期权价格对参数变化的方向

基本结论：

• 看涨 payoff：$\max(S-K,0)$；看跌 payoff：$\max(K-S,0)$。
• 欧式只能到期行权；美式可提前行权，因此“时间更多”对美式通常不减价值。

方向总结：

变量	欧式看涨	欧式看跌	美式看涨	美式看跌
$S\uparrow$	上升	下降	上升	下降
$K\uparrow$	下降	上升	下降	上升
$\tau\uparrow$	不一定单调	不一定单调	上升	上升
$\sigma\uparrow$	上升	上升	上升	上升
$r\uparrow$	上升	下降	上升	下降
$D\uparrow$	下降	上升	下降	上升

直觉：

• $\sigma$ 越大，尾部更肥，期权（凸）更值钱。
• $r$ 越大，$K e^{-r\tau}$ 越小：看涨更值钱、看跌更不值钱。
• 分红会压低标的预期价格，对看涨不利、对看跌有利。

---

Original Explanation

• A call payoff is $\max(S-K,0)$. A put payoff is $\max(K-S,0)$.
• A European option can be exercised only at maturity $T$. An American option can be exercised at any time before $T$.

The table shows the direction of change for each input variable:

Variable	European call	European put	American call	American put
$S \uparrow$ (stock)	up	down	up	down
$K \uparrow$ (strike)	down	up	down	up
$\tau \uparrow$ (time)	? uncertain	? uncertain	up	up
$\sigma \uparrow$ (vol)	up	up	up	up
$r \uparrow$ (rate)	up	down	up	down
$D \uparrow$ (dividend)	down	up	down	up

• Stock price $S$ / Strike $K$: straightforward from payoff arguments.
• Time $\tau$:
  • For a European option: not necessarily monotonic, because it can only be exercised at $T$. The effect of more time is uncertain for a pure European payoff.
  • For an American option: having more time generally increases the option value (it can be exercised at any time).
• Volatility $\sigma$: higher volatility $\Rightarrow$ more chance to benefit from payoff $\Rightarrow$ value goes up.
• Risk-free rate $r$: affects discounting of $K e^{-r\tau}$. Higher $r$ makes $K e^{-r\tau}$ smaller $\Rightarrow$ call is worth more, put is worth less.
• Dividend $D$: paying a dividend lowers the underlying stock price $\Rightarrow$ opposite effect to $S$.