Submission: Tingjun GARCH model updates

[tingjun-yang]

Hi David,

Thank you for the clarification on the requirements.

Following your instructions, I have re-optimized the model parameters using a 5M-day sample against the theoretical target parameters ($\sigma_0 = 0.2586$ and $z_{off} = 0.0214$). Using the new configuration, I achieved an $R^2$ of 0.997.

I also conducted a convergence study to analyze the relationship between sample size and fit stability. My results show that $R^2$ scales positively with the number of days, consistently stabilizing above 0.995 once the sample exceeds 1M days. This confirms the structural robustness of the parameters against stochastic noise.

Additionally, I implemented a vectorized simulation function using NumPy, which has significantly improved computational speed and allowed for faster iteration across these larger datasets.

I have updated the repository with these findings and look forward to your feedback!

Best regards，
Tingjun Yang

[tingjun-yang]

Hi @Orrell, sorry I just saw your email about the 0.01 parameter. I am going to address that and update this PR.

[tingjun-yang]

Ok @Orrell I've updated README.md and declare 0.01 as another parameter Mean Reversion (γ) so the model now has 4 parameters. The results did not change.

[Orrell]

Hi @tingjun-yang To compare your model with standard GARCH I set alpha=0.2475, beta = 0.7425, sigma = 0.095. In standard GARCH the model has omega = (1.0 - alpha - beta) * sigma2_daily while in your model the difference is that omega = (1.0 - beta) * sigma2_daily. So it isn't the same as a standard GARCH model and I was wondering if (a) you had tried optimizing the standard model to compare performance, and (b) if you had a particular motivation for making the change or was it more trial and error?

[tingjun-yang]

Hi @Orrell, Thanks for your question. I think my notation is a little confusing. I was trying to follow the GARCH model as explained in this article:
In a GARCH(1,1) model, we calculate today's variance ($\sigma_t^2$) using three specific components:
$$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2$$
Where:
$\sigma_t^2$: Today's predicted variance (how much risk we expect).
$\omega$ (Omega): The Baseline. This is a constant; the long-term average volatility.
$\alpha$ (Alpha): The "Shock" term. It looks at yesterday’s squared return ($\epsilon_{t-1}^2$). If there was a massive market crash yesterday, $\alpha$ makes today's volatility jump.
$\beta$ (Beta): The "Persistence" term. It looks at yesterday’s predicted variance ($\sigma_{t-1}^2$). If volatility was high yesterday, it’s likely to stay high today.

The stability of the GARCH(1,1) model depends on the sum of $\alpha + \beta$:

• Integrated (IGARCH): $\alpha + \beta = 1$; volatility shocks persist forever.
• Mean-Reverting: $\alpha + \beta < 1$; volatility returns to its long-run average.

My implementation utilizes the mean-reverting setting. I defined the persistence as $\lambda = \beta$ and the mean-reversion weight as $\gamma = 1 - \alpha - \beta$. To streamline the optimization, I fixed $\gamma$ at $0.01$ and focused on estimating the parameters $\mu$, $\sigma$, and $\lambda$.

[Orrell]

Hi @tingjun-yang Yes, the switch to lambda and gamma makes it harder to compare so maybe I am missing something here, but in your model you have beta (=lambda) = 0.8. Also gamma = 1 - alpha - beta = 0.01 so alpha = 1 - beta - 0.01 = 0.19. In a normal GARCH model you have omega = (1 - alpha - beta) * sigma2_daily = 0.01sigma2_daily, but in yours you have omega = (1 - beta) * sigma2_daily = 0.2sigma2_daily. Is that correct? In any case can I ask that for comparison with other GARCH models you rephrase using parameters alpha and beta (rather than lambda/gamma)? This would make interpretation a lot easier. Thanks, David

[tingjun-yang]

Hi @Orrell, Thanks for pointing that out. I see what you mean now. Yes, the initial definition of omega was an oversight. I will correct that and use the standard parameter names alpha and beta. I will reoptimize the parameters and submit a PR soon. Thanks!

[Orrell]

Thanks, it will be interesting to see whether the standard GARCH can do as well. Either way it would be very useful to have a note in the README summarizing the difference. David On Dec 28, 2025, at 3:18 PM, tingjun-yang ***@***.***> wrote: [https://avatars.githubusercontent.com/u/128106253?s=20&v=4]tingjun-yang left a comment (q-variance/challenge#16)<#16 (comment)> Hi @Orrell<https://github.com/Orrell>, Thanks for pointing that out. I see what you mean now. Yes, the initial definition of omega was an oversight. I will correct that and use the standard parameter names alpha and beta. I will reoptimize the parameters and submit a PR soon. Thanks! — Reply to this email directly, view it on GitHub<#16 (comment)>, or unsubscribe<https://github.com/notifications/unsubscribe-auth/AEBN4KLQLSAA6EC4HHIRK7L4EA3KNAVCNFSM6AAAAACQCQWPR6VHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZTMOJVGAZDAMRYGI>. You are receiving this because you were mentioned.Message ID: ***@***.***>