## Change of Variables

I have a function $f(S,t)$, which satisfies some PDE, with the condition that $f(S,T) = max\{S-1,0\}$. I introduce two new variables $x$ and $\tau$ as follows:

$S = e^x$ and $t=T-\frac{2\tau}_{\sigma^2}$, and define the function $W$ by $f(S,t) = W(x,\tau)$. Is the new initial condition going to be:

$f(S,T)=W(x,\tau(t)=T)=W(x,0)=\{e^{x}-1,0\}?$

Thanks.