^.⁰⁵⁹

σ

This is the volatility estimate that we would use in our Black-Scholes model and given this volatility, σ_bs =0.059, we would price a European call option on a 4%, 5 year maturity bond with a current value of $103.539, an exercise price of $100 (quoted price), a maturity of 21 months (1.75 years), and interest rates as shown in Exhibit 1 as,

9 Remember the definition of volatility is that it is the standard deviation of the percentage change in the underlying asset.

10 This is the time closest to the remaining time on the bond at the maturity of the option.

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625 . 97

$ 2

$ 539 . 103

$

2

$ 2

$ 539 . 103

$

5 . 1 0160 . 1

0136 . 5

. 0112 .

5 . 1 1

5 .

.₆ _{1 2} _{1 8}

=

⋅

−

⋅

−

⋅

−

=

⋅

−

⋅

−

⋅

−

=

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

e e

e UAV

e e

e

UAV ⁱ ⁱ ⁱ

( )

^⋅





























=

059 .

% 72 . 1

100

$ 21

625 . 97

$ 33

. 3

$

21

months

i r

X Maturity

UAV

Call

bs f

σ

Interest Rate Caps and Floors

Interest rate caps and floors are very popular vehicles used to modify interest rate risk.

Often a firm will issue a floating rate note and at some future date decide to limit its exposure. Interest rate caps and floors are options designed to deal with this. Let’s look at a specific example.

Interest Rate Caps

Today’s date is December 31^st, 2003. The maturity date of the cap is in 6 months, June 30^th, 2004. If the 3-month LIBOR rate on June 30^th is greater than 1.50%, the cap pays the holder 3 months worth of interest on $1,000,000 based on the interest rate differential between the 3- month LIBOR rate on June 30^th, 2004 and 1.50%. If the 3-month

LIBOR6/30 rate on June 30^th is less than 1.50%, the cap expires worthless. An additional feature of this cap is that while the payoff is determined on June 30^th the actual payment is made 3 months later on September 30^th, 2004. See table below for summary of payoff structure.

Condition at maturity 6/30/04 Payoff determined 6/30/04 but paid 9/30/04 If 3-month LIBOR6/30 > 1.50% (3-mon. LIBOR6/30 – 1.50%) x $1,000,000 x .25 Payoff =

If 3-month LIBOR6/30 < 1.50% $0

In more traditional option terms the payoff would be

( )

[

6/30

]

³¹²

9

0 6

, 25 . 000 , 000 , 1

$ ⋅ ⋅ ⁻ ^⋅

⋅

−

= Max LIBOR X e ^f

Payoff

Here we have incorporated the three- month delay in the payment.

We can value this using the Black-Scholes model. From our discussion on call options with delayed payments, we can value the cap as a straight option and then discount the value by the time of the payment delay. For the straight option part, the key inputs are the UAV and the volatility. In this case, the underlying asset is a 3-month LIBOR payment on June 30^th, 2004 equal to

[ (

LIBORT

)

^⋅^$¹^,⁰⁰⁰^,⁰⁰⁰^⋅^.²⁵

]

. We can find the value

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of this payment today, by hedging the LIBOR payment with the sell-side of a Eurodollar futures contract that matures on June 30^th, 2004 and lock in the futures rate. Once the rate is locked in, we can value it by discounting it at the risk- free rate¹¹.

From Exhibit 5, the rate we can lock in is the Eurodollar futures rate for the contract with a maturity of June 2004, F6/30 =1.52%. This is the futures rate for the time period June to September and is essentially the same as the forward LIBOR rate for this time period.

The value of this today is the present value at the 6- month spot rate, 1.12% from Exhibit 1. Note that there is a slight difference between the Eurodollar futures rates shown in Exhibit 5 and the US Treasury forward rates in Exhibit 1. Eurodollar futures are slightly higher¹².

75 . 778 , 3

$

000 , 250

$ 015115 .

25 . 000 , 000 , 1

$

% 52 . 1

25 . 000 , 000 , 1

$

5 . 0112 .

