Assume we have the discretization in space $x_1, x_2, ... , x_M$ and time $t_1, t_2, ... , t_N$ for a finite difference or finite element method for option pricing and we want to solve for the option value $u(x, t) = E[e^{-r(T-t)}(x_T-K)^{+}]$ with $K$ the strike, $T$ the maturity of the option and $r$ a constant risk free rate.
When $u(x, t)$ is evaluated on the grid $[x_1, x_2, ... , x_M] * [t_1, t_2, ... , t_N]$ should I keep the same values of $x$ for each time step or do I have to take the forward/discounted value of $x$ when moving from one time step to another?
Suppose that at $t=t_N$ I used the values $x_1, x_2, ... , x_M$ to value $u$ as $u(x_1, t_N), u(x_2, t_N), ... , u(x_M, t_N)$. Should I use :
$$x_1e^{-r(t_N-t_{N-1})} , x_2e^{-r(t_N-t_{N-1})}, ... , x_Me^{-r(t_N-t_{N-1})}$$
at the next time step $t=t_{N-1}$ to value $u$ as : $$u(x_1e^{-r(t_N-t_{N-1})}, t_{N-1}), u(x_2e^{-r(t_N-t_{N-1})}, t_{N-1}), ... , u(x_Me^{-r(t_N-t_{N-1})}, t_{N-1})$$
Or simply take :
$$u(x_1, t_{N-1}), u(x_2, t_{N-1}), ... , u(x_M, t_{N-1})$$
? What is the reasoning I should have to understand what to use?
Thank you for your help.
