1

Interest Rate Interest Rate OptionsOptions Interest rate options provide the

right to receive one interest rate and pay another.

An interest rate call pays off if the interest rate ends up above the strike rate. The holder pays the strike rate and receives the market rate, usually LIBOR.

An interest rate put pays off if the interest rate ends up below the strike rate. The holder pays the market rate – LIBOR - and receives the exercise rate.

2

Interest rate options usually are written by dealers and are tailored to the needs of a specific clientele.

The options are typically European, i.e., they can be exercised only at expiration. The exercise price, or as it is referred to: the strike rate usually is set at the current level of the spot rate; for example, the current six -month LIBOR. In this case, the options pays off on the basis of the difference between the six-month LIBOR at expiration and the exercise or strike rate.

3

The payoff of an interest rate call:

  (principal)Max{O,LIBOR-X}[n/360],

Where X = the exercise rate and n = the number of days from the

option’s expiration to the actual payment

day.* * when exercised, the payment by the

writer is made not at the option’s expiration but at a future date that corresponds to the maturity of the underlying spot instrument.

4

Example 1: A call option written on the 90-day

LIBOR with an exercise rate of 11% which is the current LIBOR, based on a principal amount of $25M and with an expiration date 30 days hence.

Upon expiration, the call holder exercises the call if the market 90-day LIBOR exceeds 11%. In this case, the payment from the call call holder will be made 90 days from the exercise date. The call holder’s profit will be:$25M[LIBOR – 11%][90/360.]

If, the 90 – day LIBOR at expiration were 13.45%, for instance, the Call holder’s profit would be:

$25M{.1345-.11}[90/360] = $153,125.

5

Hedging a Planned Hedging a Planned Loan with an Interest Loan with an Interest

Rate CallRate Call Example 2:

A firm plans to borrow $20 million in 30 days at the 90-day LIBOR+100 bps. The loan is taken out in 30 days and will mature 90 days later. The loan is paid back in one lump sum. The firm faces the risk of increasing LIBOR during the next 30 days and would like to establish a maximum rate it will pay on the loan. Thus, the firm buys an interest rate call based on 90-day LIBOR with an exercise rate at the current 90-days LIBOR: X = 10%. The payoff is based on 90 days and a 360-day year. The firm pays now the call premium = $50,000.The table below describes different possible results of the loan rate hedged by the call.

6

LIBOR in 30 days

Payoff of Call at Loan Maturity

Interest on Loan at Maturity

TotalEffectiveInterest

AnnualizeCost of loan with Call

AnnualizedCost of Loan without Call

6.00% $0 $350,000 $350,000 8.39% 7.29%

6.50 0 375,000 375,000 8.93 7.82

7.00 0 400,000 400,000 9.48 8.36

7.50 0 425,000 425,000 10.02 8.90

8.00 0 450,000 450,000 10.57 9.44

8.50 0 475,000 475,000 11.12 9.99

9.00 0 500,000 500,000 11.67 10.53

9.50 0 525,000 525,000 12.22 11.08

10.00 0 550,000 550,000 12.78 11.63

10.50 25,000 575,000 550,000 12.78 12.18

11.00 50,000 600,000 550,000 12.78 12.74

11.50 75,000 625,000 550,000 12.78 13.29

12.00 100,000 650,000 550,000 12.78 13.85

12.50 125,000 675,000 550,000 12.78 14.41

13.00 150,000 700,000 550,000 12.78 14.97

13.50 175,000 725,000 550,000 12.78 15-54

14.00 200,000 750,000 550,000 12.78 16.10

7

The results in the table are obtained as follows:

First, we put the call premium and the loan on the same

time footings.The premium of $50,000

today, will be valued:

$50,000[1+.11(30/360)]= $50,458

in 30 days. Thus, in 30 days, the firm effectively receives:

$20,000,000-$50,458= $19,949,542.

