Explanation

This is a Black-Scholes option pricing problem for a European put option. The key parameters are:

• Stock price (S): $50
• Strike price (K): $50
• Time to expiration (T): 3 months = 0.25 years
• Risk-free rate (r): 10% = 0.10
• Volatility (σ): 30% = 0.30
• Dividend yield: 0 (non-dividend-paying stock)

Black-Scholes Formula for Put Option:

$P = Ke^{-rT}N(-d_2) - SN(-d_1)$

Where: $d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$ $d_2 = d_1 - \sigma\sqrt{T}$

Calculations:

1. Calculate d₁: $d_1 = \frac{\ln(50/50) + (0.10 + 0.30^2/2) \times 0.25}{0.30 \times \sqrt{0.25}} = \frac{0 + (0.10 + 0.045) \times 0.25}{0.30 \times 0.5} = \frac{0.145 \times 0.25}{0.15} = \frac{0.03625}{0.15} = 0.2417$

2. Calculate d₂: $d_2 = d_1 - \sigma\sqrt{T} = 0.2417 - 0.30 \times 0.5 = 0.2417 - 0.15 = 0.0917$

3. Calculate N(-d₁) and N(-d₂): $N(-d_1) = N(-0.2417) = 0.4045$ $N(-d_2) = N(-0.0917) = 0.4634$

4. Calculate put option price: $P = 50 \times e^{-0.10 \times 0.25} \times 0.4634 - 50 \times 0.4045$ $P = 50 \times e^{-0.025} \times 0.4634 - 20.225$ $P = 50 \times 0.9753 \times 0.4634 - 20.225$ $P = 22.59 - 20.225 = 2.365$

Rounded to two decimal places, the put option price is $2.37, which corresponds to option A.

Verification:

This result makes sense because:

• The option is at-the-money (S = K)
• With 3 months to expiration and 30% volatility, there's significant time value
• The put-call parity relationship also confirms this result