CHAPTER 14: INTEREST RATE DERIVATIVES
END-OF-CHAPTER QUESTIONS AND PROBLEMS
1.
Fixed- for floating-rate swaps are like a series of FRAs (forward rate agreements) and like issuing a fixed- or
floating-rate bond and using the proceeds to buy the other type of bond. For example, if you enter into a
swap to pay fixed and receive floating, you have entered into a series of FRAs. On a series of future dates,
you have committed to paying interest at a specific rate and receiving interest at a rate determined at the
beginning of the interest payment period. This is exactly like a series of FRAs. Each FRA matches a swap
payment. The first swap payment, however, is not an FRA as that payment value is known. It is more like
taking out a short-term fixed rate loan and using the proceeds to lend at a different short-term rate. The
remaining swap payments, however, are identical to those on a series of FRAs. Alternatively, the swap can
be viewed as issuing a fixed-rate bond and using the proceeds to buy a floating-rate bond. The floating-rate
bond interest payments would be set at the beginning of the interest payment period and the interest is paid
at the end.
2.
An interest rate option requires that you pay a premium up front. Take an interest rate call for example.
When the option expires, you decide whether to exercise it. In effect, you have the right to receive a floating
interest rate and pay a fixed interest rate. You would choose to do that if the floating interest rate were
higher. The actual payment you receive occurs at a later date. With an FRA you pay nothing up front, but
agree that on the expiration date you will make an interest payment at a predetermined fixed interest rate and
receive an interest payment at a floating rate. A financial manager might want to buy an interest rate call or
FRA to protect against a future interest rate increase. In the case of the interest rate call, the manager can
still benefit if rates go down, but that comes at the expense of having to pay a premium up front. With an
FRA, the manager will gain if rates go up and lose if rates go down but pays no up-front premium. So the
manager saves the up-front premium by giving up the right to gain if rates fall.
3.
There are several ways to terminate a swap and you might wish to build these features into the swap
contract. One way is to allow you to have a third party take over your payments. Obviously your
counterparty would need to agree and would put some restrictions on the credit quality of this third party.
Another feature you might want is to allow you to terminate the swap by paying the counterparty the market
value or having it pay you the market value. You might also want to buy an option to terminate the swap
early. This would require an up-front premium. You can also terminate a swap by entering into an offsetting
swap but this would not require the original counterparty's approval up-front or later because your original
swap remains on the books.
4.
Most interest rate derivatives, specifically swaps and options, pay off later than the expiration or settlement
date. For example, at the expiration of an interest rate option, the underlying interest rate is compared to the
exercise rate. If the option is in-the-money, it pays off but at a later date. If the underlying rate is, for
example, the rate on a 90-day Eurodollar, the payoff will typically occur ninety days later. This is in keeping
with the fact that on a given day, the 90-day Eurodollar rate as of that date implies an interest payment that
will be made ninety days later. A similar procedure occurs on the settlement dates of a swap. On an FRA,
however, the payoff typically occurs on the expiration date. This type of payoff is called in-arrears and can
be done on a swap or option, though that is the exception. It is possible to set up an FRA to pay off with a
delayed settlement but that is the exception.
5.
A cap is nothing more than a series of interest rate call options. A cap is designed to protect against a
series of floating rate resets, such as might be encountered when one makes periodic interest payments at a
floating rate. A cap is a portfolio of interest rate options, with each option designed to protect against the
interest adjustment on a specific date. The value of a cap is the sum of the values of the individual interest
rate options that make up the cap, which are called caplets. An interest rate call that is not part of a cap
protects only against a single rate increase.
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Copyright © 2001 by Harcourt, Inc
6.
The two types of swaptions are payer swaptions and receiver swaptions. A swaption is an option to enter
into a swap. A payer swaption is the right to enter into a swap as a fixed-rate payer and a receiver swaption
is the right to enter into a swap as a fixed-rate receiver. The underlying is the rate on a specific type of swap.
