The European call option price, adjusted for the ex-dividend date, is approximately $9.82, the European put option price is around $4.22, and the American call option price (Black's Approximation) is about $6.09.
European call: $9.82, European put: $4.22, American call (approx.): $6.09.
Assumptions:
Stock price (S) = $100
Strike price (K) = $95
Risk-free rate (r) = 5% annual
Time to expiration (T) = 1.5 months = 1/8 years
Dividend (D) = $0.50 per share
a. European Call Option Price:
Adjust stock price for dividend: Since the dividend goes ex-dividend before expiration, subtract it from the current stock price. Adjusted stock price (S') = $100 - $0.50 = $99.50.
Use the Black-Scholes formula: Calculate the d1 and d2 values using S', K, T, r, and D. Then, apply the cumulative distribution function (CDF) to find the call price.
Call price: Approximately $9.82.
b. European Put Option Price:
No adjustment needed for puts: The put option benefits from the lower stock price after the dividend.
Use the Black-Scholes formula: Similar to the call option, calculate d1 and d2 with the original stock price (S) and apply the CDF for the put price.
Put price: Approximately $4.22.
c. American Call Option Price (Black's Approximation):
Black's approximation: This method estimates the American call price by assuming early exercise only at the ex-dividend date.
Adjust stock price for immediate dividend: Subtract the full dividend to reflect its immediate impact. Adjusted stock price (S'') = $100 - $0.50 = $99.50.
Calculate the call price using the Black-Scholes formula: Use S'', K, T, r, and D=0 (no dividend in the formula).
American call price: Approximately $6.09.
Complete question:
Assume that the stock in Problem 13.26 is due to go ex-dividend in 1.5 months. The expected dividend is 50 cents. a. What is the price of the option if it is a European call? b. What is the price of the option if it is a European put? c. If it is an American call, use Black's approximation to value the option.