简单记录第一次看 The Volatility Surface 时推导有问题的部分。

(P16) 给定下列等式:

Vt+12vS22VS2+ρηvS2VSv+12η2v2Vv2+rSVSrVλ(vvˉ)Vv=0 \begin{gathered} \frac{\partial V}{\partial t}+\frac{1}{2} v S^2 \frac{\partial^2 V}{\partial S^2}+ \rho \eta v S \frac{\partial^2 V}{\partial S \partial v}+\frac{1}{2} \eta^2 v \frac{\partial^2 V}{\partial v^2}+r S \frac{\partial V}{\partial S}-r V -\lambda(v-\bar{v}) \frac{\partial V}{\partial v}=0 \end{gathered} 进行如下的变量代换: x=ln(Ft,T/K)=ln(Serτ/K), x=\ln \left(F_{t, T} / K\right)=\ln \left(S e^{r \tau} / K\right) , 可得到如下的等式: Cτ+12v2Cx212vCx+12η2v2Cv2+ρηv2Cxvλ(vvˉ)Vv=0 \begin{gathered} -\frac{\partial C}{\partial \tau}+\frac{1}{2} v \frac{\partial^2 C}{\partial x^2}-\frac{1}{2} v \frac{\partial C}{\partial x}+\frac{1}{2} \eta^2 v \frac{\partial^2 C}{\partial v^2}+\rho \eta v \frac{\partial^2 C}{\partial x \partial v}-\lambda(v-\bar{v}) \frac{\partial V}{\partial v}=0 \end{gathered} 一开始看书时没明白 rSVSrVr S \frac{\partial V}{\partial S}-r V 这项是怎么被消去的。看了这个回答之后才发现我没把当期价格VV表示成终期价格CCerτC(x,ν,τ)=V(S,ν,t)e^{-r \tau} C(x, \nu, \tau)=V(S, \nu, t),所以一开始推不出书上的式子。