For example, consider a two-year payer swaption on a five-year swap on the 180-day LIBOR. It expires in
two years. At expiration, the swaption will be in-the-money if the rate on a five-year swap based on 180-day
LIBOR exceeds the exercise rate specified in the swaption. Because swap rates are directly related to
interest rates, a payer swaption is similar to an interest rate call or cap. When an interest rate call or cap is
exercised, however, it provides a payoff based on the difference between the interest rate and the exercise
rate. When a payer swaption is exercised, its permits the holder to establish a swap, paying the exercise
rate. Because swaps in the market can be established at the higher market rate, the value of the swaption
when it is exercised is determined by the value of an annuity of length equal to the length of the underlying
swap and where the annuity payments equal the difference between the market rate and the exercise rate. A
receiver swaption works just the opposite and is more like an interest rate put or floor.
7.
A forward swap is a type of forward contract and, as such, carries the obligation to enter into a swap. A
swaption is an option and, as such, is the right to enter into a swap. A forward swap, like any forward
contract, does not require an up-front payment whereas a swaption, like an option, does require an up-front
payment.
8.
A binomial term structure model has the same advantages of a binomial model for pricing options on stocks.
It permits a specification of the evolution of the underlying variable, such as the one-period interest rate.
Some instruments, such as American options, cannot typically be priced any other way. In fact some
European-style instruments cannot be priced any other way. In addition it can be structured so as to
guarantee that no arbitrage opportunities are possible across the entire term structure. It is, however, slower
and should not be used when a formula such as Black-Scholes is available.
9.
An interest rate cap provides protection against increases in the interest rate over the exercise rate at the
expense of having to pay cash up front. By combining a short position in an interest rate floor, you obtain
an interest rate collar, which will provide the same protection, but the firm can pay for it in a different way.
When a party buys an interest rate floor, it obtains protection if rates fall below the floor exercise rate. By
selling a floor, a firm receives a premium up front as compensation for the possibility that it will have to make
payments to the floor holder if rates fall below the floor exercise rate. Thus, if the buyer of a cap sells a floor
with a lower exercise rate, the payment received up front from the floor can partially or wholly offset the
payment made for the cap. The disadvantage of a collar is that the gains from falling interest rates below the
lower strike are forgone.
10.
a.
LIBOR = 14 %
The FRA payoff will be $5,000,000(.14 - .12)(90/360) = $25,000, which will be paid to the firm.
The loan interest will be $5,000,000(.14)(90/360) = $175,000.
Thus, the net interest paid when the loan matures is $150,000. This is a rate of
 $5,150,000

 $5,000,000



365/90
- 1 = .1274
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Copyright © 2001 by Harcourt, Inc
b.
LIBOR = 8 %
The FRA payoff will be $5,000,000(.08 - .12)(90/360) = -$50,000, which means that the firm will have
to pay $50,000.
The loan interest will be $5,000,000(.08)(90/360) = $100,000.
Thus, the net interest paid when the loan matures is $150,000. This is the same outcome as in a.
11.
Date
Now
90 days later
180 days later
270 days later
360 days later
LIBOR
11.5
10.5
10.2
9.6
----
Floating
Payment
-------575,000
525,000
510,000
480,000
Fixed
Payment
------550,000
550,000
550,000
550,000
Net Payment
to Fixed
---------25,000
-25,000
-40,000
-70,000
The first floating payment is determined as $20,000,000 (.115)(90/360) = $575,000. All remaining payments are
similarly calculated based on LIBOR at the beginning of the period. The fixed payments are all $20,000,000
(.11)(90/360) = $550,000.
12.
First convert the rates to discount factors:
1/(1 + .12(180/360)) = 0.9434
1/(1 + .1225(360/360)) = 0.8909
1/(1 + .1275(540/360)) = 0.8395
1/(1 + .1302(720/360)) = 0.7934
The swap rate is the coupon rate that will give a $25 million bond issue priced using the above discount
factors a value of $25 million. Set this up first with a $1 notional principal. That is,
coupon(0.9434) + coupon(0.8909) + coupon(0.8395) + (coupon + 1)(0.7934) = 1.
