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I'm dealing with option pricing models and there is a statement that says the variance of underlying asset price is propotional with time $𝑉𝑎𝑟(𝑆_{𝑚+1})=𝑆_𝑚^2𝜎^2Δ𝑡$ where $\Delta t = \frac{T}{N}$ and $\sigma$ is volatility. How this equality can be explained/proved?

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    $\begingroup$ This holds true for stochastic processes having time homogeneous Independent Increments. Are you familiar with this property? For such a process the variance over two days is the sum of the variance of day 1 and the variance of day 2 (by independence). These two variances are the same by the assumption that the process is time homogeneous (does not change in time). So the two day variance is twice the 1 day variance. $\endgroup$ – Alex C Jun 10 at 13:07
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Exactly what @Alex C said. It's the time homogeneous diffusion proprety. You can't state such an argument in models where volatility is no longer time homogeneous ( that's being time independant and depending only on the underlyings).

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