Recent work building non-linear regime models has led Quantivity back into econophysics. In doing so, coincidentally bumped into the proposed Log-Periodic Power Law (LPPL), which is a log-periodic oscillation model for describing the characteristic behavior of a speculative bubble and predicting its subsequent crash. In other words: a macro regime discovery model.

This model was independently proposed by Sornette, Johansen and Bouchaud (J. Phys. I. France 6 pp. 167-175, 1996) and Feigenbaum and Freund, (Int. J. Moder Phys. B 10: 3737, 1996 and Modern Physics Letters B 12, 1998). Econophysics fans will recollect Bouchaud from Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management and Capital Fund Management.

Mathematically, LPPL proposes price $p$ of an instrument evolves at time $t$ according to:

$p(t) = A + B ( t_{c} - t )^{z} + C ( t_{c} - t)^{z} \cos (\omega \log ( t_{c} - t ) + \Phi )$

where $t_{c}$ is most probable time of crash, $z$ is exponential growth, $\omega$ is oscillation amplitude, and the remaining variables carry no structural interpretation ($A$, $B$, $C$, and $\Phi$). In other words: an oscillating, exponential model for price evolution. Intuition underlying crash prediction is essentially an appeal to the impossibility for continuation of exponential price growth, with increasing oscillations approaching failure indicating swings in investor sentiment.

As always, an effective way to gain model intuition is via practical illustration: LPPL fitting for Hang Seng and S&P 500 crashes in 1997 / 1998 (reproduced from p. 67 of Critical Market Crashes by Sornette):

Where HK (Hang Seng) and WS (S&P 500) are the fittest LPPL for each index, up to each respective crash.

Although Quantivity admittedly does not find this model compelling (although the qualitative analysis, based on herding and imitation effects originating from behavioral finance, which motivates this model is fairly interesting), a brief study of LPPL is worth the effort as both a mathematical and sociological curiosity.

Mathematically, LPPL is interesting as it is built upon power laws—which are a favorite of econophysics, owing to their origins in statistical mechanics. The reasons for primacy of power laws are interesting, and nicely summarized by Shalizi (coincidently, note the reference to Sornette):

• Order in complexity: “power law correlations are interesting because they’re a sign of something interesting and complicated happening”
• Multiplicative growth: “power laws turn out to result from a kind of central limit theorem for multiplicative growth processes” (see also 1/f, long memory, and fractional Brownian motion)

Sociologically, LPPL is interesting for several reasons, as compared with standard quantitative finance fare:

• Literature: lively research literature over the past 15 years, including debate and rebuttal
• Partial disclosure: lack of public disclosure on the estimation methodology for LPPL by its primary proponent
• Sealed forecasting: use of sealed forecasts for a “Financial Bubble Experiment”

The combo of all three is remarkably unusual for the field, and thus fascinating in its own right. For those interested in following the evolution of LPPL, the following are representative literature:

See Sornette arXiv profile for more articles.

6 Comments leave one →
1. Scott Locklin permalink
February 8, 2011 8:28 pm

An interesting and disturbing summary of LPPL. Disturbing because my own researches have brought me ’round to this … and I hate it when people scoop me. Like you, I’m not super compelled by the model itself, in particular the way it seems to be “fit” -but it is a neat bundle of ideas. I worked on a SBIR for another of Sornette’s models, the random field Ising model -would like to get back to it. I think the use of scale-free social networks might eventually be useful, and the Bass lifecycle limit is an interesting side effect:
https://scottlocklin.wordpress.com/2010/02/02/music-molecules-and-misanthropy-econophysics-part-1/