5 . 30 / 6

6

=

⋅

=

⋅

=

⋅

=

⋅

−

⋅

−

UAV UAV

e UAV

e F

UAV ⁱ

Volatility is a bit more problematic. Exhibit 6 shows Eurodollar futures rates for different contract maturities over time. For example, the column labeled 12/03 is the Eurodollar futures contract with a maturity date of December, 2003. The first row labeled 60 months is the rate on that contract 60 months (5 years) prior to the maturity date on the contract, December 1998. The row labeled 12 months is the rate that was quoted on the December 2003 contract twelve months prior, December 2002. Hence, as we go down each column row we have the rate that was quoted for a particular contract maturity date 60 months prior, 57 months prior, etc. If we calculate the volatility¹³ of the rates across each row we have the volatility of the 3-month rate x number of months prior.

Figure 1 shows the volatility estimates of the 3-month LIBOR rates “x” number of

months prior. Here we see the usual result. Volatilities are “humped”. Near term futures rates have much more volatility than futures rates much further out in the future. Since the derivation of the Black-Scholes model assumed that the volatility of the underlying asset was constant, this presents a problem for valuing caps using this model.

For our example, the underlying 3- month LIBOR rate starts 6 months in the future and this time period in the future decreases as we get closer to the maturity. From Figure 1

11 This is essentially the same approach we used in Valuation to value each of the hedged payments when valuing a swap. As you will see, the only major difference here is that we will use the US Treasury spot rates to discount the hedged payment.

12 The underlying asset of a Eurodollar futures contract is an interbank US$ deposit.

They are essentially risk-free but since these deposits do not have a government guarantee, they trade at a slightly higher rate. I have chosen to discount the Futures rate at the US risk-free spot rate. Sometimes you will see the LIBOR spot rate used. Generally, this does not result in a different cap value.

13 Volatility is defined as the standard deviation of the percentage changes or the natural logarithm of the price relatives. Note that since these are quarterly observations we would annualize the quarterly standard deviation by multiplying by the square root of 4.

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we also see that the volatility of the underlying 3-month LIBOR rate also decreases. In order to deal with this, most practitioners effectively cheat. They use average of the volatilities. Again for our example the volatility for the 3-month LIBOR rate 6 months in the future is about .35 but this decrease to about .31 at maturity. Looking at Figure 1 we might use an average volatility of .331 to account for the decline in volatility over the life of the cap. This seems a bit ad hoc but we do similar things every time we estimate volatility for an option.

Now that we have a UAV and volatility, we can value this cap. The UAV is $3,778.75, the exercise price¹⁴ is $3,750, time to maturity is 6 months, the risk- free rate is the 6- month spot rate from Exhibit 1, 1.12%, the volatility is .331 and again from Exhibit 1, the forward rate for months 6 through 9 to account for the delayed payment is 1.48%.

73 . 373

$ 12 . 375

$ 12

. 375

$ 331

.

% 12 . 1

750 , 3

$ 6

75 . 778 , 3

$

25 . 0148 12 .

12 3

3 91 2

9

6 = ⋅ = ⋅ =





























=

⋅

= ⁻ ^⋅ e⁻ ^⋅ e⁻ ^⋅

months

r X Maturity

UAV

e

Call ^f

f f

σ

We could have valued the cap using the rates and then multiplied the result by the dollar amounts involved. With rates, the UAV would be

% 5115 . 1

% 52 .

1 ^.⁰¹¹²^.⁵

5 . 30 / 6

6 = ⋅ =

⋅

=F e⁻ ^⋅ e⁻ ^⋅

UAV ⁱ

001494 . 0 0015005 .

0 0015005

. 0

331 .

% 12 . 1

015 . 6

015115 .

25 . 0148 12 .

12 3

3 912

9

6 = ⋅ = ⋅ =





























=

⋅

= ⁻ ^⋅ e⁻ ^⋅ e⁻ ^⋅

months

r X Maturity

UAV

e

Call ^f

f f

σ

73 . 373

$ 25 . 000 , 000 , 1

$ 001494 .

0 ⋅ ⋅ =

= Call

This is the same as the dollar amount approach. Most often you see the cap valued and quoted with rates and then valued based on the dollar amount involved.

Floors

If interest rate caps are calls then floors are puts. Floors pay the holder the difference between the LIBOR rate and the exercise price when LIBOR is less than the exercise price. These are puts and we value them using put-call parity.