8

Next, consider two outcomes, one ofwhich leaves the call out-of-the-moneyand one of which leaves the call in-the-money.The Call is out-of-the-money.LIBOR at Expiration is 6 %. Interest onthe loan:

$20,000,000[.07(90/360)] = $350.000.Total effective interest: $350,000Amount borrowed: $19,949,542Amount paid back: $20,350,000Effective annual rate:

8.39% OR, 0839.1542,949,19$

000,350,20$ 90

365

9

The call is in-the-moneyLIBOR at Expiration is 14%. Interest on the loan:  $20,000,000[.15(90/360)] =

$750,000.Call is in-the-money, exercised, and

pays:$20,000,000(.14-.10)(90/360)

= $200.000.Total effective interest: $750,000-$200,000=$550,000.Amount borrowed: $19,949,542Amount paid back: $20,550.000Effective annual rate:

12.78% OR, 1278.1542,949,19$

000,550,20$ 90

365

10

The next figure illustrates the cost of the planned fixed-rate loan with and without the interest rate call. The fixed-rate loan plus the call creates a maximum cost of 12.78%, which is reached if LIBOR ends up at 10% or above. Note that this payoff graph looks similar to a covered call or short put. The difference is that previously, the loss/profit profile was based on the underlying asset price. Here, the payoff is based on an interest rate.

11

The Cost of Planned Loan with and

without the Interest Rate Call

6

8

10

12

14

16

18

5 6 7 8 9 10 11 12 13 14

Loan

Loan plus theInterest Rate Call

LIBOR at Expiration (Percent)

Ann

ualiz

ed C

ost o

f L

oan

(Per

cent

)

12

AN INTEREST RATE PUT: Next we illustrates an interest rate put.

Avery common use of an interest rate put is by a bank that lends at LIBOR plus possibly a spread. It thus, faces the risk of a decline in LIBOR before the loan is given out. In general the payoff from an interest rate put is:(principal)Max{O,X-LIBOR}[n/360],

Where X = the exercise rate and n = the number of days from the

option’s expiration to payment day.

13

Example 3: A bank will lend $10 million in 90

days at 180-day LIBOR + 150 bps. The loan will mature 180 days from the day the loan is given out and is paid back in one lump sum.

The bank buys an interest rate put for $26,500 with a strike rate of 9%, thereby putting a floor to the rate it will receive. The put will pay off to the bank in 90 days from now, the 180-day LIBOR is below the strike rate of 9%. The payoff is based on 180 days and a 360-day year. The exercise price X is the current LIBOR is 9%.

The following table describe some possible results of the loan with the protective interest rate put.

14

LIBOR in 90 days

Payoff of Put at Loan Maturity

Interest on Loan at Maturity

Total Effective Interest

Annualized Return on Loan with Put

Annualized Return on Loan without Put

5.00% $200,000 $325,000 $525,000 10.32% 6.70%

5.50 175,000 350,000 525,000 10.32 7.22

6.00 150,000 375,000 525,000 10.32 7.75

6.50 125,000 400,000 525,000 10.32 8.28

7.00 100,000 425,000 525,000 10.32 8.81

7.50 75,000 450,000 525,000 10.32 9.34

8.00 50,000 475,000 525,000 10.32 9.87

8.50 25,000 500,000 525,000 10.32 10.40

9.00 0 525,000 523,000 10.32 10.93

9.50 0 550,000 550,000 10.86 11.47

10.00 0 575,000 575,000 11.39 12.00

10.50 0 600,000 600,000 11.92 12.54

11.00 0 625,000 625,000 12.46 13.08

11.50 0 650,000 650,000 13.00 13.62

12.00 0 675,000 675,000 13.54 14.16

12.50 0 700,000 700,000 14.08 14.71

13.00 0 725,000 725,000 14.62 15.25

15

Here, the bank plans to lend in 90 days at LIBOR+150 bps. The amount of the loan is $10 million, and the loan will be for 180 days and be paid back in one lump sum. The payoff is based on 180 days and a 360-day year. The put premium is $26,500. Its payoff is($10M)Max(0,.09-LIBOR)180/360First, we compound the premium forward for 90 days at today's LIBOR + 150bps:

 $26,500[l + .105(90/360)] = $27,196.

The effective proceeds paid by the bank when the loan is taken out are:

$10,000,000 + $27,196 = $10,027,196.

