BS：三个月 ATM put（S=40, vol=30%） At-the-money put 专题 Finance / 金融 难度 L4 来源 QuantQuestion 题目详情 How much should a three months at- the- money put on an asset with spot price $40 and volatility 30% be worth? Assume, for simplicity, that interest rates are zero and that the asset does not pay dividends. 解析 Black–Scholes（r=0,q=0r=0,q=0r=0,q=0）： P=S N(−d1)−K N(−d2),d1=ln(S/K)+12σ2TσT, d2=d1−σT.P=S\,N(-d_1)-K\,N(-d_2),\quad d_1=\frac{\ln(S/K)+\frac12\sigma^2T}{\sigma\sqrt{T}},\ d_2=d_1-\sigma\sqrt{T}.P=SN(−d1)−KN(−d2),d1=σTln(S/K)+21σ2T, d2=d1−σT. 本题 S=K=40S=K=40S=K=40，σ=0.30\sigma=0.30σ=0.30，T=0.25T=0.25T=0.25，因此 σT=0.30⋅0.5=0.15,d1=0.5⋅0.32⋅0.250.15=0.075,d2=−0.075.\sigma\sqrt{T}=0.30\cdot 0.5=0.15,\quad d_1=\frac{0.5\cdot 0.3^2\cdot 0.25}{0.15}=0.075,\quad d_2=-0.075.σT=0.30⋅0.5=0.15,d1=0.150.5⋅0.32⋅0.25=0.075,d2=−0.075. 所以 P=40(N(0.075)−N(−0.075))≈40⋅0.0598≈2.39.P=40\bigl(N(0.075)-N(-0.075)\bigr)\approx 40\cdot 0.0598\approx \boxed{2.39}.P=40(N(0.075)−N(−0.075))≈40⋅0.0598≈2.39.