The solution is found as
1 − 0.7934
= .0596 .
0.9434 + 0.8909 + 0.8395 + 0.7934
Thus, on the basis of $25 million, the swap payment would be .0596($25,000,000) = $1,490,000. Quoted on an
annual basis, the swap rate would be .0592(2) = .1192.
13.
First convert these rates to discount factors:
1/(1 + .0975(90/360)) = 0.9762
1/(1 + .09875(270/360)) = 0.9310
1/(1 + .10(450/360)) = 0.8889
First we find the present value of the fixed coupons as the present value of a fixed rate bond. The fixed
coupon is $10,000,000(.12/2) = $600,000. The present value (including the principal) is
600,000(0.9762) + 600,000(0.9310) + 10,600,000(0.8889) = 10,566,660.
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Copyright © 2001 by Harcourt, Inc
The upcoming floating payment is $10,000,000(.1025/2) = $512,500. We need the present value of this and
the present value of the remaining floating payments. But on the next coupon date, we know that the
present value of the remaining floating payments plus the final principal will be $10 million. Thus, we need
calculate only
512,500(0.9762) + 10,000,000(0.9762) = 10,262,303.
Thus, the value of the swap is 10,262,303 – 10,566,660 = -304,357.
14.
First we must solve for the rate on a four-year swap at the swaption’s expiration. This is the solution to the
following equation:
c(1.10)-1 + c(1.105)-2 + c(1.109)-3 + (c + 1)(1.112)-4 = 1.000
Solving gives c = .1111. With a strike rate of 11.75 percent, the receiver swaption expires in-the-money.
Then to determine the payoff value of the swaption, we must find the value of a four-year annuity of (.1175 .1111) = .0064, using the 1-, 2-, 3- and 4-year zero coupon rates.
(.1175 - .1111)[(1.10)-1 + (1.105)-2 + (1.109)-3 + (1.112)-4] = .0199.
Based on a $20 million notional principal, the value of the swaption at expiration is $398,000. In other words,
this swaption allows the holder to enter into a four-year swap receiving 11.75 percent, while in the market,
four-year swaps pay 11.11 percent. Thus, the holder can use the swaption to enter into the swap, receiving
11.75 percent, and simultaneously enter into a swap in the market paying 11.11 percent. Alternatively, the
two parties to the swaption can settle by having the writer pay the holder of the swaption cash of $398,000.
As another alternative, the holder of the swaption can simply enter into the swap receiving promised
payments of 11.75 percent and hold the position.
15.
LIBOR at 6 percent
The option expires worthless. At the beginning of the loan the option premium has a future value of
32,000(1 + (.09+.025)(45/360)) = 32,460. The interest owed will be 10,000,000(.085)(180/360) = 425,000. Thus
you effectively borrowed 10,000,000 - 32,460 = 9,967,540 and effectively paid back 10,000,000 + 425,000 =
10,425,000. The annualized rate is (10,425,000/9,967,540)365/180 - 1 = .0953.
LIBOR at 12 percent
The option is worth (.12 - .09)(180/360)(10,000,000) = 150,000. Again, the option premium has a future value
of 32,460. The interest owed will be 10,000,000 (.145)(180/360) = 725,000. Again, you borrowed 9,967,540 and
effectively paid back 10,000,000 + 725,000 - 150,000 = 10,575,000.
The annualized rate is
(10,575,000/9,967,540)365/180 - 1 = .1275.
16.
LIBOR at 6.5 percent
The option is worth (.095 - .065)(90/360)(25,000,000) = 187,500. The option premium has a future value of
60,000(1 + (.095+.01)(30/360)) = 60,525. The interest received will be 25,000,000(.075)(90/360) = 468,750. Thus
you effectively lent 25,000,000 + 60,525 = 25,060,525 and effectively received 25,000,000 + 468,750 + 187,500 =
25,656,250 for an annualized rate of (25,656,250/25,060,525)365/90 - 1 = .10.
