Explanation

This is a Black-Scholes option pricing problem with dividends. The Black-Scholes formula for a European call option on a dividend-paying stock is:

$C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$

Where:

• $S_0 = 50$ (current stock price)
• $K = 50$ (strike price)
• $T = 0.5$ (6 months = 0.5 years)
• $r = 0.04$ (risk-free rate)
• $q = 0.02$ (dividend yield)
• $\sigma = 0.18$ (volatility)

First, calculate $d_1$ and $d_2$:

$d_1 = \frac{\ln(S_0/K) + (r - q + \sigma^2/2)T}{\sigma\sqrt{T}} = \frac{\ln(50/50) + (0.04 - 0.02 + 0.18^2/2) \times 0.5}{0.18 \times \sqrt{0.5}}$ $d_1 = \frac{0 + (0.02 + 0.0162) \times 0.5}{0.18 \times 0.7071} = \frac{0.0181}{0.1273} = 0.1422$

$d_2 = d_1 - \sigma\sqrt{T} = 0.1422 - 0.18 \times 0.7071 = 0.1422 - 0.1273 = 0.0149$

Now calculate the normal CDF values:

• $N(d_1) = N(0.1422) \approx 0.5565$
• $N(d_2) = N(0.0149) \approx 0.5059$

Finally, compute the call price: $C = 50 \times e^{-0.02 \times 0.5} \times 0.5565 - 50 \times e^{-0.04 \times 0.5} \times 0.5059$ $C = 50 \times 0.9900 \times 0.5565 - 50 \times 0.9802 \times 0.5059$ $C = 27.53 - 24.79 = 2.74$

The closest answer is $2.75 (Option B), though there may be slight rounding differences in calculation.