14 X =1.50%⋅$1,000,000⋅.25

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UAV e

X Call Put

e X Call Put

UAV

T r

f

−

⋅ +

=

⋅ +

= +

⋅

−

⋅

−

For example, a floor with the same parameters would have a value of

06 . 324

$

75 . 778 , 3

$ 750

, 3

$ 75 . 373

$ ^.⁰¹¹²^.⁵

=

−

⋅ +

=

−

⋅ +

= ⁻ ^⋅ ⁻ ^⋅

Floor Put

e UAV

e X Call

Put ^r^f ^T

Caps and Caplets

In the example above, we dealt with a single option that is termed a “caplet”. Usually a cap is a series of caplets with the same exercise price but different maturity dates.

Suppose that we use the example above but now the cap is a series of 4 caplets that mature on the following dates:

Mar. 31^st, 2004 Jun. 30^th, 2004 Sept. 30^th, 2004 Dec. 31^st, 2004 The UAV and other inputs for each caplet would be

Maturity Mar. 31^st, 2004 Jun. 30^th, 2004 Sept. 30^th, 2004 Dec. 31^st, 2004 Maturity

(T yrs.) .25 .5 .75 1.00

Futures rate (F)

(Exhibit 5) 1.26% 1.52% 1.74% 1.98%

Spot rate (i ) _T

(Exhibit 1) 1.00% 1.12% 1.24% 1.36%

T i_t

e F

UAV = ⋅ ⁻ ^⋅ ^1.2569% ^1.5115% ^1.7239% ^1.9533%

Exercise Price 1.50% 1.50% 1.50% 1.50%

Volatility ? ? ? ?

Rf (spot rate) 1.00% 1.12% 1.24% 1.36%

Call Value ? ? ? ?

Payment delay .25 .25 .25 .25

Delay period 3/04-6/04 6/04-9/04 9/04-12/04 12/04-3/05 Forward rate

delay period (f) (Exhibit 1)

1.24% 1.48% 1.72% 1.96%

Caplet Value ? ? ? ?

Most of the inputs are readily available and fairly well defined. However, as is usually the case, the volatility estimate is the most problematical. The non constant volatility shown in figure 1 presents a real challenge in valuing caps. There are two approaches that people take. The first is to use an average or “spot” volatility which matches the

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maturity of each caplet. The other is to use an average of the average and use the same

“flat” volatility for each of the caplets.

Figure 2 shows the individual volatilities, the “spot” volatilities (average) and the “flat”

volatilities for data in exhibit 6. If we use the “spot” volatilities from Figure 2 to value the caplets we wo uld get:

Maturity Mar. 31^st, 2004 Jun. 30^th, 2004 Sept. 30^th, 2004 Dec. 31^st, 2004 Maturity

(T yrs.) .25 .5 .75 1.00

Futures rate (F)

(Exhibit 5) 1.26% 1.52% 1.74% 1.98%

Spot rate (i ) _T

(Exhibit 1) 1.00% 1.12% 1.24% 1.36%

T i_t

e F

UAV = ⋅ ⁻ ^⋅ ^1.2569% ^1.5115% ^1.7239% ^1.9533%

Exercise Price 1.50% 1.50% 1.50% 1.50%

Volatility .329 .331 .327 .3052

Rf (spot rate) 1.00% 1.12% 1.24% 1.36%

Call Value 0.0001671 0.0015005 0.0032367 0.0052438

Payment delay .25 .25 .25 .25

Delay period 3/04-6/04 6/04-9/04 9/04-12/04 12/04-3/05 Forward rate

delay period (f) (Exhibit 1)

1.24% 1.48% 1.72% 1.96%

Caplet Value 0.0001666 0.0014949 0.0032228 0.0052182 Caplet $ Value

(x $250,000) $41.66 $373.73 $805.70 $1,304.55

Cap Value

(sum of caplets) $2,525.65

The value ¹⁵of the cap (4 caplets) is $2,525.65.

15 Another alternative would have been to use a flat volatility. In this case, we would use the same volatility for all of the caplets. For example, if we used a flat volatility of .32176, the value of the cap would have been $2,525.65. Firms vary in their approach. Some use spot volatilities and others use flat volatilities.

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Swaption

A swap is an exchange of payments. Swaps exist in ma ny forms. You can see a range of swaps from plain- vanilla interest rates swaps, where there is a simple exchange of fixed rate payments for floating rate payments based on a nominal amount, to more

complicated swaps, which offer exchange of returns on commodities in exchange for returns on exotic stock indexes. In either case, these contracts have value.