14-4
Copyright © 2001 by Harcourt, Inc
LIBOR at 12.5 percent
The option expires worthless. Again, the option premium has a future value of 60,525. The interest received
will be 25,000,000(.135(90/360)) = 843,750. Thus you lent (25,000,000+60,525) = 25,060,525 and effectively
received 25,000,000 + 843,750 = 25,843,750 for an annualized return of (25,843,750/25,060,525)365/90 - 1 = .1329.
17.
Date
6/28
9/28
12/31
3/31
6/29
Days
-92
94
90
90
LIBOR
10.00
11.00
11.65
12.04
----
Interest
Due
-----511,111
574,444
582,500
602,000
Cap
Principal
Payment Repayment
(70,000)
0
0
0
52,222
0
82,500
0
102,000 20,000,000
Net
Cash Flow
19,930,000
(511,111)
(522,222)
(500,000)
(20,500,000)
Without Cap Net
Cash Flow
20,000,000
(511,111)
(574,444)
(582,500)
(20,602,000)
Interest Due: On 9/28 you owe 20,000,000(.10)(92/360) = 511,111 based on LIBOR on 6/28. All remaining
figures are calculated similarly based on the number of days during the period.
Cap Payment: 70,000 paid for cap on 6/28. Cap pays off 20,000,000(.11 - .10)(94/360) = 52,222 on 12/31 based
on LIBOR on 9/28. Remaining figures are calculated similarly.
Principal Repayment: 20,000,000 paid on 6/29 to pay off loan.
Net Cash Flow: -Interest Due + Cap Payment - Principal Repayment. Figure on 6/28 also reflects receipt of
20,000,000.
Without Cap Net Cash Flow: -Interest Due - Principal Repayment.
The internal rate of return with the cap is
19,930,000 =
511,111 522,222 500,000
20,500,000
+
+
+
2
3
(1 + y)
(1 + y )
(1 + y )
(1 + y )4
y = .026, annualized to (1.026)4 - 1 = .108
18.
Date
4/15
7/16
10/15
1/16
4/16
Days
--92
91
93
90
LIBOR
8.00
7.90
7.70
8.10
----
Interest
Received
------715,556
698,931
696,208
708,750
Floor
Payment
(60,000)
0
8,847
27,125
0
Principal
Repayment
0
0
0
0
35,000,000
Net
Cash Flow
(35,060,000)
715,556
707,778
723,333
35,708,750
Without Floor Net
Cash Flow
(35,000,000)
715,556
698,931
696,208
35,708,750
Interest Received: On 7/16, 35,000,000(.08)(92/360) = 715,556. All remaining figures calculated similarly
based on LIBOR at beginning of period and number of days in period.
Floor Payment: 60,000 paid for floor on 4/15. On 10/15 floor pays off (.08 - .079)(91/360)(35,000,000) = 8,847.
All remaining figures calculated similarly. Floor does not pay off on 4/16 because LIBOR on 1/16 is above 8.
Principal Repayment: 35,000,000 principal is paid off on 4/16.
Net Cash Flow: Interest Received + Floor Payment + Principal Repayment. On 4/15 also reflects payment of
35,000,000.
14-5
Copyright © 2001 by Harcourt, Inc
Without Floor Net Cash Flow: Interest Received + Principal Repayment.
The internal rate of return with the floor is
35,060,000 =
715,556
707,778 723,333 35,708,750
+
+
+
(1 + y)
(1 + y ) 2 (1 + y )3
(1 + y ) 4
y = .0199, annualized to (1.0199)4 - 1 = .0821.
19.
a.
A collar. Selling the floor reduces the effective cost of the cap. The firm gives up the chance to
gain if interest rates fall.
b.
