股票翻倍：call 会怎么变

没有单一答案，取决于期权的 moneyness、到期、波动率等。

• 深度实值（deep ITM）：由 put-call parity 可近似 $C\approx S-Ke^{-rT}$，因此当 $S$ 翻倍时 $C$ 近似从 $S-Ke^{-rT}$ 变为 $2S-Ke^{-rT}$，即大致翻倍。
• 平值附近（ATM）：翻倍会使期权从 ATM 变为深 ITM，价格的百分比变化通常会远大于 100%。
• 深度虚值（deep OTM）：原本很便宜，翻倍可能让它变得有显著价值，价格可能上升多个数量级。

定性总结：翻倍一定会使 call 价格上升，但上涨幅度强烈依赖于初始 moneyness。

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英文解析

The value of a deep in the money call on a non-dividend-paying asset can be approximated, e.g., by using the Put-Call parity, as
$C \approx S - Ke^{-rT}$,
where $K$ and $T$ are the strike and the maturity of the option, and $r$ is the constant risk-free rate.

Thus, if the spot price $S$ doubles, the call option will be even deeper in the money, and therefore its value will be approximately
$2S - Ke^{-rT}$.

In other words, the value of deep in the money calls roughly doubles if the spot price doubles.

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If the option is around at the money, the percentage change generated by the doubling of the stock price is about one order of magnitude larger since the option will become deep in the money.

If the option is deep out of the money, then it trades for fractions of cents. The doubling of the spot price would result in changing the price of the option by several orders of magnitude.

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As a numerical example, consider a six-month call option with strike $20$ on a non-dividend-paying underlying asset with volatility $25\%$. Assume that the risk-free rate is constant at $5\%$.
The Black-Scholes values of the call option corresponding to several spot prices of the underlying asset can be found below:

Spot Price	Option Price
10	0.000045
20	1.65
40	20.49
80	60.49
400	380.49
800	780.49

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If the call option is deep out of the money and the spot price doubles from $10$ to $20$, the value of the call increases from $0.000045$ to $1.65$, i.e., by more than four orders of magnitude.

If the call option is at the money and the spot price doubles from $20$ to $40$, the value of the call increases from $1.65$ to $20.49$, i.e., more than tenfold.

If the call option is deep in the money, and the spot price doubles from $40$ to $80$, the value of the call increases from $20.49$ to $60.49$, i.e., by a factor of $2.95$.

If the call option is even deeper in the money and the spot price doubles from $400$ to $800$, the value of the call increases from $380.49$ to $780.49$, i.e., the call approximately doubles in value.

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Moreover, if the call option is deep in the money, its value is very close to
$S - K e^{-rT}$,
i.e., the value of the spot price of the underlying asset minus the present value of the strike.

For all the spot prices greater than $40$, the estimate $C \approx S - Ke^{-rT}$ is very accurate.

Thus, if the call option is at the money and the spot price doubles, the value of the call option increases by the same amount as the increase in the spot price.

For example, if the spot price doubles from $40$ to $80$, the value of the call increases by $40$, from $20.49$ to $60.49$, which is exactly the increase in the spot price.