Let us examine the valuation of a plain- vanilla interest rate swap¹⁶. Consider a 3- year swap where you receive a fixed rate of 2.328% and pay 3- month LIBOR. Today is December 31, 2003 and the first payment is due March 31^st, 2004. This means that the first fixed payment will be

00 . 820 , 5 4 $

02328 000 .

, 000 , 1

04 $

/

3 = ⋅ =

F

and the floating Rate LIBOR payment made on March 31^st, 2004 will be based on the current 3- month spot LIBOR rate of 1.02% and would be

550 , 2 4 $

0102 000 . , 000 , 1

04 $

/

3 = ⋅ =

L .

The June LIBOR payment would be based on the 3-month LIBOR rate, LIBOR3/04, as of March 31^st, 2004 and the payment would be

000 4 , 000 , 1

$ ³^/⁰⁴

04 / 6

LIBOR

L = ⋅ .

The value of the fixed payment is just the present value of the payments discounted¹⁷ at the LIBOR spot rates. The value of the LIBOR payments is determined by assuming that the LIBOR payments are hedged using Eurodollar futures contracts and locking in the Eurodollar futures rate. These hedged payments are discounted using the LIBOR spot rates.

Exhibit 7 shows the Eurodollar futures rates from Exhibit 5 and the corresponding

LIBOR spot rates derived¹⁸ from those rates. We use the Eurodollar futures to hedge and

16 For a fuller explanation, see Valuation of “Plain-Vanilla” Interest Rate Swaps (UVA -F-1121).

17 In valuing swaps, the convention is to v alue the swaps using the LIBOR spot rates derived from LIBOR Eurodollar futures contracts.

18 The relationship between LIBOR spot and the Futures rates is as follows:

1 1 1

1

1 4 1 4

1 4 



 

 +

 ⋅



 

 +

 =



 

 + spot^t ^t spot^t⁻ ^t⁻ ^t⁻ f^t

and solving for the Libor spot rates in Exhibit 7 yields,

(12)

this results in the hedged payments shown in Exhibit 8. These payments are discounted at the LIBOR spot rate and the value of the LIBOR payments is then $67,740. Also included in Exhibit 8 are the fixed payments, and discounting these at the LIBOR spot rates yields a value of $67,740. The values of the LIBOR payments and the Fixed payments are equal. This is usually the result when a swap is first initiated or priced.

A swaption gives the holder the right to enter into a swap at a set fixed rate at some future date. For example, consider a swaption that gave the holder the right one year from today to enter into a 3- year swap to pay fixed of 2.328% and receive LIBOR on a notional amount of $1,000,000. It is essentially the same as the swap above except that the start date of the swap is December 31^st, 2004.

This swaption will have value at maturity if at maturity, December 31^st, 2004, the 3-year swap rate is higher than 2.328%. Suppose that the 3- year swap rate at maturity is 3.00%.

The holder of the swaption would realize the value of the swaption by exercising the swaption and paying 2.328% and receiving LIBOR. At the same time, the holder would enter into a new swap agreement to pay LIBOR and receive the fixed rate of 3.00%. This would result in a fixed payment of 0.672% (3.00% - 2.328%=0.672%), every 3 months for the next three years.

The key to valuing a swaption is to look at the payoff and realize that this is a series of call options that pay off the difference between the 3- year swap rate on December 31^st, 2004 and the exercise fixed rate. All mature on December 31^st, 2004 but one has a delay of 3 months (pays off March 31^st, 2005), another has a delay of 6 months (pays off June 30^th, 2005), and so on. Hence, the key is to value the basic call option and incorporate the payoff delay in each call.

We will use the Black-Scholes model to value the basic call. Our approach will be to do everything in terms of rates. As usual, the inputs are Underlying Asset Value, Exercise Price, Time to Maturity, Risk-free rate, and volatility. The easy inputs are X=2.328%, T=1 year, and from Exhibit 1, r_f =1.36%. The exercise price, time to maturity and risk- free rate are fairly straight forward. The UAV is a bit more difficult. What we need here is the one year forward swap rate. This is the rate quoted today for a 3-year swap that begins one year from today. Exhibit 9 shows the calculation of the one-year forward 3- year swap rate. Based on the current Futures rates and the implied LIBOR spot rates, the rate today that should be quoted is 3.284%. Using the same analogy as we did for the interest rate caps, the UAV for the swaption should be

% 2396 . 3

% 284 . 3

% 284 .

