I'm learning about SDE's and I need to simulate the black scholes equation. I asked already several things about it, but I'm still puzzeled by a few things.
I obtained (with the help of forum members) the following equations for the black scholes equation:
SDE:
$d S_{t}=\alpha S_{t} d t+\sigma_{t} S_{t} d w_{t}^{1}$
Euler scheme:
$f_{t+\Delta t}=f_{t}+\left(\alpha-\frac{1}{2} \sigma_{t}^{2}\right) \Delta t+\sigma_{t} z_{t}^{1} \sqrt{\Delta t}$
When I write convert this back with:
$S_{t}=\exp \left(f_{t}\right)$ or $f_{t}=\ln \left(S_{t}\right)$
The exact solution is obtained:
$S_{t+d t}=S_{t} \exp \left(\left(\alpha-\frac{1}{2} \sigma_{t}^{2}\right) d t+\sigma_{t} z_{t}^{1} \sqrt{d t}\right)$
When I want to realise these tracks I don't see why I should use the euler scheme instead of the exact solution.
In want to emperically show the order of convergence in the strong and in the weak sense. Let's focus on the order of convergence in the weak sense, thus the the error of the mean. I understand that I need to realise a lot of tracks for several dt. And for each dt I need to calculate the average on a time T. But to what do I compare these averages to in order to calculate the error? Which formula do I need to use in order to generate the tracks the $S_t$ formula or the $f_t$ formula and why?
I searched for them, but haven't found them. Are there maybe examples available of order to convergence calculations for the black scholes equation?
Tim