16

The put is out-of-the-moneyLIBOR at Expiration is 12%

Interest on the loan: $10,000,000[.135(180/360)]

= $675,000.Again, the put premium value in 90 days:

$26,500[1 + .105(90/360) = $27,196

Total effective interest: $675,000Amount paid out on loan: $10,027,196Amount repaid on loan: $10,675,000Effective annual rate:

 

13.54%. or, 1354.1196,027,10

000,675,10 180

365

17

The put is in-the-money:LIBOR at Expiration is 7%Interest on the loan:

$10,000,000[.085(180/360)] = $425,000.

Put is exercised, and the bank receives:  $10,000,000(.09 - .07)(180/360)

= $100,000.Total effective interest received:

$425,000 + $100,000 = $525,000.Amount paid out on loan: $10,027,196Amount repaid on loan: $10,525,000Effective annual rate:

10.32%. or, 1032.1196,027,10

000,525,10 180

365

18

The next figure illustrates the return on the loan with and without the interest rate put.

Notice that the payoff looks like a call or protective put.

In previous examples, these were bullish strategies that paid off if an asset price rose. Here they pay off

ifan LIBOR increases. The PUT provides a minimum return of

10.32%,which is reached if LIBOR is at 9

percentor below.

19

Return on Planned Loan with and without interest Rate Put

6

8

10

12

14

16

18

5 6 7 8 9 10 11 12 13 14

Loan

Loan+Interest Rate Put

LIBOR at Expiration (Percent)Ann

ualiz

ed C

ost o

f L

oan

(Per

cent

)

20

Pricing Interest Rate Options

The Black model requires the forward price of the underlying asset, the exercise price, the risk-free rate, the time to expiration, and the volatility of the forward price. Here, we use the forward rate for the forward price, the strike rate for the exercise price, and the volatility of the forward rate for the volatility of the forward price. The risk-free rate and the time to expiration are the same variables as before. Because of the delay between the option expiration and the day the payoff is actually made, the computed price must be discounted using the forward rate.

21

Consider example 2. The interest rate call we examined expires in 30 days and has a strike rate of 10%.Suppose that the 30-day continuously compounded risk-free rate is 8% and the volatility - the standard deviation - of the forward LIBOR- is σ = 0.2. Let the 30-day forward LIBOR for 90-day deposits be 11.01 percent. The time to expiration is 30/365, or T = .0822. Let c be the premium obtained from the Black model, then the total cost of the option is:

[c](principal)(n/360)

n = the number of days in the underlying LIBOR instrument

22

Calculating the Black Price for Calculating the Black Price for an Interest Rate Callan Interest Rate Call

Lf0 = .1101;X=.10;r=.08;σ=.2;T=.0822

d2 = 1.71 -.2 .0822 = 1.65.N(1.71) =.9564.N(1.65) =.9505.

c = {e–(.08)(.0822)[.1101(.9564) - .10(.9505)]}e -.1101 (90/365)

= .0099.

($20,000,000)(90/360)(.0099) = $49,500.

Notice that this amount is very close to the $50,000 charged by the dealer.

71.10822.2.

0822.)2(.5.10.

1101.ln

d

2

1

23

A similar approach is used for an interest rate put.

By employing the put-call parity for options on futures,:

P = C - (F0 - X)e –rT

one easily obtains the put price once the call price is obtained. Alternatively, a direct computation of the Black's model for put options is:

P = e –rT[XN(-d2) -Lf0N(-dl)].

24

Interest Rate Caps, Interest Rate Caps, Floors, and CollarsFloors, and Collars

An interest rate cap is a series of European interest rate calls that pay off at dates corresponding to the interest payment dates on a loan. Each option is a separate interest rate call. These individual component options are called caplets.

At each interest payment date of a cap, the holder of the cap decides whether to exercise the call based on whether the market LIBOR has risen above the exercise rate.

A price is paid up front for the cap. The price corresponds to the sum of the prices of the series of calls that make up the cap.

25

A cap example: On January 2, a Firm borrows $25 million over one year. It will make payments on April 2, July 2, October 2, and next January 2. On each date, starting with January 2, LIBOR in effect on that day will be the interest- rate paid over the next three months. The current LIBOR is 10%. The firm wishes to fix its loan rate at or below 10%, so it buys a cap for an up-front cap premium of $70,000. The payoffs are based on the exact number of days and a 360-day year. At each interest payment date, the cap will be worth:

$25M(n/360)Max{0, LIBOR - .10}.

