三个月 ATM 期权的心算近似

Interviewer

专题: Finance / 金融
难度: L4
来源: QuantQuestion

个人笔记

题目详情

一只 $100$ 美元股票，三个月 ATM call，隐含波动率 $40\%$ ，设 $r=0$ 且无分红。10 秒内估算 call 价值；若改为 put，答案如何变化？

英文原题

Interviewer: “You are fully familiar with Black–Scholes pricing, aren’t you?”
Interviewee: (confidently, after a slight pause) “Yes indeed.”

Interviewer:
“What is the value of a three-month at-the-money (i.e., $S=X$ ) call option on a $100 stock when the implied vol is $40\%$ ?
Please assume $r=0$ (it is the least important ingredient anyway) and assume also that the stock pays no dividends.
You have 10 seconds to perform the calculation in your head.
Now tell me how your answer changes if it is instead a put.”

10-second answer (mental math)

Given $S=X=100$ , $T=\tfrac{1}{4}$ , $\sigma=0.40$ , $r=0$ :

$\sigma\sqrt{T} = 0.40 \times \sqrt{0.25} = 0.20$ , so
$d_1 = \tfrac{\sigma\sqrt{T}}{2} = 0.10$ , $d_2 = -0.10$ .
ATM with $r=0$ :
$C = S\big(2N(d_1)-1\big) \approx 100\,(2N(0.10)-1) \approx 7.97 \approx 8.$
By put–call parity with $r=0$ and $S=X$ ,
$P = C \approx 8.$

Answer: Call ≈ $8. Put ≈ $8 (no change).

解析

平值、 $r=0$ 、无分红时，

C\approx P\approx \frac{S\sigma\sqrt{T}}{\sqrt{2\pi}} \approx 0.4S\sigma\sqrt{T}.

本题 $S=100$ ， $\sigma=0.40$ ， $T=1/4$ ，所以 $\sqrt{T}=1/2$ 。因此

C\approx 0.4\cdot100\cdot0.40\cdot\frac12=8.

由 put-call parity， $r=0$ 且 $S=K$ 时 ATM put 与 call 同价：

P=C\approx \boxed{8}.

英文解析

For an at-the-money option with $r=0$ and no dividends,

C\approx P\approx \frac{S\sigma\sqrt{T}}{\sqrt{2\pi}} \approx 0.4S\sigma\sqrt{T}.

Here $S=100$ , $\sigma=0.40$ , and $T=1/4$ , so $\sqrt{T}=1/2$ . Hence

C\approx 0.4\cdot100\cdot0.40\cdot\frac12=8.

By put-call parity, when $r=0$ and $S=K$ , the ATM put has the same value:

P=C\approx \boxed{8}.