call 的 Greeks

Black–Scholes（连续分红率 $q$）：

$$C=Se^{-q\tau}N(d_1)-Ke^{-r\tau}N(d_2),\quad \tau=T-t,$$ $$d_1=\frac{\ln(S/K)+(r-q+\tfrac12\sigma^2)\tau}{\sigma\sqrt{\tau}},\quad d_2=d_1-\sigma\sqrt{\tau}.$$

令 $\varphi$ 为标准正态密度：

• $\boxed{\Delta=\partial_S C=e^{-q\tau}N(d_1)}$
• $\boxed{\Gamma=\partial_{SS}C=\frac{e^{-q\tau}\varphi(d_1)}{S\sigma\sqrt{\tau}}}$
• $\boxed{\mathcal{V}=\partial_{\sigma}C=Se^{-q\tau}\varphi(d_1)\sqrt{\tau}}$（vega）
• $\boxed{\rho=\partial_r C=K\tau e^{-r\tau}N(d_2)}$
• $\boxed{\Theta=\partial_t C=-\frac{Se^{-q\tau}\varphi(d_1)\sigma}{2\sqrt{\tau}}-rKe^{-r\tau}N(d_2)+qSe^{-q\tau}N(d_1)}$

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英文解析

$$C=Se^{-q\tau}N(d_1)-Ke^{-r\tau}N(d_2),\quad \tau=T-t,$$ $$d_1=\frac{\ln(S/K)+(r-q+\tfrac12\sigma^2)\tau}{\sigma\sqrt{\tau}},\quad d_2=d_1-\sigma\sqrt{\tau}.$$

• $\boxed{\Delta=\partial_S C=e^{-q\tau}N(d_1)}$
• $\boxed{\Gamma=\partial_{SS}C=\frac{e^{-q\tau}\varphi(d_1)}{S\sigma\sqrt{\tau}}}$
• $\boxed{\mathcal{V}=\partial_{\sigma}C=Se^{-q\tau}\varphi(d_1)\sqrt{\tau}}$（vega）
• $\boxed{\rho=\partial_r C=K\tau e^{-r\tau}N(d_2)}$
• $\boxed{\Theta=\partial_t C=-\frac{Se^{-q\tau}\varphi(d_1)\sigma}{2\sqrt{\tau}}-rKe^{-r\tau}N(d_2)+qSe^{-q\tau}N(d_1)}$