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Are the log-periodic power law models used to predict financial market crashes continuous or discrete-time processes?

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    $\begingroup$ Could you add a reference to the Log-periodic power law for those (such as me) that are unfamiliar with it? $\endgroup$ – Bob Jansen Oct 23 '16 at 15:03
  • $\begingroup$ I believe the OP is referring to the work of the [in]famous Didier Sornette. $\endgroup$ – Alex C Oct 23 '16 at 20:09
  • $\begingroup$ Yeah sorry, heres a reference arxiv.org/pdf/cond-mat/9907270v1.pdf $\endgroup$ – Darragh Oct 25 '16 at 3:40
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The model is continuous in t, but you can use is as an approximation (with discrete prices as inputs) without adjustments.

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The log distribution of returns is $$\frac{1}{2\gamma}\text{sech}\left[\frac{\pi}{2}\left(\frac{x-\mu}{\gamma}\right)\right]$$ when bankruptcy, mergers and the budget constraint are ignored. I think the actual shocks in Sornette are actually liquidity shocks and because the bid-ask spread is ignored in most modelling, it looks like changes of underlying volatility. There is enough empirical data to support the issue that I think a careful study needs done. Also, modeling things in logs is a bit more difficult than to do so in raw form. We don't use punch cards anymore. No one buys log(3 shares) +log($7/share).

You should treat them as continuous. Do note that as you add variables to the distribution above, that you do not add covariates. The distribution involved lacks anything resembling a covariance matrix. $\gamma$ just turns into $\gamma'$.

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