The current price of a stock is $50. This includes the value of a dividend of $2 that will be paid three months from now. In three months, the net of dividend stock price will rise by a factor of 1.2 or fall by a factor of 0.8. The risk-free rate of interest is 0.0834% per month (simple interest). Using the technique of Schroder (1988), what is the price of a six-month American call option at a strike of $51?
A) $3.62
B) $4.12
C) $4.62
D) $5.12

Question

Quizplus · Accepted Answer

Using the Schroder model, we can obtain the following calculations:

1. First, calculate the risk-neutral probabilities for the two possible outcomes at three months:
p = (1 + rf – d – u) / (u – d) = (1 + 0.0834% – 0.02 – 1.2) / (1.2 – 0.8) = 0.42915
q = 1 – p = 0.57085

2. Calculate the call option prices at three months:
C_u = max(S_u – K, 0) = max(60 – 51, 0) = 9
C_d = max(S_d – K, 0) = max(40.0 – 51, 0) = 0

3. Calculate the risk-neutral expected call option price at six months:
C = (p * C_u + q * C_d) / (1 + rf) = (0.42915 * 9 + 0.57085 * 0) / (1 + 0.0834%) = 3.956

4. The American call option price is the maximum of the European call option price and the early exercise value. At any time, the early exercise value is the difference between the stock price and the strike price (if positive). At maturity, the early exercise value is zero. To calculate the early exercise value at three months, we compare the option value C with the intrinsic value I:
I = max(S – K, 0) = max(50 – 51, 0) = 0
If C > I, the option holder should not exercise early. If C ≤ I, the option holder should exercise early.

5. We can now calculate the American call option price at three months:
C_3 = max(C, I) = max(3.956, 0) = 3.956

6. The American call option price at six months is the discounted risk-neutral expected value of the option payoff at six months, or the early exercise value at three months (whichever is higher):
C_6 = max(C, I + C_3 / (1 + rf)) = max(3.956, 0 + 3.956 / (1 + 0.0834%)) = 4.621

Therefore, the price of a six-month American call option at a strike of $51 is $4.62, which is closest to choice C ($4.62).

The Current Price of a Stock Is $50