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31 December, 18:40

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 $40*1.1=$44 or to $40*0.9=$36 at the end of the period and if the stock price at the start of a period is $44, it will go to $44*1.1=$48.44 or to $44*0.9=$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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  1. 31 December, 18:49
    0
    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
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