S(0) = current stock price = $50
X = Strike price in put = $54
t = time period = 3 months = 3/12 = 0.25
r = continuous compounded risk free rate = 10%
up move = u = 1.05
down move = d = 0.95
probability of up move (p) = (ert - d) / (u - d) = (e0.1*0.25 - 0.95) / (1.05 - 0.95) = 75.32%
probability of down move = 1 - p = 1 - 0.7532 = 24.68%
value of put option at expiry = max(X - S(t), 0)
since, this is an American option, value of the option at every node = max(PV of future option payoffs, X - S(t))
$52.5 * 1.05 = $55.125 |
value(put) = max(54 - 55.125, 0) = $0 |
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$50 * 1.05 = $52.5 |
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$50 |
$52.5 * 0.95 = $49.875 |
value(put) = max(54 - 49.875, 0) = $4.125 |
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$50 * 0.95 = $47.5 |
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$47.5 * 0.95 = $45.125 |
value(put) = max(54 - 45.125, 0) = $8.875 |
Therefore,
value(u,u) = $0
value(u,d) = $4.125
value(d,d) = $8.875
value(u) = max(54 - 52.5, (e-0.1*0.25 * ((0.7532 * 0) + (0.2468 * 4.125))))
= max(1.5, (0.9753 * (0 + 1.0181)))
= max(1.5, 0.9929)
value (u) = 1.5
value(d) = max(54 - 47.5, (e-0.1*0.25 * ((0.7532 * 4.125) + (0.2468 * 8.875))))
= max(5.5, (0.9753 * (3.1070 + 2.1904)))
= max(5.5, 5.1665)
value (d) = 5.5
value(0) = max(54 - 50, (e-0.1*0.25 * ((0.7532 * 1.5) + (0.2468 * 5.5))))
= max(4, (0.9753 * (1.1298 + 1.3574)))
= max(4, 2.4258)
value (0) = $4
Therefore, value of the 6 month American put option today is $4.
max value(d)= 6.5
the answer will be the same but i think it is needed correction there.
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