Date
1/15
4/16
7/15
10/14
1/15
Days
--91
90
91
93
Interest
Paid
------758,333
851,250
773,500
686,650
LIBOR
10.00
11.35
10.20
8.86
--
Cap
Payment
(150,000)
0
101,250
15,167
0
Floor
Payment
115,000
0
0
0
(10,850)
Principal
Repayment
0
0
0
0
30,000,000
With Collar
Net Cash Flow
29,965,000
(758,333)
(750,000)
(758,333)
(30,697,500)
All calculations are done the same way as in problems 17 and 18 with one exception. In problem 18
the firm bought the floor. Here the firm sells the floor so it receives cash up front and pays out
cash whenever LIBOR is less than 9. Thus on 1/15 (at maturity) the floor pays 30,000,000(.09 .0886)(93/360) = 10,850.
The internal rate of return is
29,965,000 =
758,333 750,000 758,333 30,697,500
+
+
+
(1 + y)
(1 + y ) 2 (1 + y )3
(1 + y ) 4
y = .0249, annualized to (1.0249)4 - 1 = .1034.
c.
The firm needs to adjust the strike on the floor such that its premium offsets the $150,000 premium
on the cap. The floor premium is $115,000 when the strike is 9 percent. To raise the premium, it
would have to raise the floor, meaning that the firm would have to give up the benefits of falling
rates starting at a rate higher than 9 percent. This would then be a zero cost collar.
14-6
Copyright © 2001 by Harcourt, Inc
20.
f0 = .1322
X = .12
rc = .1128
d1 =
σ = .17
ln(.1322/. 12) + (.17 ) 2 /2(.1233)
T = 45/365 = .1233
= 1.65
.17 .1233
N(1.65) = .9505
d 2 = 1.65 - .17 .1233 = 1.59
N(1.59) = .9441
C = (e
-.1128(.1233)
[.1322(.95 05) - .12(.9441) ]) e-.1322(180/365) = .0114
For every $1,000,000 of calls sold, this will amount to $1,000,000(.0114)(180/360) = $5,700.
21.
For the first caplet:
f0 = .08
X = .07
rc = .071
d1 =
σ = .166
ln(.08/.07 ) + (.166 ) 2 /2(.2493)
T = 91/365 = .2493
= 1.65
.166 .2493
N(1.65) = .9505
d 2 = 1.65 - .166 .2493 = 1.57
N(1.57) = .9418
C = (e
-.071(.249 3)
[.08(.9505 ) - .07(.9418) ]) e -.08(90/365) = .0097
For $10,000,000 notional principal, this will amount to $10,000,000(.0097)(90/360) = $24,250.
For the second caplet:
f0 = .082
X = .07
rc = .073
d1 =
σ = .166
ln(.082/.0 7) + (.166 ) 2 /2(.4986)
T = 182/365 = .4986
= 1.41
.166 .4986
N(1.41) = .9207
d 2 = 1.41 - .166 .4986 = 1.29
N(1.29) = .9015
C = ( e-.073(.4986) [.082(.920 7) - .07(.9015) ]) e-.082(90/365) = .0117
For $10,000,000 notional principal, this will amount to $10,000,000(.0117)(90/360) = $29,250.
The total cost of the cap is, thus, $24,250 + $29,250 = $53,500.
14-7
Copyright © 2001 by Harcourt, Inc
22.
a.
The binomial tree is as follows, with the probabilities in parentheses:
Time 0
Time 1
Time 2
Time 3
Time 4
16.86 %
(.0915)
15.68 %
(.1664)
12.41 %
(.3025)
9.47 %
(.55)
6%
13.46 %
(.2995)
12.31 %
(.4084)
9.13 %
(.4950)
6.28 %
(.45)
10.16 %
(.3675)
9.04 %
(.3341)
5.95%
(.2025)
6.95 %
(.2005)
5.86 %
(.0911)
3.83 %
(.0410)
In each node of the tree, the probability is calculated by raising .55 to the power of the number of
times the rate went up to reach that node, times .45 to the power of the number of times the rate
went down to reach that node, times the number of ways the rate could get to that node. The
number of ways the rate could get to each cell is discussed in the book.
b.
The solution for the in-arrears FRA is the rate FRA 2 in the following equation:
.3025(.1241 - FRA 2) + .4950(.0913 - FRA 2) + .2025(.0595 - FRA 2) = 0
Solving gives FRA 2 = .0948.