In the formula, LIBOR is the rate that was in effect at the beginning of each quarter.

26

For the first quarter, the firm will payLIBOR of 10 percent in effect on January 2. Thus, on April 2 it will owe $25,000,000(.10)(91/360) = $631,944,based on 91 days from January 2 to April 2 . Then, on April 2, LIBOR is 10.68%. This is greater than 10% thus,the cap will pay off at the next interest payment date and the holder of the

cap will receive a payment of:

$25M(91/360)(.1068 - .10)= $42,972.

27

This will help offset the interest of $674,917, based on a rate of 10.68% for 91 days from April 2 to July 2. On July 2, LIBOR is 12.31%, so the cap will pay off on October 2. The net effect of these cash flows is seen in the table below. On January 2, the firm received $25 million from the lender but paid out $70,000 for the cap for a net cash inflow of $24,930,000. It made periodic payments as shown and on the next January 2, made the final interest payment less the cap payoff and repaid the principal. Notice that because of the cap, the interest payments differ only because of the different number of days in each quarter and not because of the rate. The interest rate is capped at LIBOR of 10 percent.

28

If we wish to know what annualized rate the firm actually paid, we essentially must solve for the internal rate of return, which requires a computer or financial calculator. We are solving for the cash flow that equates the present value of the four payouts to the initial receipt:

.y)(1

9$25,638,88

y)(1

$638,889

y)(1

$631,944

y)(1

$631,8440$24,930,00

43

21

The solution is y = .026. Annualizing y gives a rate of (1.026) 4 –1 = .108 or, 10.8%.

29

Solving for the internal rate of return for the cash flows of the un capped loan gives an annual rate of .117 or 11.7. Thus, the cap saved the firm 90 basis points, because during the life of the loan, interest rates generally were higher than they were at the time the loan was initiated. Of course, if rates had fallen, the firm would have benefited less because the caplets would have been out-of-the-money and would not have been exercised, but the premium was paid up front.

The next two tables summarize an hypothetical outcome, assuming different LIBOR rates for the duration of the loan and the cap.

30

Summary of the interest Rate Summary of the interest Rate Cap.Cap.

Scenario: On January 2, a firm takes out a $25M, one-year loan with interest paid quarterly at LIBOR. The firm buys an interest rate cap with a strike of 10 percent for a premium of $70,000. The payoffs are based on the exact number of days and a 360-day year.n = the number of days in the quarter.I= Interest due.

Date, n

LIBOR I Cap Payment

1.2 10.00% -$70,000

4.2 91

10.68% $631,944 0

7.2 91

12.31% $674,917 $42,972

10.2 92

11.56% $786,472 $147,583

1.2 92

$738,556 $99,667

31

The next table summarizes The cash flow with and without the cap:

Date

Principal Payment

Net cash flow with the cap

Net cash flow without

the cap

1.2 0 $24,930,000

-$25,000,000

4.2

0 -$631,944 -$631,944

7.2

0 -$631,944 -$674,917

10.2

0 -$638,889 -$786,472

1.2

$25,000,000

-$25,638,88

9

-$25,738,556Effective annual rate paid on the loan:

With cap: 10.8%; Without cap: 11.7%.

Note: This is but one of infinite number of possible outcomes to the cap. It is used only in order to illustrate how the payments are determined and not their likely amounts.

32

Interest Rate FloorsInterest Rate Floors

The lender in a floating-rate loan may want protection against falling rates. This type of protection can be purchased with an

interest rate floor, which is a series of interest rate put options expiring at the interest payment dates. Each component put is called a floorlet.

33

At each interest payment date, the payoff of an interest rate floor tied to LIBOR with an exercise rate of, say,X%, payoffs based on the exact number of days, n and a 360-day Year and a notional principal N will be: 

(N)Max(0,X - LIBOR)(n/360).

As previously, LIBOR is determined at the beginning of the interest payment period.