3 ⋅ ^{1 2} ¹^.⁰ = ⋅ ^.⁰¹³⁶¹^.⁰ =

= e⁻ ^⋅ e⁻ ^⋅

UAV_swaption ⁱ ^months .

4

4 1 4 1

1

1 1

1 ⋅













 −









 



 

 +

⋅



 

 +

= ⁻

−

t t t t

t t

f spot spot

(13)

The other difficult input is volatility. Figure 3 shows the 1-year, 3-year and 5-year interest rate swap rates for the last 5 years. There has been a great deal of volatility. This is reflected in the volatilities calculated from this data. The 1-yr swap rate volatility is .277, the 3-year swap rate volatility is .350 and the 5-year is .298. Since the underlying asset is a 3-year swap rate¹⁹, we can use the .350. With this, we can value the basic call as follows:

005848 . 0

35 .

% 36 . 1

% 00 . 3 12

% 2396 . 3

=





























=

months r

X Maturity

UAV

Call

f

σ

The $ value of the basic call option would be $1,461.97.

Now if the 12 payment differentials were all paid at the maturity of the option, December 2004, the value of the swaption would be just 12 times the basic call option value.

However, while the payment differential is determined on December 2004, each of the twelve payments is made at some future date. Hence, the value of the swaption is actually the value of a series of these basic call options each with a different delayed payment. The underlying swap involves 12 payments. The first begins March 2005, and each payment has a different delay. Exhibit 10 shows the series of swap payment dates and the delay relative to December 2004. Each payment date contributes an amount equal to the basic call value discounted for the delay. For example, the September 2005 payment date is the basic call value of $1,461.97 discounted back for 9 months to December 2004. The discount rate is based on the forward rates²⁰ from December 04 to March 05, March 05 to June 05 and June 05 to September 05. If we do this for each payment, the value of the swaption is $16,759.68.

Summary

In this note we have discussed the valuation of options on bonds, interest rates and swaps.

In each case, there are certain adaptations that are required but the basic process remains the same.

19 The underlying asset is actually the one-year forward 3-year swap rate. F rom our discussion of interest rate caps, we know that the volatility of the spot rate, like the 3-year swap rate, can be different than the volatility of the 3-year swap rate one year forward. For our purposes here, we will ignore this distinction.

20 The forward rates are the risk-free continuously compounded rates from Exhibit 1.

(14)

Exhibit 1.

Forward Rates and Spot Rates (US Treasury) (continuous compounded)

Forward Spot Rates

End of month

From To

Forward rate

Time (months)

time (years)

