Interpreting Log-Periodic Power Laws as Continuous-Time Models

Summary

The discussion addresses whether log-periodic power law (LPPL) models used in crash prediction should be treated as continuous- or discrete-time processes. Its main answer is that the model is defined continuously in time, while discrete price observations can be used as an approximation without modifying the model.

A second response raises cautions about interpreting the underlying distribution and shocks. It suggests liquidity shocks may be mistaken for volatility changes when bid-ask spreads are omitted, and notes that adding variables does not automatically justify adding covariates to a distribution without a covariance structure. These comments are presented as opinions, not as a derivation or empirical test. The document gives no data or evidence validating LPPL crash forecasts, so it clarifies the time convention but does not establish predictive performance or offer a full modeling procedure.

Key ideas

• LPPL models are formulated as continuous-time processes.
• Discrete price observations can be used to approximate the model without adjustments, according to the response.
• Omitting bid-ask spreads may cause liquidity effects to be interpreted as volatility changes.
• The discussion cautions against adding covariates where the assumed distribution has no covariance structure.

Tags

# Log-periodic power law model: is it a continuous or discrete-time process?


# Log-periodic power law model: is it a continuous or discrete-time process?












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

## Answer by onlyvix.blogspot.com (score 1)

https://quant.stackexchange.com/a/30700

The model is continuous in t, but you can use is as an approximation (with discrete prices as inputs) without adjustments.

## Answer by Dave Harris (score 0)

https://quant.stackexchange.com/a/36972

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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