A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10% (meaning, precisely, if the stock price at the start of a period is $40, it will go to $401.1=$44 or to $400.9=$36 at the end of the period and if the stock price at the start of a period is $44, it will go to $441.1=$48.44 or to $440.9=$39.6 at the end of the period). The risk-free interest rate is 12% per annum with continuous compounding. a. What is the value of a six-month European put option with a strike price of $42? b. What is the value of a six-month American put option with a strike price of $42? c. What is the value of a six-month American put option with a strike price of $45? What do you conclude about whether or not it is optimal to exercise this American option immediately (Hint: What would be the value of this American option if it were to be exercised immediately)

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Business

Sammy Simmons

5 September, 15:51

A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10% (meaning, precisely, if the stock price at the start of a period is $40, it will go to $401.1=$44 or to $400.9=$36 at the end of the period and if the stock price at the start of a period is $44, it will go to $441.1=$48.44 or to $440.9=$39.6 at the end of the period). The risk-free interest rate is 12% per annum with continuous compounding. a. What is the value of a six-month European put option with a strike price of $42? b. What is the value of a six-month American put option with a strike price of $42? c. What is the value of a six-month American put option with a strike price of $45? What do you conclude about whether or not it is optimal to exercise this American option immediately (Hint: What would be the value of this American option if it were to be exercised immediately)

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Presley · Answer 1

The Risk neutral probability is given by

e rt - D / U-D

U=1.1

D=0.9

R=0.12

T=3/12

π u = e∧ 0.12 ∗ 3 / 12 - 0.9 / 1.1 - 0.9

=0.652

π d = 1 - 0.652 = 0.348

The values of american and european options at each node is given in the following table.

0.652

0

0.81 48.4

0.652

0.81

American option value 2.54 44

probability 0.652/0.3478'

Option value 2.12 2.4

Futures price 40 6 39.6

0.3478

4.76

36

0.3478

9.6

32.4

Time period 0 3 6

the value at up node at 3 months is given by = (0.652∗ 0) + (0.3478 ∗ 2.4) / e ∧0.12 ∗ 3 / 12 = 0.81

Hence, value of european put option = $2.12

Value of American put option = 2.54