Financial Risk Manager Part 1

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Explanation:

To price the call option on a dividend-paying stock, we first calculate the adjusted stock price by subtracting the present value of the expected dividend.

PV of Dividend = D * e^(-r * t_d) PV(D) = 2 * e^(-0.05 * 0.25) = 2 * 0.98758 = 1.975
Adjusted Stock Price (S') = S - PV(D) = 50 - 1.975 = 48.025
Compute d1 and d2 for the Black-Scholes formula: d1 = [ln(S' / K) + (r + (σ^2)/2) * T] / (σ * √T) d1 = [ln(48.025 / 50) + (0.05 + 0.2^2 / 2) * 1] / (0.2 * 1) d1 = [-0.04033 + 0.07] / 0.2 = 0.02967 / 0.2 = 0.14835 d2 = d1 - σ * √T = 0.14835 - 0.2 = -0.05165
Find N(d1) and N(d2) from standard normal distribution: N(d1) = N(0.148) ≈ 0.5588 N(d2) = N(-0.052) ≈ 0.4793
Calculate Call Price (C): C = S' * N(d1) - K * e^(-rT) * N(d2) C = (48.025 * 0.5588) - (50 * e^(-0.05) * 0.4793) C = 26.836 - (47.561 * 0.4793) = 26.836 - 22.796 = 4.04.

Explanation:

To price the call option on a dividend-paying stock, we first calculate the adjusted stock price by subtracting the present value of the expected dividend.

PV of Dividend = D * e^(-r * t_d) PV(D) = 2 * e^(-0.05 * 0.25) = 2 * 0.98758 = 1.975
Adjusted Stock Price (S') = S - PV(D) = 50 - 1.975 = 48.025
Compute d1 and d2 for the Black-Scholes formula: d1 = [ln(S' / K) + (r + (σ^2)/2) * T] / (σ * √T) d1 = [ln(48.025 / 50) + (0.05 + 0.2^2 / 2) * 1] / (0.2 * 1) d1 = [-0.04033 + 0.07] / 0.2 = 0.02967 / 0.2 = 0.14835 d2 = d1 - σ * √T = 0.14835 - 0.2 = -0.05165
Find N(d1) and N(d2) from standard normal distribution: N(d1) = N(0.148) ≈ 0.5588 N(d2) = N(-0.052) ≈ 0.4793
Calculate Call Price (C): C = S' * N(d1) - K * e^(-rT) * N(d2) C = (48.025 * 0.5588) - (50 * e^(-0.05) * 0.4793) C = 26.836 - (47.561 * 0.4793) = 26.836 - 22.796 = 4.04.

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Q.57 Suppose that the current stock price is USD 50 and the risk-free rate of interest is 5% per year. A dividend of USD 2 is expected on an ex-dividend date three months from now. If the stock volatility is 20% per year and the strike price is USD 50, what is the price of a 1-year call option on the stock?

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Last updated: June 22, 2026 at 14:11

USD 4.21.

USD 5.22.

USD 3.96.

USD 4.04