3 个月 ATM call 的心算近似

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.”

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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).