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You are generating a price series, in time steps $p_{dt},p_{2dt},p_{3dt},...$. I assume? So if you do $\text{Median}\{p_{dt},p_{2dt},p_{3dt},..\}$ then you will get a value biased towards $0$ (for the parameters you gave, and the sample size you have). If you want to look at the distributional characteristics of the price, then you have to do so per each ...

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I suggest that you can use the Pathways - Euler Method to estimate vega, for sure with this method you can approximate the vega value.

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