Answer: $2.75

## 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.

Author: LeetQuiz Editorial Team