I have the standard black-scholes PDE for a put option (paying no dividends):
$\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 P}{\partial S^2}+rS\frac{\partial P}{\partial S}-rP=0$
Also the standard payoff for a binary put option $P_b(S,t)$ has the payoff:
$$ P_b(S,t=T)= \begin{cases} 0 &if &S>X,\\ K &if & S<X \end{cases} $$
Somehow I need to show that $P_b(S,t) = K\frac{\partial P}{\partial X}$ where $P(S,t;X)$ is the value of a 'vanilla put option'.
I understand that if you differentiate the payoff of a vanilla call at expiry then you will get $0$ if $S>X$ and $1$ if $S<X$ and therefore if you simply multiply this by K you get the payoff of the binary put at expiry.
However how would you show $P_b(S,t) = K\frac{\partial P}{\partial X}$ for all t and not just at expiry? The only hint that I have been given is that I need to differentiate the black-scholes pde with respect to X, but surely this is just zero?
Thank you in advanced