*These analyses are researched by D. Sornette and W.-X. Zhou.*

Based on a theory of cooperative herding and imitation working both in bullish as well as in bearish regimes that we have developed in a series of papers, we have detected the existence of a clear signature of herding in the decay of the US S&P500 index since August 2000 with high statistical significance, in the form of strong log-periodic components.

Please refer to the following paper for a detailed description: D. Sornette and W.-X. Zhou, The US 2000-2002 Market Descent: How Much Longer and Deeper? Quantitative Finance 2 (6), 468-481 (2002) (e-print at http://arXiv.org/abs/cond-mat/0209065).

Why Stock Markets Crash: For a general presentation of the underlying concepts, theory, empirical tests and concrete applications, with a discussion of previous predictions, see the recent book, Why Stock Markets Crash.

__NEW:__ Testing the Stability of the 2000-2003 US Stock Market Antibubble.

Since August 2000, the USA as well as most other western markets have depreciated almost in synchrony according to complex patterns of drops and local rebounds. We have proposed to describe this phenomenon using the concept of a log-periodic power law (LPPL) antibubble, characterizing behavioral herding between investors leading to a competition between positive and negative feedbacks in the pricing process. Here, we test the possible existence of a regime switching in the US s&P 500 antibubble. First, we find some evidence that the antibubble might be on its way to cross-over to a shift in log-periodicity described by a so-called second-order log-periodicity previously documented for the Japanese Nikkei index in the 1990s (see last figure of this webpage). Second, we develop a battery of tests to detect a possible end of the antibubble which suggest that the antibubble is still alive and may still continue well in the future. Our tests provide quantitative measures to diagnose the end of the antibubble, when it will come. Such diagnostic is not instantaneous and requires probably three to six months within the new regime before assessing its existence with confidence. In conclusion, our prediction that the s&P 500 is going to plunge progressively from the summer 2003 to bottom in 2004 seems to remain basically intact, possibly with a few month delay extending almost to the end of 2003 if the shift to the second-order log-periodicity is confirmed.

Fig. 1 shows 9 years of the evolution of the Japanese Nikkei index and almost 8 years of the USA s&P500 index, compared to each other after a translation described in the update of September 17, 2000 has been performed. The years are written on the horizontal axis (and marked by a tick on the axis where January 1 of that year occurs). This figure illustrates an analogy noted by several observers that our work has made quantitative. The oscillations with decreasing frequency which decorate an overall decrease of the stock markets are observed only in very special stock markets regimes, that we have terms log-periodic "anti-bubbles". By analyzing the mathematical structure of these oscillations, we quantify them into one (or several) mathematical formula(s) that can then be extrapolated to provide the prediction shown in the following figures. Note that extrapolating is often a risky endeavor and needs to be justified. In our case, the extrapolations, which give the forecasts, are based on the belief that these equations offered below embody the major forces in the market at the macroscopic scale. This leads to the possibility of describing several probable scenarios. We do not believe in the existence of deterministic trajectories but we aim at targeting the most probable future paths.

Fig. 2 shows the predictions of the future of the US s&P 500 index performed on Aug. 24, 2002. The continuous line is the fit and its extrapolation, using our theory assuming the persistence of the antibubble based on investor herding and crowd behavior. The dashed line is the fit and its extrapolation by including in the function a second log-periodic harmonic. The two fits are performed using the index data from Aug. 9, 2000 to Aug. 24 2002 that are marked as black dots. The blue dots show the daily price evolution from Aug. 25, 2002 to Jan. 16, 2004. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

Fig. 3 shows the new predictions of the future of the US s&P 500 index using all the data from Aug. 9, 2000 to Jan. 16, 2004, illustrated by (continuous and dashed) black lines. Again, the continuous line is the fit and its extrapolation using the super-exponential power-law log-periodic function derived from the first order Landau expansion of the logarithm of the price, while the dashed line is the fit and its extrapolation by including in the function a second log-periodic harmonic. We also present the two previous fits (red lines) performed on Aug. 24, 2002 (shown in Fig. 2) for comparison. The blue dots show the daily price evolution from Aug. 9, 2000 to Jan. 16, 2004. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

Fig. 4 extends Figs. 2 and 3 by performing a sensitivity analysis on the simple log-periodic formula (continuous lines in Figs. 2 and 3), in order to assess the reliability and range of uncertainty of the prediction. Using the fit shown in black solid lines in Fig. 3, we have generated 10 realizations of an artificial s&P 500 by adding GARCH noise to the black solid line. The innovations of the used GARCH noise have been drawn from a Student distribution with 3 degrees of freedom with a variance equal to that of the residuals of the fit of the real data. The fits are shown as the bundle of 10 curves in magenta. The typical width of the blue dots gives a sense of the variability that can be expected around this most probable scenario. The real s&P 500 price trajectory is shown as the red wiggly line.