The solution for the delayed settlement FRA is the rate FRA 2 in the following equation:
.3025(.1241 - FRA 2)(1.1241)-1 + .4950(.0913 - FRA 2)(1.0913)-1
+ .2025(.0595 - FRA 2)(1.0595)-1 = 0
Solving gives FRA 2 = .0943.
c.
Swap pricing is easily done using only the time 0 zero coupon discount rates. It is the solution c to
the following:
c(1.06)-1 + c(1.07)-2 + (c + 1.00)(1.078)-3 = 1.00
Solving gives c = .0771.
14-8
Copyright © 2001 by Harcourt, Inc
d.
The payoffs of the put in the five states at time 4 are
Max(0,.10 - .1686) /1.1686= 0
Max(0,.10 - .1346) /1.1346= 0
Max(0,.10 - .1016) /1.1016= 0
Max(0,.10 - .0695) /1.0695= .0285
Max(0,.10 - .0383) /1.0383= .0594
The prices in the four states at time 3 are
[.55(0) + .45(0)]/1.1568 = 0
[.55(0) + .45(0)]/1.1231 = 0
[.55(0) + .45(.0285)]/1.0904 = .0118
[.55(.0285) + .45(.0594)]/1.0586 = .0401
The prices in the three states at time 2 are
[.55(0) + .45(0)]/1.1241 = 0
[.55(0) + .45(.0118)]/1.0913 = .0049
[.55(.0118) + .45(.0401)]/1.0595 = .0232
The prices in the two states at time 1 are
[.55(0) + .45(.0049)]/1.0947) = .0020
[.55(.0049) + .45(.0232)]/1.0628 = .0124
The price at time 0 is, therefore,
[.55(.0020) + .45(.0124)]/1.06 = .0063
e.
To price a three-period cap, we must price 1-, 2- and 3-period interest rate calls or caplets.
For the three-period caplet, we have the following:
The payoffs at time 3 are as follows:
Max(0,.1568 - .09)/1.1568 = .0577
Max(0,.1231 - .09)/1.1231 = .0295
Max(0,.0904 - .09)/1.0904 = .0004
Max(0,.0586 - .09)/1.0586 = 0
The prices at time 2 are as follows:
[.55(.0577) + .45(.0295)]/1.1241 = .0400
[.55(.0295) + .45(.0004)]/1.0913 = .0150
[.55(.0004) + .45(.0)]/1.0595 = .0002
14-9
Copyright © 2001 by Harcourt, Inc
The prices at time 1 are as follows:
[.55(.0400) + .45(.0150)]/1.0947 = .0263
[.55(.0150) + .45(.0002)]/1.0628 = .0078
The price at time 0 is, therefore,
[.55(.0263) + .45(.0078)]/1.06 = .0170
For the two-period caplet, we have the following:
The payoffs at time 2 are as follows:
Max(0,.1241 - .09)/1.1241 = .0303
Max(0,.0913 - .09)/1.0913 = .0012
Max(0,.0595 - .09)/1.0595 = 0
The prices at time 1 are as follows:
[.55(.0303) + .45(.0012)] 1.0947= .0157
[.55(.0012) + .45(.0)]/1.0628 = .0006
The price at time 0 is, therefore,
[.55(.0157) + .45(.0006)]/1.06 = .0084
For the one-period caplet, we have the following:
The payoffs at time 1 are as follows:
Max(0,.0947 - .09)/1.0947 = .0043
Max(0,.0628 - .09)/1.0628 = 0
The price at time 0 is, therefore,
[.55(.0043) + .45(0)]/1.06 = .0022
The three-period cap is worth the sum of the prices of the three component caplets,
.0170 + .0084 + .0022 = .0276
f.
We must find the rate on a two-period swap in each of the three states at time 2. In the upper state
c(1.1241)-1 + (c + 1.00)(1.1233)-2 = 1.00
Solving for c gives .1233. In the middle state,
c(1.0913)-1 + (c + 1.00)(1.0908)-2 = 1.00
14-10
Copyright © 2001 by Harcourt, Inc
Solving for c gives .0908. In the lower state,
c(1.0595)-1 + (c + 1.00)(1.0579)-2 = 1.00
Solving for c gives .0579.
