无限期限

如果标的不支付股息，即连续股息率 $q=0$，那么执行价为 $K$、现价为 $S$、无风险利率为 $r$、波动率为 $\sigma$ 的无限到期欧式看涨期权，在 Black-Scholes 模型下的极限价格为：

$$\boxed{C_{T \to \infty} = S}$$

没有股息时，一份超长期看涨期权几乎等同于持有股票：在足够长的时间范围内，标的价格大概率会超过任意固定的行权价；同时，在极远未来支付固定行权价的现值会趋近于 0，因为 $Ke^{-rT} \to 0$。因此期权价格趋近于股票现价 $S$。

如果标的支付连续股息率 $q > 0$，则看涨期权价格的极限为

$$\boxed{C_{T\to\infty} = 0}$$

在正股息率下，股票会持续派发价值，而看涨期权持有人拿不到这些股息；在无限期视角下，这部分损失会把期权价值压到 0。

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Original Explanation

If the underlying pays no dividends (continuous dividend yield $q=0$), the Black-Scholes value of a European call with strike $K$, spot $S$, interest rate $r$, and volatility $\sigma$ tends to:

$$\boxed{C_{T \to \infty} = S}$$

With no dividends, owning a very long-dated call is almost as good as owning the stock: over an arbitrarily long horizon the stock will — with high probability — reach levels far above any fixed strike and, importantly, the present value of paying the fixed strike at that very far future time is essentially zero (because $Ke^{-rT} \to 0$ according to the Black-Scholes model). So the call's price approaches the stock price $S$.

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If the underlying pays a continuous dividend yield $q > 0$, the call price instead tends to

$$\boxed{C_{T\to\infty} = 0}$$

With positive dividends the stock pays out value over time that the call-holder does not receive, and the present value of those foregone dividends over an infinite horizon wipes out the option's value — the call tends to zero.