Fig. 5 extends Figs. 2 and 3 by performing a sensitivity analysis on the log-periodic formula with a second log-periodic harmonic (dashed lines in Figs. 2 and 3), in order to assess the reliability and range of uncertainty of the prediction. Using the fit shown in dashed solid lines in Fig. 3, we have generated 10 realizations of an artificial s&P 500 by adding the GARCH noise (described in the previous caption of Fig. 4) to the dashed solid line. We have then fitted each of these 10 synthetic noisy clones of the s&P 500 by our log-periodic formula. This yields the 10 curves shown here in magenta. The real s&P 500 price trajectory is shown as the red wiggly line.

Fig. 6 analyses the VIX index by fitting it with our simple log-periodic formula. The VIX index is one of the world's most popular measures of investors' expectations about future stock market volatility (that is, risk). Note that a new methodology for constructing the VIX index has been effective on Sep. 22, 2003. See http://www.cboe.com/micro/vix/index.asp. For historical data, see http://www.cboe.com/micro/vix/historical.asp. The (new) VIX time series is shown as the red wiggly curve. We have followed the same procedure as for Figs. 4 and 5: (i) we fit the real VIX data with our simple log-periodic formula; (ii) we then generate 10 synthetic time series by adding GARCH noise to the fit; (iii) we redo a fit of each of the 10 synthetic time series by the simple log-periodic formula and thus obtain the bundle of 10 predictions shown as the magenta lines.

Fig. 7 shows the s&P500 as a function of time since the plateau of the summer 2003, following a first rally after the minimum reached at the end of the first quarter of 2003. One can observe three approximately linear growth regimes with increasing slopes shown by the colored lines. Alternatively, this can be represented by an accelerating growth rate: in logarithmic scale such that an exponential growth with constant rate qualifies as a straight line, we still observe this upward curvature, characteristic of a super-exponential acceleration. Comparing with all our previous analyses previously published, such a behavior suggests the existence of a positive feedback and of herding behavior pushing the prices up. In general, such a regime should shift to another phase as it is unsustainable. In the present case, the new regime could take the form of a consolidation followed by a progressively growing correction as predicted in our previous figures. Or, as occurred following October 1997, a further accelerating growth could resume following a consolidating plateau. We are not able to make more precise statements with this approach, as the log-periodic patterns are not present in this putative bubble and the accelerating properties are not yet well-developed.

CONCLUDING REMARKS:

In our previous upage of December 16, 2003, we have defined two probabilities extracted from our paper entitled "Testing the Stability of the 2000-2003 US Stock Market Antibubble" available on this website: P1 is the probability that the fit confirms the continuation of the antibubble and P2 is the probability to the fit could incorrectly classify the present regime of the stock market as an antibubble. We calculated their values for seven different future scenarios [*] until Mid-February 2004, as shown in Table 2 (PDF document). Although there are still one months to go, it is now probable that the market is somewhere between scenario (i) and (ii). If confirmed, all the above updated predictions will turn out to be wrong. We should be able to confirm or deny this definitively in Feb. 2004. Fig. 7 adds another twist to the problem as a possible bubble within the antibubble may be growing.

[*]

(i) a random walk taking the s&P 500 to the value 1200;

(ii) a random walk taking the s&P 500 to 1100;

(iii) a random walk taking the s&P 500 to 1000;

(iv) a random walk taking the s&P 500 to 900;

(v) a random walk taking the s&P 500 to 800;

(vi) a continuation of the antibubble with noise obtained by a GARCH process as described above;

(vii) a continuation of the antibubble with noise obtained by drawing at random the residuals over six previous months.

**THIS IS AN EXPERIMENT PERFORMED IN REAL TIME AND WE WILL CONTINUE UPDATING EVERY MONTH.**

**REMEMBER THAT this analysis is for academic purposes only and must not be construed as investment or trading advice.**