# How underlying asset price variance is connected with time

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?

• 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. – Alex C Jun 10 '19 at 13:07

## 1 Answer

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).