Renormalization group analysis of the 2000–2002 anti-bubble in the US S&P500 index: explanation of the hierarchy of five crashes and prediction

Abstract

We propose a straightforward extension of our previously proposed log-periodic power law model of the "anti-bubble" regime of the USA market since the summer of 2000, in terms of the renormalization group framework to model critical points. Using a previous work by Gluzman and Sornette (2002) on the classification of the class of Weierstrass-like functions, we show that the five crashes that occurred since August 2000 can be accurately modelled by this approach, in a fully consistent way with no additional parameters. Our theory suggests an overall consistent organization of the investors forming a collective network which interact to form the pessimistic bearish "anti-bubble" regime with intermittent acceleration of the positive feedbacks of pessimistic sentiment leading to these crashes. We develop retrospective predictions, that confirm the existence of significant arbitrage opportunities for a trader using our model. Finally, we offer a prediction for the unknown future of the US S&P500 index extending over 2003 and 2004, that refines the previous prediction of Sornette and Zhou (2002).

Is there a real-estate bubble in the US?

“…Figure 8 presents the dependence of ω for the LPPL model and of ω 0 for the 2nd-order Landau LPPL model as a function of t start . We find ω ≈ 10 and ω 0 oscillating between ω ≈ 10 and ω/2, as we expect from the generic existence of harmonics (see our previous extended discussions in [10,14]). The stability of ω 0 and the compatibility between the two descriptions is a signature of the robustness of the signal.…”

Significance of log-periodic signatures in cumulative noise

“…Notice that there are now several peaks of comparable size at frequencies 1.7, 3.4, 7.4 and 8.4. The fact that these lie close to the harmonics of 1.7 compares well to the results of [ZS03].…”

“…This complements evidence for a more sophisticated modelling of the S&P 500 anti-bubble by Sornette and Zhou in [ZS03].…”

Fundamental factors versus herding in the 2000–2005 US stock market and prediction

“…of the fit residuals, there is another diagnostic for the bad quality of Model 1: the three different contributions (LPPL, VIX and interest rate r(t)) are large compared with their sum, suggesting the existence of spurious (in the sense of "over-fitting") compensations between these terms. In contrast, since the LPPL term was shown to fit very well the data over the period from 2000 to 2003 [25,26,28,30], the LPPL term should be the leading contribution while the other factors should have been perturbations, perhaps growing with time. In other words, Model 1 is not a perturbation of the LPPL model used previously in [25,26,28,30].…”