为啥在计算S0的时候用的不是T为1.5，futures不是1.5年之后到期嘛？

问题如下：

A derivatives dealer actively trades options on various underlying assets with its clients. The firm wants to apply the Black-Scholes-Merton (BSM) model to price a call option on a futures contract. Relevant data is provided below:

· Current futures price: EUR 63

· Strike price of the option: EUR 68

· Time to expiration of the option: 6 months

· Time to maturity of the underlying futures contract: 18 months

· Continuously compounded annual risk-free interest rate: 3%

· N(d₁): 0.4678

· N(d₂): 0.3449

Which of the following is closest to the value of this option estimated using the BSM model?

选项：

A.

EUR 5.75

B.

EUR 5.93

C.

EUR 6.36

D.

EUR 6.81

解释：

B is correct:

The option on futures using the BSM model is expressed as follows:

c = F0e^−rTN(d₁) − Ke^−rTN(d₂)

where:

F₀ = current futures price = EUR 63

K = strike price of the option on futures = EUR 68

T = time to expiration of option = 0.5

r = risk-free interest rate = 3%

N(d₁) = 0.4678

N(d₂) = 0.3449

Therefore,

c=63 * e^{−0.03 * 0.5} * (0.4678) − 68 * e^{−0.03 * 0.5} * (0.3449)=EUR 5.9286

A is incorrect. This values the option using the time to maturity of the futures contract rather than the time to expiration of the option.

C is incorrect. This omits the “e^−rT” from the futures price term in the equation.

D is incorrect. This multiplies the futures price by e^qT instead of e^-qT in the equation.