It appears that Blacks approach should understate the true option value. This is
because the holder of the option has more alternative strategies for deciding when to
exercise the option than the two strategies implicitly assumed by the approach. These
al
Problem 13.16.
Show that the BlackScholesMerton formula for a call option gives a price that
tends to
as
.
T 0
max (S0 K 0)
ln( S0 K ) ( r 2 2)T
T
ln( S0 K ) r 2 2
T
T
, the second term on the right hand side tends to zero. The first term tends
d1
As
t
Problem 13.12.
Assume that a non-dividend-paying stock has an expected return of
of
and a volatility
. An innovative financial institution has just announced that it will trade a
derivative that pays off a dollar amount equal to
1 ST
ln
T S0
at time
.
e 02501 e05001 19265
Also:
S0 680735 K 65 032 r 01 T 06667
d1
ln(680735 65) (01 0322 2)06667
05626
032 06667
d 2 d1 032 06667 03013
N (d1 ) 07131 N (d 2 ) 06184
and the call price is
680735 07131 65e 0106667 06184 1094
or $10.94.
DerivaGem can be used t
Since
ln 40 3689
, the required probability is
3689 3687
1 N
1 N (0008)
0247
From normal distribution tables N(0.008) = 0.5032 so that the required
probability is 0.4968.
a) In this case the required probability is the probability of the stock price b
Problem 13.15.
A call option on a non-dividend-paying stock has a market price of
$250
. The stock
price is $15, the exercise price is $13, the time to maturity is three months, and the
risk-free interest rate is 5% per annum. What is the implied volatili
or $5.06.
Problem 13.14.
What is the price of a European put option on a non-dividend-paying stock when the
stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum,
the volatility is 35% per annum, and the time to maturity
Problem 13.9.
A stock price has an expected return of 16% and a volatility of 35%. The current
price is $38.
a) What is the probability that a European call option on the stock with an
exercise price of $40 and a maturity date in six months will be exerci
Problem 13.17.
Explain carefully why Blacks approach to evaluating an American call option on a
dividend-paying stock may give an approximate answer even when only one dividend
is anticipated. Does the answer given by Blacks approach understate or oversta
Practice
Questions
Problem 13.8.
A stock price is currently $40. Assume that the expected return from the stock is 15%
and its volatility is 25%. What is the probability distribution for the rate of return
(with continuous compounding) earned over a one-y
95% confidence intervals for
ln ST
are therefore
ln S0 (
2
)T 196 T
2
and
2
)T 196 T
2
are therefore
ln S0 (
95% confidence intervals for
ST
eln S0 (
2
2)T 196 T
and
eln S0 (
S0e(
2
2)T 196 T
and
S0 e(
2
2)T 196 T
i.e.
2
2)T 196 T
Problem 13.11.
Problem 13.18.
Consider an American call option on a stock. The stock price is $70, the time to
maturity is eight months, the risk-free rate of interest is 10% per annum, the exercise
price is $65, and the volatility is 32%. A dividend of $1 is expected a
Problem 13.20.
Show that the BlackScholesMerton formulas for call and put options satisfy put
call parity.
The BlackScholesMerton formula for a European call option is
c S0 N (d1 ) Ke rT N (d 2 )
so that
c Ke rT S 0 N (d1 ) Ke rT N (d 2 ) Ke rT
or
c Ke rT
a) re
. In a risk-neutral world this becomes
r 2 2
. The value of the
2
derivative at time zero is therefore:
2 rT
r e
2
Problem 13.13.
What is the price of a European call option on a non-dividend-paying stock when the
stock price is $52, the strike
f the options as 7.54, indicating that the time value is still positive ( 054 ). Keeping
the number of time steps equal to 50, trial and error indicates the time value
disappears when the stock price is reduced to 21.6 or lower. (With 500 time steps this
Relative performance stock options clearly provide a better way of rewarding senior
management for superior performance. Some companies have argued that, if they
introduce relative performance options when their competitors do not, they will lose
some of
Problem 9.10.
Suppose that a European put option to sell a share for $60 costs $8 and is held until
maturity. Under what circumstances will the seller of the option (the party with the
short position) make a profit? Under what circumstances will the optio
Figure S9.4
Profit from trading strategy in Problem 9.12
Problem 9.13.
Explain why an American option is always worth at least as much as a European
option on the same asset with the same strike price and exercise date.
The holder of an American option ha
Problem 9.17.
Consider an exchange-traded call option contract to buy 500 shares with a strike
price of $40 and maturity in four months. Explain how the terms of the option
contract change when there is
a) A 10% stock dividend
b) A 10% cash dividend
c) A
Problem 9.9.
Suppose that a European call option to buy a share for $100.00 costs $5.00 and is
held until maturity. Under what circumstances will the holder of the option make a
profit? Under what circumstances will the option be exercised? Draw a diagram
Practice Questions
Problem 9.8.
A corporate treasurer is designing a hedging program involving foreign currency
options. What are the pros and cons of using (a) the NASDAQ OMX and (b) the overthe-counter market for trading?
The NASDAQ OMX offers options w
Problem 9.16.
The treasurer of a corporation is trying to choose between options and forward
contracts to hedge the corporations foreign exchange risk. Discuss the advantages
and disadvantages of each.
Forward contracts lock in the exchange rate that will
Problem 9.11.
Describe the terminal value of the following portfolio: a newly entered-into long
forward contract on an asset and a long position in a European put option on the
asset with the same maturity as the forward contract and a strike price that i
Problem 9.14.
Explain why an American option is always worth at least as much as its intrinsic
value.
The holder of an American option has the right to exercise it immediately. The
American option must therefore be worth at least as much as its intrinsic
(5)
(Forward Rate Agreement)
FRA T1 T2 T1 T2
FRA T1 FRA T2
FRA
Notation
Effective Date
from now
Termination Date
from now
Underlying
Rate
1x3
1 month
3 months
2 months LIBOR
1x7
1 month
7 months
6 months LIBOR
3x6
3 months
6 months
3 months LIBOR
3x9
(2)
(The Management of Net Interest Income)
Table 4.6
/
/ / /
Table 4.6
Maturity (Years)
Deposit Rate
Mortgage Rate
1
3%
6%
5
3%
6%
55
(3)
(The Management of Net Interest Income)
56
28
(4)
(The Management of Net Interest Income)