Skip to content
All library documents

Testing the Square-Root-Time Diffusion Pattern in Market Returns

Article Quant Q&A · Author: Azur
Read the original

Summary

The question asks whether the square-root-of-time scaling associated with diffusion in the heat equation also appears in financial price models connected to Black–Scholes. The proposed quantity is the typical distance traveled over a time interval, which scales with the square root of elapsed time under standard diffusion assumptions. The response points to the Lo–MacKinlay variance ratio test as a way to examine this relationship in market returns and names a methodological resource.

The exchange offers a research direction rather than a derivation, simulation, or empirical result. A variance ratio test can assess return variance across different horizons under assumptions related to random walks, but it does not establish that real markets follow the Black–Scholes model or that a measured pattern has a physical interpretation. The discussion does not specify data selection, test implementation, or controls, so anyone applying the idea would need to define the return series and horizon comparisons and interpret findings with those limitations in mind.

Key ideas

  • Diffusion models imply that typical displacement grows with the square root of elapsed time.
  • The question connects this scaling intuition to Black–Scholes and financial price models.
  • The suggested empirical tool is the Lo–MacKinlay variance ratio test.
  • The exchange provides a pointer to a method, not a simulation or evidence that market prices obey the model.
  • Empirical interpretation depends on the data and assumptions used in the test.

Tags

StatisticsEquitiesDerivatives pricingBacktesting
Full text
# Trying to measure "radius of diffusion" in the stock market


# Trying to measure "radius of diffusion" in the stock market












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 :)

## Answer by Azur (score 1, accepted)

https://quant.stackexchange.com/a/54659

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/

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.