34

An Interest Rate Floor:An Example

Suppose that on December 16 a bank makes a one-year, $15 million loan with payments made at 3-month LIBOR on March 16, June 16, September 15, and next December 16. Currently, it is December 16, and the 3-month LIBOR is 7.93%. Thus, on March 16 the bank will receive: $15M[.0793(90/360)) = $297,375 in interest.Assume that the new rate on March 16 is 7.50% percent. Thus, the floor is in-the-money and will pay off: 

$15M(.08 - .075)(92/360) = $19,167

on the next interest payment day.

35

This pay off from the floor will add to the interest payment of $287,500, which is lower because of the fall in interest rates. The complete results for the one-year loan are shown in the table below. The floor is in-the-money and thus is exercised on each of the last three interest payment dates. This is because in this example, the interest rates were lower than 8% during the entire year.

36

The lender paid out $15,000,000 up front to the borrower and another $30,000 for the floor. Following the same procedure as in the cap, we can solve for the periodic rate that equates the present value of the inflows to the outflow. This rate turns out to be about 1.97 percent. Annualizing this gives a rate of (1.0197) 4 - 1 = .081 or 8.1%. The cash flows without the floor yield an annualizes return associated with these cash flows is 7.4%. Thus, the floor boosted the bank’s return by 70 basis points. Of course, in a period of rising rates, the bank will gain less from the increase in interest rates.

37

Summary of the interest Rate Summary of the interest Rate floor.floor.

Scenario: On December 16, a bank makes a $25M, one-year loan with interest paid quarterly at LIBOR. The bank buys an interest rate floor with a strike of 8% for a premium of $30,000. The payoffs are based on the exact number of days and a 360-day year.n = the number of days in the quarter.I= Interest due.

Date, n LIBOR I Floor Payment

12.16 7.93% -$30,000

3.16 90 7.50% $297,375 0

6.16 91 7.06% $287,500 $19,167

9.16 92 6.06% $267,692 $35,642

12.16 92

$232,300 $74,367

38

The next table summarizes The cash flow with and without the floor:

Date

Principal Payment

Net cash flow with the floor

Net cash flow without

the cap

12.16

0 -$15,030,000

-$15,000,000

3.16

0 $279,375 $297,375

6.16

0 $306,667 $287,500

9.15 0 $303,333 $267,622

12.16

$15,000,000

$15,306,667

$15,233,300Effective annual rate received on the loan: With floor: 8.1%;

Without floor: 7.4%.Note: This is but one of infinite number of possible outcomes to the floor. It is used only in order to illustrate how the payments are determined and not their likely amounts.

39

Interest Rate Interest Rate CollarsCollarsConsider a firm planning to borrow

money that decides to purchase an interest rate cap. In so doing, the firm is trying to place a ceiling on the rate it will pay on its loan if rates increase. If rates fall, the firm can gain by paying lower rates. In some cases, however, a firm will find it more advantageous to give up the right to gain from falling rates in order to lower the cost of the cap. One way to do this is to sell a floor. That is, the firm sells the floor in order to finance the cap.The combination of a long interest rate cap and short interest rate floor is called an interest rate collar. The premium received from selling the floor helps finance part or all of the purchased cap.

40

It is even possible to structure the exercise rates on the collar so that the premium received from the sale of the floor exactly equals the premium paid for the purchase of the cap. This is called a zero cost collar. If interest rates fall, the options that comprise the floor will be exercised.The net effect of the collar is that the strategy will establish both a floor and a ceiling on the interest cost.

The existence of both limited gains and losses should remind you of a money spread with options.

41

Interest Rate Interest Rate Zero Cost CollarsZero Cost Collars

Example: a zero cost collar.Consider a firm borrowing $50M overTwo years buys a cap for $250,000with an exercise price of 10% and sellsa floor for $250,000 with an exerciseprice of 8.5%.The loan begins on March 15 and willrequire payments at approximately 91-day intervals at LIBOR.Remember that the principal amountof $50M is paid on the final date –March 14, two years hence.

42

Summary of the interest Rate Summary of the interest Rate Collar.Collar.

n = the number of days in the quarter.I= Interest due.

