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Assume that the stock in Problem 13.26 is due to go ex-dividend in 1.5 months. The expected dividend is 50 cents.

What is the price of the option if it is a European call?

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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.

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