Black-Scholes 微分方程改寫
我已經看到 Black-Scholes 方程
$$ \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+ rS\frac{\partial V}{\partial S}-rV=0 $$
也可以寫成以下形式：
$$ \frac{\partial V}{\partial t}+\frac{\partial}{\partial S} \left( \frac{1}{2}\sigma^2S^2\frac{\partial V}{\partial S}\right)+\left(rS-\frac{\partial}{\partial S}\left(\frac{1}{2}\sigma^2S^2\right)\right)\frac{\partial V}{\partial S}-rV=0. $$
以下兩項如何相等？
$$ \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+ rS\frac{\partial V}{\partial S} = \frac{\partial}{\partial S} \left( \frac{1}{2}\sigma^2S^2\frac{\partial V}{\partial S}\right)+\left(rS-\frac{\partial}{\partial S}\left(\frac{1}{2}\sigma^2S^2\right)\right)\frac{\partial V}{\partial S} $$
注意： $$ \frac{\partial}{\partial S} \left( \frac{1}{2}\sigma^2S^2\frac{\partial V}{\partial S}\right) =\sigma^2S\frac{\partial V}{\partial S}+ \frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2} $$ 上面的第一項與第二個括號中的第二項相抵消。