Black Scholes PDE

Question

How to show that $V_1(S,t)=S\frac{\partial V(S,t)}{\partial S} $ satisfies Black-Scholes PDE given as $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ ?

Answer 1

Let $V(S,t)$ --- solution of the PDE. Then $$ S\frac{\partial}{\partial S}\left(\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV\right) = 0 $$ $$ S\left(\frac{\partial^2 V}{\partial t \partial S} + \frac{\sigma^2 }{2}\left(2S\frac{\partial^2V}{\partial S^2} + S^2\frac{\partial^3V}{\partial S^3}\right) + r\frac{\partial V}{\partial S} + rS\frac{\partial^2 V}{\partial S^2} - r\frac{\partial V}{\partial S} \right) = 0 $$ $$ \frac{\partial}{\partial t}S\frac{\partial V}{\partial S} + \frac{\sigma^2 S^2 }{2}\left(2\frac{\partial^2V}{\partial S^2} + S\frac{\partial^3V}{\partial S^3}\right) + rS^2\frac{\partial^2 V}{\partial S^2} = 0. $$ Here we note that $\frac{\partial^2}{\partial S^2}\left(S\frac{\partial V}{\partial S}\right) = 2\frac{\partial^2V}{\partial S^2} + S\frac{\partial^3V}{\partial S^3}$ and $rS\frac{\partial}{\partial S}\left(S\frac{\partial V}{\partial S} \right) = rS\frac{\partial V}{\partial S} + rS^2\frac{\partial^2 V}{\partial S^2}$.

And finally $$ \frac{\partial V_1}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V_1}{\partial S^2} + rS\frac{\partial V_1}{\partial S} -rV_1 = 0. $$