Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / B / 8 / a / Solution

Set $x=B^2/S$ and $q=(S/B{)}^{\alpha }$, so $W=qV(t,x)$. Direct differentiation givesSubstitute into the Black-Scholes equation and divide by $q$:To recover the same operator on $V$, its first-derivative coefficient must be $r$. This givesFor this choice the last coefficient is $-r$, since $\frac{1}{2}{\sigma }^2(\alpha -1)+r=0$. Hence $\mathcal{L}W=q\mathcal{L}V=0$, proving the required invariance. The transformation is the stock-inversion reflection for the Black-Scholes equation.