Quick Discretization question for finite difference and finite element methods

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.

• The points in $x,t$ space are absolutely fixed and known at the outset of your calculation. Depending on your solution scheme (explicit, implicit / Crank-Nicholson) you ‚simply‘ add weighted neighbouring function values. Hth? Commented Nov 14, 2020 at 6:10
• Thank you. This is very helpful! Maybe you can help me with my other question? quant.stackexchange.com/questions/59347/…. Thanks again Commented Nov 14, 2020 at 11:35