as of the

end of Spot rate

Dec-03 Mar-04 1.00% 3 0.25 Mar-04 1.00%

Mar-04 Jun-04 1.24% 6 0.50 Jun-04 1.12%

Jun-04 Sep-04 1.48% 9 0.75 Sep-04 1.24%

Sep-04 Dec-04 1.72% 12 1.00 Dec-04 1.36%

Dec-04 Mar-05 1.96% 15 1.25 Mar-05 1.48%

Mar-05 Jun-05 2.20% 18 1.50 Jun-05 1.60%

Jun-05 Sep-05 2.44% 21 1.75 Sep-05 1.72%

Sep-05 Dec-05 2.68% 24 2.00 Dec-05 1.84%

Dec-05 Mar-06 2.92% 27 2.25 Mar-06 1.96%

Mar-06 Jun-06 3.16% 30 2.50 Jun-06 2.08%

Jun-06 Sep-06 3.40% 33 2.75 Sep-06 2.20%

Sep-06 Dec-06 3.64% 36 3.00 Dec-06 2.32%

Dec-06 Mar-07 3.88% 39 3.25 Mar-07 2.44%

Mar-07 Jun-07 4.12% 42 3.50 Jun-07 2.56%

Jun-07 Sep-07 4.36% 45 3.75 Sep-07 2.68%

Sep-07 Dec-07 4.60% 48 4.00 Dec-07 2.80%

Dec-07 Mar-08 4.84% 51 4.25 Mar-08 2.92%

Mar-08 Jun-08 5.08% 54 4.50 Jun-08 3.04%

Jun-08 Sep-08 5.32% 57 4.75 Sep-08 3.16%

Sep-08 Dec-08 5.56% 60 5.00 Dec-08 3.28%

Dec-08 Mar-09 5.80% 63 5.25 Mar-09 3.40%

Mar-09 Jun-09 6.04% 66 5.50 Jun-09 3.52%

Jun-09 Sep-09 6.28% 69 5.75 Sep-09 3.64%

Sep-09 Dec-09 6.52% 72 6.00 Dec-09 3.76%

Dec-09 Mar-10 6.76% 75 6.25 Mar-10 3.88%

Mar-10 Jun-10 7.00% 78 6.50 Jun-10 4.00%

Jun-10 Sep-10 7.24% 81 6.75 Sep-10 4.12%

Sep-10 Dec-10 7.48% 84 7.00 Dec-10 4.24%

(15)

Exhibit 2.

Yield Volatilities

US Treasury Constant Maturity

3-year 5-year 7-year 10-year

Yield (%) %? Yield (%) %? Yield (%) %? Yield (%) %?

Jan-01 5.06 4.99 5.16 5.12

Feb-01 4.59 (0.0929) 4.78 (0.0421) 5.01 (0.0291) 5.1 (0.0039)

Mar-01 4.48 (0.0240) 4.67 (0.0230) 4.88 (0.0259) 4.87 (0.0451)

Apr-01 4.37 (0.0246) 4.66 (0.0021) 4.92 0.0082 4.98 0.0226

May-01 4.5 0.0297 4.94 0.0601 5.17 0.0508 5.3 0.0643

Jun-01 4.49 (0.0022) 4.94 - 5.24 0.0135 5.39 0.0170

Jul-01 4.47 (0.0045) 4.88 (0.0121) 5.21 (0.0057) 5.37 (0.0037)

Aug-01 4.09 (0.0850) 4.62 (0.0533) 4.9 (0.0595) 5.11 (0.0484)

Sep-01 3.91 (0.0440) 4.46 (0.0346) 4.72 (0.0367) 4.85 (0.0509)

Oct-01 3.18 (0.1867) 3.9 (0.1256) 4.33 (0.0826) 4.55 (0.0619)

Nov-01 2.91 (0.0849) 3.66 (0.0615) 4.01 (0.0739) 4.24 (0.0681)

Dec-01 3.28 0.1271 4.04 0.1038 4.51 0.1247 4.75 0.1203

Jan-02 3.59 0.0945 4.38 0.0842 4.84 0.0732 5.07 0.0674

Feb-02 3.62 0.0084 4.37 (0.0023) 4.78 (0.0124) 5.02 (0.0099)

Mar-02 3.73 0.0304 4.43 0.0137 4.82 0.0084 4.98 (0.0080)

Apr-02 4.29 0.1501 4.93 0.1129 5.29 0.0975 5.44 0.0924

May-02 3.78 (0.1189) 4.49 (0.0892) 4.86 (0.0813) 5.08 (0.0662)

Jun-02 3.72 (0.0159) 4.36 (0.0290) 4.75 (0.0226) 5.06 (0.0039)

Jul-02 3.35 (0.0995) 4.08 (0.0642) 4.54 (0.0442) 4.85 (0.0415)

Aug-02 2.6 (0.2239) 3.46 (0.1520) 4.04 (0.1101) 4.47 (0.0784)

Sep-02 2.5 (0.0385) 3.22 (0.0694) 3.78 (0.0644) 4.14 (0.0738)

Oct-02 2.11 (0.1560) 2.75 (0.1460) 3.34 (0.1164) 3.72 (0.1014)

Nov-02 2.14 0.0142 2.92 0.0618 3.54 0.0599 4.01 0.0780

Dec-02 2.51 0.1729 3.31 0.1336 3.89 0.0989 4.22 0.0524

Jan-03 1.99 (0.2072) 2.78 (0.1601) 3.36 (0.1362) 3.83 (0.0924)

Feb-03 2.17 0.0905 3.05 0.0971 3.58 0.0655 4.01 0.0470

Mar-03 1.89 (0.1290) 2.66 (0.1279) 3.21 (0.1034) 3.68 (0.0823)

