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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

Tim
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  • @UBM I think they should all contain t – Tim Dec 19 '20 at 19:13
  • do you know the form of $\sigma_t$? is $\sigma_t$ a function? – UBM Dec 19 '20 at 19:54
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    @UBM yes! I didn't include that function, since I thought it was not too important for this question, but please have a look here: https://math.stackexchange.com/questions/3941429/how-is-the-simulation-done-of-the-black-scholes-model/3941449#3941449 – Tim Dec 19 '20 at 23:17
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    In the title of this question you say Black-Scholes model (in this case $\sigma$ would be constant). However, in the other question $\sigma$ follows an Ito process. This is important because the (exact) solution ${S_t; 0 \leq t \leq T }$ depends on the form of $\sigma.$ – UBM Dec 20 '20 at 01:09
  • @UBM I don't understand. Ito is used to find a solution for the SDE right? why is the sigma constant in the exact solution, but not in the euler scheme? – Tim Dec 20 '20 at 09:24

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