Suppose VS's stock price is currently $20.In the next six months it will either fall to $10 or rise to $30.What is the current value of a put option with an exercise price of $15? The six-month risk-free interest rate is 5% per six-month period.

Question

Quizplus · Accepted Answer

To calculate the value of the put option, we need to find the probability of the stock rising or falling, and then discount the payoffs accordingly.
The risk-neutral probabilities of a stock rising or falling can be calculated as follows:
q_u = (1 + r - d) / (u - d) = (1.05 - 1) / (1.5 - 0.5) = 0.5
q_d = 1 - q_u = 0.5
where r is the risk-free rate, u is the factor the stock price goes up by (1.5 in this case, i.e. 30/20), and d is the factor the stock price goes down by (0.5 in this case, i.e. 10/20).
The payoffs of the put option at each node can be calculated as follows:
At the up node (stock price of $30), the put option is worth $0 (since the option holder would not exercise the option and buy the stock for $15 when it can be bought for $30).
At the down node (stock price of $10), the put option is worth $5 (since the option holder can exercise the option and sell the stock for $15 when it can only be sold for $10).
Therefore, the current value of the put option is:
$5 x 0.5 + $0 x 0.5 = $2.50 (not discounted)
Discounting this value with the risk-free rate for six months gives:
$2.50 / (1 + 0.05) = $2.14
So the answer is B.

Suppose VS's Stock Price Is Currently $20