The payoffs of the swaption at time 2 are as follows:
Max(0,.1233 - .09)[(1.1241)-1 + (1.1233)-2] = .0560
Max(0,.0908 - .09)[(1.0913)-1 + (1.0908)-2] = .0014
Max(0,.0579 - .09)[(1.0595)-1 + (1.0579)-2] = 0
Note that the swaption payoff is a two-period annuity. The values of the swaption at time 1 are
[.55(.0560) + .45(.0014)]/1.0947 = .0287
[.55(.0014) + .45(0)]/1.0628 = .0007
The price of the swaption is, therefore,
[.55(.0287) + .45(.0007)]/1.06 = .0152
g.
At time 2, we see that the rates for the underlying swap are .1233, .0908 and .0579. The rate on the
forward swap will be the probability-weighted average of these rates where the probabilities are
those given in the tree in a. for the respective states at time 2. Thus, the forward swap rate is
.3025(.1233) + .4950(.0908) + .2025(.0579) = .0940
23.
f0 = .0979
X = .10
rc = .0838
d1 =
σ = .1465
T = 74/365 = .2027
ln(.0979/. 10) + (.1465 ) 2 /2(.2027)
= - 0.29
.1465 .2027
N(-0.29) = .3859
d 2 = - 0.29 - .1465 .2027 = - 0.36
N(-0.36) = .3594
P = (e
-.0838(.2027)
[ .10(1 - .3594) - .0979(1 - .3859]) e-.0979(90/365) = .0038
For $22,000,000 of calls sold, this will amount to $22,000,000(.0038)(90/360) = $20,900.
24.
The payoffs at time 4 are as follows:
Max(0,.11 - .1686)/1.1686 = 0
Max(0,.11 - .1346)/1.1346 = 0
Max(0,.11 - .1016)/1.1016 = .0076
Max(0,.11 - .0695)/1.0695 = .0379
Max(0,.11 - .0383)/1.0383 = .0691
The values of the put at time 3 are as follows:
[.55(0) + .45(0)]/1.1568 = 0
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Copyright © 2001 by Harcourt, Inc
[.55(0) + .45(.0076)]/1.1231 = .0030
[.55(.0076) + .45(.0379)]/1.0904 = .0195
[.55(.0379) + .45(.0691)]/1.0586 = .0491
We must check to determine if the put is worth more exercised at each of these nodes. At the top two
nodes, the put is out-of-the-money. At the second-from-bottom and bottom nodes, we have
Max(0,.11 - .0904)/1.0904 = .0180, which is less than .0195 so do not exercise
Max(0,.11 - .0586)/1.0586 = .0486, which is less than .0491 so do not exercise
At time 2, the values of the put are as follows:
[.55(0) + .45(.0030)]/1.1241 = .0012
[.55(.0030) + .45(.0195)]/1.0913 = .0096
[.55(.0195) + .45(.0491)]/1.0595 = .0310
We must check for early exercise. In the middle and bottom nodes, we have
Max(0,.11 - .0913)/1.0913 = .0171 so replace .0096 with .0171
Max(0,.11 - .0595)/1.0595 = .0477 so replace .0310 with .0477
At time 1, the values of the put are as follows:
[.55(.0012) + .45(.0171)]/1.0947 = .0076
[.55(.0171) + .45(.0477)]/1.0628 = .0290
We must check each node for early exercise. We have
Max(0,.11 - .0947)/1.0947 = .0140 so replace .0076 with .0140
Max(0,.11 - .0628)/1.0628 = .0444 so replace .0290 with .0444
At time 0, the price of the put is
[.55(.0140) + .45(.0444)]/1.06 = .0261
If exercised now, the put would be worth
Max(0,.11 - .06)/1.06 = .0472
So the current value of the put is .0472, which would be obtained by exercising it immediately.
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Copyright © 2001 by Harcourt, Inc