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A stock price is currently $50. Over each of the next two three-month periods it is...

A stock price is currently $50. Over each of the next two three-month periods it is expected to go up by 6% or down by 6%. The risk-free interest rate is 6% per annum with continuous compounding. What is the value of a six-month American put option with a strike price of $53?

Solutions

Expert Solution

Price of six-month American Put Option = $3.73

Since it is an American Option, it can be exercised at any point of time and hence we take the put option value as the maximum value that the option buyer will get by exercising the option

At T= 3months

The stock price at Upper3 = Current Stock Price0 * (1 + 6%) = $53
The stock price at Lower3 = Current Stock Price0 * (1 - 6%) = $47

At T=6 months
The stock price at Upper6 = stock price at Upper3 * (1 + 6%) = $56.18
The stock price at Lower6 = stock price at Lower3 * (1 - 6%) = $44.18
The stock price at Middle6 = stock price at Lower3 * (1 + 6%) = $49.82

Value of Put Option = Max (0, K - ST)
where K = Strike Price
ST = Strike Price at T

Since it is an American Option therefore, the Value of Put Option at T = Max (K - ST, PV of Probability weighted average of future value of Put Option)

Probability of Up Move = 50% or 0.5
Probability of Down Move = 1 - Probability of Up Move
= 1 - 0.5
= 0.5

Since the risk-free rate is continuously compounding, therefore, we will discount it using e-rt
where r = Rate
t = time period


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