衍生品定价

一般定价：选计价资产（numeraire）$N_t$，则

$$\boxed{V_0=N_0\,\mathbb{E}^{\mathbb{Q}^N}\left[\frac{\text{Payoff}_T}{N_T}\right]}.$$

取股票为计价资产 $N_t=S_t$（stock numeraire），可以把含 $S_T$ 的 payoff 简化，并得到如 $N(d_1)$、$N(d_2)$ 的概率解释：

$$\mathbb{Q}^S(S_T>K)=N(d_1),\qquad \mathbb{Q}(S_T>K)=N(d_2).$$

因此 vanilla call 的价格可写为

$$\boxed{C_0=S_0N(d_1)-Ke^{-rT}N(d_2)}.$$

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英文解析

General pricing by a numeraire: choose a numeraire asset $N_t$. Then

$$\boxed{V_0=N_0\,\mathbb{E}^{\mathbb{Q}^N}\left[\frac{\text{Payoff}_T}{N_T}\right]}.$$

Taking the stock as numeraire, $N_t=S_t$, can simplify payoffs involving $S_T$ and gives the probability interpretations of $N(d_1)$ and $N(d_2)$:

$$\mathbb{Q}^S(S_T>K)=N(d_1),\\qquad \mathbb{Q}(S_T>K)=N(d_2).$$

Therefore the vanilla call price is

$$\boxed{C_0=S_0N(d_1)-Ke^{-rT}N(d_2)}.$$