In a risk-neutral world, the probability that a European call option will be exercised is given by $N(d_2)$.

The formula for $d_2$ in the Black-Scholes-Merton model is: $d_2 = \frac{\ln(S / K) + (r - \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}$

Given the inputs:

• $S = 50$
• $K = 48$
• $T = 6/12 = 0.5$ years
• $r = 8\% = 0.08$
• $\sigma = 25\% = 0.25$

First, calculate the terms: $\ln(S/K) = \ln(50 / 48) \approx 0.040822$ $r - \frac{\sigma^2}{2} = 0.08 - \frac{0.25^2}{2} = 0.08 - 0.03125 = 0.04875$ Denominator = $\sigma\sqrt{T} = 0.25 \times \sqrt{0.5} \approx 0.176777$

Now find $d_2$: $d_2 = \frac{0.040822 + (0.04875 \times 0.5)}{0.176777} = \frac{0.040822 + 0.024375}{0.176777} = \frac{0.065197}{0.176777} \approx 0.3688$

Using standard normal distribution tables, the value for $N(d_2)$ when $d_2 = 0.37$ is approximately 0.6443. Option C is 0.6441, which matches this calculated value very closely (arising from slight rounding differences in table lookups).