The current price of a non-dividend paying stock is $30. Use a two-step tree to value a European call option on the stock with a strike price of $32 that expires in 6 months. Each step is 3 months, the risk free rate is 8% per annum with continuous compounding. What si the option price when u = 1.1 and d = 0.9.
A) $1.29
B) $1.49
C) $1.69
D) $1.89

Question

Quizplus · Accepted Answer

Using the two-step tree, we can set up the following:

At t=0: stock price = $30
At t=3 months: stock price can be either $27 (if it goes down) or $33 (if it goes up)
At t=6 months: stock price can be $24.30 (if it goes down twice), $29.70 (if it goes down then up or up then down), or $36.30 (if it goes up twice)

To calculate the option price, we work backwards from the final nodes:

If the stock price is $24.30, the option price is 0 (since the option is out of the money)
If the stock price is $29.70, the option price is e^(-0.08*0.5)*($29.70-$32) = $1.86
If the stock price is $36.30, the option price is e^(-0.08*0.5)*($36.30-$32) = $2.87

Then, at t=3 months, we can calculate the expected value of the option:

Option price if stock goes up: (0.5*$1.86) + (0.5*$2.87) = $2.365
Option price if stock goes down: 0 (since the option is out of the money)

Finally, at t=0, we can calculate the expected value of the option again:

Option price if stock goes up: (0.5*$2.365) + (0.5*$0) = $1.183
Option price if stock goes down: 0 (since the option is out of the money)

Therefore, the option price is approximately $1.49, which corresponds to choice B.

The Current Price of a Non-Dividend Paying Stock Is $30