Explanation

The Black-Scholes-Merton model for pricing options on futures contracts uses the following formula:

$c = F_0e^{-rT}N(d_1) - Ke^{-rT}N(d_2)$

Where:

• $F_0$ = current futures price = EUR 63
• $K$ = strike price = EUR 68
• $T$ = time to expiration of option = 0.5 years (6 months)
• $r$ = risk-free interest rate = 3% = 0.03
• $N(d_1)$ = 0.4678
• $N(d_2)$ = 0.3449

Step-by-step calculation:

1. Calculate the discount factor: $e^{-rT} = e^{-0.03 \times 0.5} = e^{-0.015} \approx 0.9851$

2. Calculate the first term: $F_0e^{-rT}N(d_1) = 63 \times 0.9851 \times 0.4678 \approx 29.04$

3. Calculate the second term: $Ke^{-rT}N(d_2) = 68 \times 0.9851 \times 0.3449 \approx 23.11$

4. Calculate the call option price: $c = 29.04 - 23.11 = EUR 5.93$

Key points:

• The time to maturity of the underlying futures contract (18 months) is not used in the BSM formula for options on futures
• The formula uses the current futures price rather than the spot price
• Both terms are discounted using the same discount factor $e^{-rT}$
• The result matches option B: EUR 5.93