A stock is currently selling for $50. The stock price could go up by 10 percent or fall by 5 percent each month. The monthly interest rate is 1 percent (periodic rate). Calculate the price of a call option on the stock with an exercise price of $50 and a maturity of two months. (Use the two-stage binomial method.)
A)$5.10
B)$2.71
C)$4.78
D)$3.62

Question

Quizplus · Accepted Answer

Using the two-stage binomial method, we first need to calculate the possible prices of the stock after two months. There are three possible outcomes: 
1. The stock goes up twice (10%) = $55 
2. The stock goes up then down (10% and then 5%) = $52.25 
3. The stock goes down twice (5%) = $47.5

Next, we calculate the two possible payoffs of the call option after two months: 
1. The stock goes up twice, so the option is worth $5 (the stock price minus the exercise price of $50). 
2. The stock goes up then down, so the option is worth $2.25 (the stock price after the first month is $55*1.01 = $55.55, but since it falls to $52.25, the option is only worth $2.25). 
3. The stock goes down twice, so the option is worthless (since it is cheaper to buy the stock on the market than to exercise the option at $50).

Using backward induction, we calculate the value of the call option at the end of the first month for each of the two possible future stock prices: 
1. If the stock price goes up to $55*1.01 = $55.55, the option is worth max($0, $55.55 - $50) = $5.55. 
2. If the stock price falls to $55*0.95 = $52.25, the option is worth max($0, $52.25 - $50) = $2.25.

Finally, we calculate the value of the call option today using the weighted average of the two possible values at the end of the first month, discounted at the monthly interest rate: 
Value of call option today = [(0.5 * $5.55) + (0.5 * $2.25)] / 1.01 = $2.71

Therefore, the answer is B) $2.71.

A Stock Is Currently Selling for $50