三个月平值看跌期权估值

平值、$r=0$、无分红时，短期限 Black-Scholes 近似为

$$P_{ATM}\approx \frac{S\sigma\sqrt{T}}{\sqrt{2\pi}}\approx 0.4S\sigma\sqrt{T}.$$

本题 $S=K=40$，$\sigma=0.30$，$T=1/4$，所以

$$P\approx 0.4\cdot 40\cdot 0.30\cdot \frac12=2.40.$$

若用精确 Black-Scholes 公式，

$$d_1=0.075,\quad d_2=-0.075,\quad P=40\bigl(N(0.075)-N(-0.075)\bigr)\approx 2.39.$$

因此该 put 约值 $\boxed{2.4}$。

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英文解析

For an at-the-money option with $r=0$ and no dividends, the short-maturity Black-Scholes approximation is

$$P_{ATM}\approx \frac{S\sigma\sqrt{T}}{\sqrt{2\pi}}\approx 0.4S\sigma\sqrt{T}.$$

Here $S=K=40$, $\sigma=0.30$, and $T=1/4$, so

$$P\approx 0.4\cdot 40\cdot 0.30\cdot \frac12=2.40.$$

Using the exact Black-Scholes formula gives

$$d_1=0.075,\quad d_2=-0.075,\quad P=40\bigl(N(0.075)-N(-0.075)\bigr)\approx 2.39.$$

So the put is worth about $\boxed{2.4}$.