Ponzo Wave Theory

In the following model we look at recent market (or stock) oscillations and try to identify some critical T-value below which we could get unbounded solutions and possibly a cRaSh. This application is based on the idealized wave theory of Dr. Peter Ponzo:

P(n+1) = [ 2-(2π/T)^2 ] P(n) – P(n-1) + (2π/T)^2 P0

* π = 3.141592…….http://www.wolframalpha.com/input/?i=pi
* T is some kind of period (like T = 5 days)
* P0 is some parameter (maybe a 10 day Moving Average)
* P(n) is today’s stock price and P(n-1) is the price yesterday
* P(n+1) is the next stock price in the sequence … namely tomorrow’s price

The attached spreadsheet application includes a data feeder,optimizer, & sensitivity analysis generator.

A question that we address in the embedded pdf is whether this strong increase in a relatively short period of time sooner or later has to terminate with a correction in the Dow that can be sizeable.

October 4, 2009: Dshort.com published an interesting article – “Is the Stock Market Cheap?”

Log-Periodic Self-Similarity

I find fascinating the scientific study of complex systems made by Prof. Didier Sornette in the book “Why Stock Markets Crash: Critical Events in Complex Financial Systems”. He boldly applies his varied experience in leading-edge physical and statistical modeling techniques to propose a general theory of how, why, and when stock markets crash.

In this paper Dr. Sornette updates the methodology and introduce an oscillatory component and a function that mimics the general upward trend (before the crash):

Sornette(t) = A + E(t) Osc(t):
A – B (tc – t)α / SQRT[1 + { (tc - t)/Δt }2α] [1 + C COS[ w Log(tc-t) + (Δw/2α) Log(1+{ (tc - t)/Δt }2α) ]]

• where tc is the time of the crash and B, C, w, α, Δt, and Δw are constants.

Note that E(t) = – B (tc – t)α / SQRT[1 + { (tc - t)/Δt }2α] rises more rapidly as the time of the crash approaches.

Note that Osc(t) = 1 + C COS[ w Log(tc-t) + (Δw/2α) Log(1+{ (tc - t)/Δt }2α) ] oscillates more rapidly as the time of the crash approaches.

Warning!

I have tried using this log periodic model for the prediction of crashes but i found it challenging to fit to data (too many free variables). If you would like to play with the parameters so as to mimic the behaviour of the S&P… before the crash click here!

References:

Critical market crashes – Author: Didier Sornette
Paper

World stock market: approaching trend reversal? – Authors: Stanislaw Drozdz, Pawel Oswiecimka
Paper

Financial Bubbles, Real Estate bubbles, Derivative Bubbles, and the Financial and Economic Crisis – Authors: Didier Sornette, Ryan Woodard.
Paper
Video Lecture

The Chinese Equity Bubble: Ready to Burst – Authors: K. Bastiaensen, P. Cauwels, D. Sornette, R. Woodard, W.-X. Zhou
Paper
Bloomberg Article

The 2006-2008 Oil Bubble and Beyond – Authors: D. Sornette, R. Woodard, W.-X. Zhou
Paper

Why Stock Markets Crash: Critical Events in Complex Financial Systems by Didier Sornette
Book

Crashes as Critical Points – Authors: Anders Johansen, Olivier Ledoit, Didier Sornette
Paper

Large financial crashes – Authors: Didier Sornette, Anders Johansen
Paper