Date, n LIBOR I Cap Paymen

t

Cap Payment

3.15 10.50%

-$250,00

0

$250,000

6.16 92

11.56%

$1,341,667

0 0

9.14 91

11.75%

$1,461,056

$197,167

0

12.14 92

9.06% $1,485,069

$221,181

0

3.15 91

9.50% $1,145,083

0 0

6.14 91

7.62%1

$1,200,694

0 0

9.14 92

8.31% $973,667 0 -$112,444

12.15 92

7.93% $1,061,833

0 -$24,278

3.14 89 $980,236 0 -$70,458

43

Summary of the interest Rate Summary of the interest Rate Collar.Collar.

n = the number of days in the quarter.I= Interest due.

Date Net cash flow with

collar

Net cash flow with cap only

Net cash flow

without cap and without

floor

3.15 $50,000,000 $49,750,000

$50,000,000

6.16 -$1,341,667 -$1,341,667

-$1,341,667

9.14 -$1,263,889 -$1,263,889

-$1,461,036

12.14 -$1,263,889 -$1,263,889

-$1,485,069

3.15 -$1,145,083 -$1,145,083

-$1,145,083

6.14 -$1,200,694 -$1,200,694

-$1,200,694

9.14 -$1,086,111 -$973,667 -$973,667

12.15 -$1,086,111 -$1,061,833

-$1,061,833

3.14 -$51,030,694 -$50,980,23

6

-$50,980,23

6

44

By now, you should be able to verify the numbers in the tables. The interest paid on June 15 is based on LIBOR on March 15 of 10.5% and 92 days during the period. The cap pays off on September 14 and December 14 because those are the ends of the periods in which LIBOR at the beginning of the period turned out to be greater than 10%. The floor pays off on September 14 and December 15 of the next year and on March 14 of the following year, the due date on the loan. Note that when the floor pays off, the firm makes, rather than receives, the payment.

45

Following the procedure previously described to solve for the internal rate of return we can solve for the borrowing rate: With the collar 9.82%With the cap only 9.91% Without unhedged 10.08%.

It is seen that the cap by itself would have helped lower the firm's cost of borrowing from 10.08% to 9.91%.By selling the floor and thus creating a collar, the cost of the loan was lowered even more to 9.82%.

This completed the collar example.

46

Interest Rate Interest Rate Options and SwapsOptions and Swaps

Now that we have introduced caps andfloors as options on interest rates, it ishelpful to see how they are related toswaps.Let us consider a simple,

One - Payment swap,which is basically an FRA. Suppose you pay, the floating rate, LIBOR, and receive the swap rate. Thepayoff will be:

Swap Rate - LIBOR.Now suppose in addition you buy acap and sell a floor, both with astrike rate of X.

47

The cap payoff is:0 for LIBOR X.LIBOR - X for LIBOR > X.

The floor payoff is: 

- (X - LIBOR) for LIBOR X.0 for LIBOR > X.

 Thus, the combinedswap, cap and floor payoff is: 

Swap Rate - X for LIBOR X. Swap Rate - X for LIBOR > X. 

48

The swap rate is set in the market. It usually is determined by the term structure of interest rates. The strike rate X, however, is chosen by the investor. If one chooses X to be equal to the market-determined swap rate, the transaction becomes not only risk-free but also guarantees a payoff of zero. In that case, the transaction must have zero cash flow up front. Since the swap has no initial cash flow, then the long cap and short floor must have no initial cash flow. Thus, in this case, the floor premium must equal the cap premium.The implication of this result is that:

a swap is equivalent to a combination of a long cap and

short floor where the strikes on the cap and floor are equivalent

and equal to the swap rate.

49

The cap-floor-swap parity.

A pay-fixed and receive-floating swap

a long cap and short floor.or

A pay-floating, receive-fixed swap

a short cap and long floor.

50

A final caveat:Remember that the above are forsingle payment swaps and caps andfloors that have just one caplet andfloorlet. When the swap has multiplepayments, as it usually does, the capand floor must have correspondingmultiple caplets and floorlets. Eachcaplet and floorlet must have a strikeequal to the swap rate. While the sumof the overall premiums for the capand floor must be equal, it is not thecase that the premium for an individual caplet and its correspondingfloorlet will be equal.