Apr-03 1.92 0.0159 2.78 0.0451 3.35 0.0436 3.84 0.0435

May-03 1.93 0.0052 2.82 0.0144 3.37 0.0060 3.88 0.0104

Jun-03 1.61 (0.1658) 2.37 (0.1596) 2.94 (0.1276) 3.43 (0.1160)

Jul-03 1.66 0.0311 2.48 0.0464 3.06 0.0408 3.56 0.0379

Aug-03 2.38 0.4337 3.37 0.3589 4.01 0.3105 4.44 0.2472

Sep-03 2.51 0.0546 3.46 0.0267 4 (0.0025) 4.45 0.0023

Oct-03 1.93 (0.2311) 2.84 (0.1792) 3.4 (0.1500) 3.96 (0.1101)

Nov-03 2.44 0.2642 3.34 0.1761 3.88 0.1412 4.4 0.1111

Dec-03 2.64 0.0820 3.46 0.0359 3.98 0.0258 4.4 -

Monthly Yield Volatities 0.1388 0.1110 0.0935 0.0776

Annual Yield Volatities 0.4809 0.3844 0.3237 0.2689

(16)

Exhibit 3

Price, y-t-m and Modified Duration for 5 year 4% coupon bond

Bond PV* PV**

time

beg. time end

Forward rate

as of the end of

Spot rate

Cash Flow

Cash

Flow CC y-t-m Duration Dec-03 Mar-04 1.00% 0.25 Mar-04 1.00%

Mar-04 Jun-04 1.24% 0.50 Jun-04 1.12% 2 1.99 1.97 0.02 Jun-04 Sep-04 1.48% 0.75 Sep-04 1.24%

Sep-04 Dec-04 1.72% 1.00 Dec-04 1.36% 2 1.97 1.94 0.04 Dec-04 Mar-05 1.96% 1.25 Mar-05 1.48%

Sep-08 Dec-08 5.56% 5.00 Dec-08 3.28% 102 86.57 86.91 8.39 Bond Price

103.539

103.539 9.180

Y-T-M 3.228%

Modified Duration (Yrs) 4.45

*This present value (PV) is based on continuous compounding (CC).

**This present value is based on a semi-annual y-t-m.

(17)

Exhibit 4

Forward Price, Forward y-t-m and Forward Modified Duration for 5 year 4% coupon bond

Bond PV PV

time

beg. time end Forward rate

as of the end of

Spot rate

Cash

Flow Time forward spot

Cash

Flow CC y-t-m Duration Dec-03 Mar-04 1.00% 0.25 Mar-04 1.00%

Mar-04 Jun-04 1.24% 0.50 Jun-04 1.12% 2 Jun-04 Sep-04 1.48% 0.75 Sep-04 1.24%

Sep-04 Dec-04 1.72% 1.00 Dec-04 1.36% 2 Dec-04 Mar-05 1.96% 1.25 Mar-05 1.48%

Mar-05 Jun-05 2.20% 1.50 Jun-05 1.60% 2 Jun-05 Sep-05 2.44% 1.75 Sep-05 1.72%

Sep-05 Dec-05 2.68% 2.00 Dec-05 1.84% 2 0.25 2.68% 1.99 1.98 0.0098 Dec-05 Mar-06 2.92% 2.25 Mar-06 1.96% 0.50 2.80%

Mar-06 Jun-06 3.16% 2.50 Jun-06 2.08% 2 0.75 2.92% 1.96 1.94 0.0289 Jun-06 Sep-06 3.40% 2.75 Sep-06 2.20% 1.00 3.04%

Sep-08 Dec-08 5.56% 5.00 Dec-08 3.28% 102 3.25 4.12% 89.22 89.32 5.7703 Forward Bond Price 100.61 100.61 6.10

Forward Y-T-M 4.13%

Forward Modified Duration (Yrs) 2.99

(18)

Exhibit 5.

Eurodollar Futures

Maturity Futures rate

Mar-04 1.26%

Jun-04 1.52%

Sep-04 1.74%

Dec-04 1.98%

Mar-05 2.22%

Jun-05 2.46%

Sep-05 2.71%

Dec-05 2.95%

Mar-06 3.19%

Jun-06 3.43%

Sep-06 3.67%

Dec-06 3.90%

Mar-07 4.17%

Jun-07 4.39%

Sep-07 4.63%

Dec-07 4.87%

Mar-08 5.12%

Jun-08 5.36%

Sep-08 5.59%

Dec-08 5.83%