Black Scholes to Heat Equation

Question

Equation (2) was derived by setting r=0 in the Black-Scholes equation for the Bachelier model (1).

Can someone please help me understand all the steps for how we get from the heat equation under time reversal (2) to (3) and then show me how to verify that the equation still holds? I cannot understand what exactly using $\eta$ achieves. Thanks!

Answer 1

Let $n=\sigma^2(T-t)$

$$dn=-\sigma^2dt$$

$\frac{\partial {V}}{\partial {t}} = \frac{\partial {V}}{\partial {n}}\frac{d {n}}{d {t}}$

$\frac{\partial {V}}{\partial {t}} = -\sigma^2\frac{\partial {V}}{\partial {n}}$

since $$\frac{\partial {V}}{\partial {t}} = -\frac{1}{2}\sigma^2\frac{\partial^2 {V}}{\partial {S^2}} $$

so

$$-\sigma^2\frac{\partial {V}}{\partial {n}} = -\frac{1}{2}\sigma^2\frac{\partial^2 {V}}{\partial {S^2}} $$

$$\frac{\partial {V}}{\partial {n}} -\frac{1}{2}\frac{\partial^2 {V}}{\partial {S^2}} =0$$