Suppose Carol's stock price is currently $20.In each six-month period it will either fall by 50% or rise by 100%.What is the current value of a one-year call option with an exercise price of $15? The six-month risk-free interest rate is 5% per six-month period.[Use the two-stage binomial method.]
A)$8.73
B)$10.03
C)$16.88
D)$13.33

Question

Quizplus · Accepted Answer

Using the two-stage binomial method, we construct a binomial tree with two levels representing the two six-month periods. At each node, we calculate the option value using the risk-neutral probabilities of an up or down movement. The risk-neutral probability of an up movement is calculated as $(1 + r - d)/(u - d)$, where $r$ is the risk-free interest rate, $u$ is the up factor (i.e. 2 in this case), and $d$ is the down factor (i.e. 0.5 in this case). The risk-neutral probability of a down movement is simply 1 minus the risk-neutral probability of an up movement.

The binomial tree for this problem looks like:

```
            40
        /       \
      20         80
    /    \      /   \
  10     40   40    160
 /  \   /  \  / \    / \
5   20 20  80 20 80  80 320
```
At expiration, if the stock price is above the exercise price of $15, the option is worth the difference between the stock price and the exercise price. If the stock price is below the exercise price, the option is worthless. We work backwards on the tree to calculate the option value at each node.

At time 1, the option value at the two leaf nodes is 5 and 25, respectively, depending on whether the stock price goes up or down. The risk-neutral probabilities of an up or down movement at time 0 are calculated as follows:

$$p_u = \frac{(1 + r - d)}{(u - d)} = \frac{(1 + 0.05 - 0.5)}{(2 - 0.5)} = 0.368$$

$$p_d = 1 - p_u = 0.632$$

Using the risk-neutral probabilities, we calculate the option value at time 0 as follows:

$$C_0 = e^{-rt}(p_uC_u + p_dC_d)$$

where $C_u$ and $C_d$ are the option values at the two nodes at time 1, and $t$ is the time to expiration. Plugging in the numbers, we get:

$$C_0 = e^{-0.05}(0.368 * 25 + 0.632 * 5) \approx 10.03$$

Therefore, the current value of a one-year call option with an exercise price of $15 is approximately $10.03. The best choice is (B).

Suppose Carol's Stock Price Is Currently $20