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Good evening!

I'm quite new to quantitative finance (coming from the math world!), so please excuse me if I'm not familiar with every concept!

I am currently studying the Black-Scholes equation, and how it can be transformed to the heat equation. So a question arised. I know that the "radius of diffusion" (ie the average distance a particle of heat travels in a set time $t$) is of order $\sqrt{t}$

Now, I assumed this property would transfer to some specific financial derivatives, since the Black-Scholes equation is equivalent to the heat equation. But, while it is quite easy to verify the property experimentally in the "physics model"; for a novice like me, simulating it financially seems "very out of reach". And I'd like to try and simulate numerically such a property; say, by modelling it in a "perfect hypothetical stock model" so I can get the hang of it. But I don't really know where to start... It's still very blurry for me.

So I wanted to know if anyone had any resources, or knew of good models verifying this "$\sqrt{t}$" property! Or simply where to look at. I tried looking for a few keywords but didn't get anywhere :/ So any idea is welcome.

Thanks for your time :)

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    $\begingroup$ BS in theory is used for vanilla European option pricing, that is the "derivatives" you referred to. Would expect the radius of diffusion concept to correspond to the volatility of the underlying asset. $\endgroup$ – John Jun 1 '20 at 19:23
  • $\begingroup$ So volatility of the market would be spreading proportionally to $\sqrt{t}$, on average? That means I'd need to find a way to measure the growth in volatility w.r.t time! $\endgroup$ – Azur Jun 1 '20 at 19:43
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    $\begingroup$ This property of financial markets (the $\sqrt t$ property) has been checked empirically under the name of Variance Ratio Test of market efficiency. It is not perfectly satisfied in practice, only approximately. (Sorry to disappoint you: Black Scholes Merton theory is not perfect description of reality ;) $\endgroup$ – noob2 Jun 1 '20 at 19:51
  • $\begingroup$ Thanks, I've just googled it and started looking into it. So pardon my additional question I just want to make sure I understand: so the VR test is used to verify if a stock model does indeed grow within a "$\pm\sqrt{t}$-bounded region (under some perfect conditions)? :) that would be the perfect for my simulations! :D $\endgroup$ – Azur Jun 1 '20 at 19:57
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Answer was given by noob2. The property can be checked by a "Variance ratio test of market efficiency" (Lo & MacKinley).

Here's a nice methodology I found: https://mingze-gao.com/measures/lomackinlay1988/

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