CONSIDER AN AMERICAN CALL OPTION AND AN AMERICAN PUT OPTION, EACH...
22.
Consider an American call option and an American put option, each with 3 months to maturity, written on a
non-dividend-paying stock currently priced at USD 40. The strike price for both options is USD 35 and the
risk-free rate is 1.5%. What are the lower and upper bounds on the difference between the prices of the call
and put options?
Scenario
Lower Bound (USD)
Upper Bound (USD)
A
5.13
40.00
B
5.00
5.13
C
34.87
40.00
D
0.13
34.87
a.
Scenario A
b.
Scenario B
c.
Scenario C
d.
Scenario D
Correct Answer: b
Rationale:
The put-call parity in case of American options leads to the inequality:
S
0
– X
≤
(C – P)
≤
S
0
– Xe
-rT
The lower and upper bounds are given by—
= 40 – 35
≤
(C – P)
≤
40 – 35e
-0.015 x 3/12
= 5
≤
(C – P)
≤
5.13
Alternatively, the upper and lower bounds for American options are given by
Option
Minimum Value
Maximum Value
American Call
c
≥
max(0, S
0
- Xe
-rT
) = 5.13
S
0
= 40
American Put
p
≥
max(0, X - S
0
) = 0
X= 35
Subtracting the put values from the call values in the table above, we get the same result—
= 5
≤
C – P
≤
5.13
(Note- the minimum and maximum values are obtained by comparing the results of the subtraction of the put price
from the call price. For instance, in this example, the upper bound is obtained by subtracting the minimum value of
the American put option from the minimum value of the American call option and vice versa).
Section:
Financial Markets and Products
Reference:
John Hull, Options, Futures and Other Derivatives, 9th Edition, Chapter 11, “Properties of Stock Options.”
Learning Objective:
Identify and compute upper and lower bounds for option prices on non-dividend and dividend
paying stocks. Explain put-call parity and apply it to the valuation of European and American stock options.
40
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2015 Financial Risk Manager (FRM®) Practice Exam