call 的 Greeks Greeks - Solution 专题 Finance / 金融 难度 L4 来源 QuantQuestion 题目详情 Calculate the greeks Δ,Γ,V,ρ,Θ\Delta ,\Gamma ,\mathcal{V},\rho ,\ThetaΔ,Γ,V,ρ,Θ for a call option. 解析 Black–Scholes（连续分红率 qqq）： C=Se−qτN(d1)−Ke−rτN(d2),τ=T−t,C=Se^{-q\tau}N(d_1)-Ke^{-r\tau}N(d_2),\quad \tau=T-t,C=Se−qτN(d1)−Ke−rτN(d2),τ=T−t, d1=ln(S/K)+(r−q+12σ2)τστ,d2=d1−στ.d_1=\frac{\ln(S/K)+(r-q+\tfrac12\sigma^2)\tau}{\sigma\sqrt{\tau}},\quad d_2=d_1-\sigma\sqrt{\tau}.d1=στln(S/K)+(r−q+21σ2)τ,d2=d1−στ. 令 φ\varphiφ 为标准正态密度： Δ=∂SC=e−qτN(d1)\boxed{\Delta=\partial_S C=e^{-q\tau}N(d_1)}Δ=∂SC=e−qτN(d1) Γ=∂SSC=e−qτφ(d1)Sστ\boxed{\Gamma=\partial_{SS}C=\frac{e^{-q\tau}\varphi(d_1)}{S\sigma\sqrt{\tau}}}Γ=∂SSC=Sστe−qτφ(d1) V=∂σC=Se−qτφ(d1)τ\boxed{\mathcal{V}=\partial_{\sigma}C=Se^{-q\tau}\varphi(d_1)\sqrt{\tau}}V=∂σC=Se−qτφ(d1)τ（vega） ρ=∂rC=Kτe−rτN(d2)\boxed{\rho=\partial_r C=K\tau e^{-r\tau}N(d_2)}ρ=∂rC=Kτe−rτN(d2) Θ=∂tC=−Se−qτφ(d1)σ2τ−rKe−rτN(d2)+qSe−qτN(d1)\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)}Θ=∂tC=−2τSe−qτφ(d1)σ−rKe−rτN(d2)+qSe−qτN(d1)