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ATM call：用价格标准差估价

Black-Scholes world

专题
Finance / 金融
难度
L4

题目详情

Assume a :contentReference[oaicite:0]{index=0} world.
You have a one-year European call option on a stock.

There are no dividends, the interest rate is assumed to be zero, and the option is struck at-the-money (i.e., strike equals current spot).

The current spot is 100100.

The standard deviation of terminal stock price (conditional on current stock price) is 100100.

Question: Is the call price closer to 11, 55, or 1010?

解析

把“到期价格的标准差（美元）”记为 σ$=σST\sigma_{\$}=\sigma S\sqrt{T}。在 r≈0r\approx 0 且 ATM 时，近似

C≈σ$2π≈0.4 σ$.\boxed{C\approx \frac{\sigma_{\$}}{\sqrt{2\pi}}\approx 0.4\,\sigma_{\$}}.

因此若标准差为 10 美元，call 约 10/2π≈410/\sqrt{2\pi}\approx 4；标准差为 20 美元则约 8。