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Interest Rate Options

- 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.

- 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.

- The payoff of an interest rate call
- (principal)MaxO,LIBOR-Xn/360,
- Where
- X the exercise rate and
- n the number of days from the options

expiration to the actual payment

day. - when exercised, the payment by the writer is

made not at the options expiration but at a

future date that corresponds to the maturity of

the underlying spot instrument.

- 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 holders profit will be - 25MLIBOR 1190/360.
- If, the 90 day LIBOR at expiration were 13.45,

for instance, the Call holders profit would be - 25M.1345-.1190/360 153,125.

Hedging a Planned Loan with an Interest Rate Call

- Example 2
- A firm plans to borrow 20 million in 30 days at

the 90-day LIBOR100 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.

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- 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
- 50,0001.11(30/360)
- 50,458
- in 30 days. Thus, in 30 days, the firm

effectively receives - 20,000,000-50,458
- 19,949,542.

- Next, consider two outcomes, one of
- which leaves the call out-of-the-money
- and one of which leaves the call in-the-
- money.
- The Call is out-of-the-money.
- LIBOR at Expiration is 6 . Interest on
- the loan
- 20,000,000.07(90/360) 350.000.
- Total effective interest 350,000
- Amount borrowed 19,949,542
- Amount paid back 20,350,000
- Effective annual rate

- The call is in-the-money
- LIBOR 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
- 750,000-200,000550,000.
- Amount borrowed 19,949,542
- Amount paid back 20,550.000
- Effective annual rate

- 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.

The Cost of Planned Loan with and without the

Interest Rate Call

Annualized Cost of Loan (Percent)

Loan

Loan plus the Interest Rate Call

LIBOR at Expiration (Percent)

- AN INTEREST RATE PUT
- Next we illustrates an interest rate put. A
- very 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)MaxO,X-LIBORn/360,
- Where
- X the exercise rate and
- n the number of days from the

options expiration to payment day.

- 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.

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- Here, the bank plans to lend in 90 days at

LIBOR150 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)180/360
- First, we compound the premium forward for 90

days at today's LIBOR 150bps - 26,500l .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.

- The put is out-of-the-money
- LIBOR at Expiration is 12
- Interest on the loan
- 10,000,000.135(180/360)
- 675,000.
- Again, the put premium value in 90 days
- 26,5001 .105(90/360)
- 27,196
- Total effective interest 675,000
- Amount paid out on loan 10,027,196
- Amount repaid on loan 10,675,000
- Effective annual rate

- 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 10,027,196
- Amount repaid on loan 10,525,000
- Effective annual rate

- 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 if
- an LIBOR increases. The PUT
- provides a minimum return of 10.32,
- which is reached if LIBOR is at 9 percent
- or below.

Return on Planned Loan with and without interest

Rate Put

Annualized Cost of Loan (Percent)

Loan

Loan Interest Rate Put

LIBOR at Expiration (Percent)

- 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.

- 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 s

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

Calculating the Black Price for an Interest Rate

Call

- Lf0 .1101X.10r.08s.2T.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.

- 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 rTXN(-d2) -Lf0N(-dl).

Interest Rate Caps, Floors, 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.

- 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)Max0, LIBOR - .10.
- In the formula, LIBOR is the rate that was in

effect at the beginning of each quarter.

- For the first quarter, the firm will pay
- LIBOR 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.

- 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.

- 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

The solution is y .026. Annualizing y gives a

rate of (1.026) 4 1 .108 or, 10.8.

- 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.

Summary of the interest Rate 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.

- The next table summarizes The cash flow with and

without the cap

- Effective 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.

Interest 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.

- 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.

- 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.

- 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.

- 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

banks return by 70 basis points. Of course, in

a period of rising rates, the bank will gain less

from the increase in interest rates.

Summary of the interest Rate 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.

- The next table summarizes The cash flow with and

without the floor

- Effective 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.

Interest Rate Collars

- Consider 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.

- 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.

Interest Rate Zero Cost Collars

- Example a zero cost collar.
- Consider a firm borrowing 50M over
- Two years buys a cap for 250,000
- with an exercise price of 10 and sells
- a floor for 250,000 with an exercise
- price of 8.5.
- The loan begins on March 15 and will
- require payments at approximately 91-
- day intervals at LIBOR.
- Remember that the principal amount
- of 50M is paid on the final date
- March 14, two years hence.

Summary of the interest Rate Collar.

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

Summary of the interest Rate Collar.

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

- 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.

- 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.

Interest Rate Options and Swaps

- Now that we have introduced caps and
- floors as options on interest rates, it is
- helpful to see how they are related to
- swaps.
- 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. The
- payoff will be
- Swap Rate - LIBOR.
- Now suppose in addition you buy a
- cap and sell a floor, both with a
- strike rate of X.

- The cap payoff is
- 0 for LIBOR ? X.
- LIBOR - X for LIBOR gt X.
- The floor payoff is
- - (X - LIBOR) for LIBOR ? X.
- 0 for LIBOR gt X.
- Thus, the combined
- swap, cap and floor payoff is
- Swap Rate - X for LIBOR ? X.
- Swap Rate - X for LIBOR gt X.

- 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.

- 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.

- A final caveat
- Remember that the above are for
- single payment swaps and caps and
- floors that have just one caplet and
- floorlet. When the swap has multiple
- payments, as it usually does, the cap
- and floor must have corresponding
- multiple caplets and floorlets. Each
- caplet and floorlet must have a strike
- equal to the swap rate. While the sum
- of the overall premiums for the cap
- and floor must be equal, it is not the
- case that the premium for an
- individual caplet and its corresponding
- floorlet will be equal.