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It isn't strictly speaking possible to convert a log vol to a normal vol, although it may be possible to get a rough idea. I am assuming you only have the vol of log returns but not the actual time series here. If you had the original time series, then you would just calculate the standard deviation of the prices to get the normal vol. I assume this is ...

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The log-return of a stock over a period $\Delta t$ starting at $t=0$ is defined as: $$r_{\Delta t} = \ln \left( \frac{S_{\Delta t}}{S_0} \right)$$ Thus you should compute $S_{\Delta t}$ as $$S_{\Delta t} = S_0 \exp ( r_{\Delta t} )$$ when you are given the $\Delta t$-period log-return i.e. the one which you sample as you propose above. Thus no ...

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You mix up several things: if you sample from Brownian motion, then $$B_{t+\Delta t} - B_t$$ is normally distributed with variance $\Delta t$. Thus if you sample a standard normal $Z$ (with variance 1) then you can use $$\sqrt{\Delta t} Z$$ as sample for $B_{t+\Delta t} - B_t$ in order to get the correct variance. Recall that constant factors enter ...

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