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.

This figure 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 updates 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 two 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. 1 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 capturing investor herding and crowd behavior. The theory takes into account the competition between positive feedback (self-fulfilling sentiment), negative feedbacks (contrariant behavior and fundamental/value analysis) and inertia (everything takes time to adjust). Technically, we use what we call a "super-exponential power-law log-periodic function" derived from a first order Landau expansion of the logarithm of the price. 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 Oct. 17, 2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

Fig. 2 shows the new predictions of the future of the US S&P 500 index using all the data from Aug. 9, 2000 to Oct. 17, 2003, 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. 1) for comparison, so as to provide an estimation of the sensitivity of the prediction and of its robustness as the price evolves. The blue dots show the daily price evolution from Aug. 9, 2000 to Oct. 17, 2003. 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 predictions of the future of the US S&P 500 index obtained by applying the so-called 'zero-phase' Weierstrass-type function, which is another child of our general theory of imitation and herding between investors. As for the previous figures, our theory takes into account the competition between positive feedback (self-fulfilling sentiment), negative feedbacks (contrariant behavior and fundamental/value analysis) and inertia (everything takes time to adjust). This 'zero-phase' Weierstrass-type function adds one additional ingredient: it attempts to capture the existence of 'critical' points within the anti-bubble, corresponding to accelerating waves of imitation within the large scale unraveling of the herding anti-bubble. The continuous black line is the forward prediction using all the data from Aug. 9, 2000 to Oct. 17, 2003, while the dashed black line is the retroactive prediction using the data from Aug. 9, 2000 to Aug. 24, 2002. Both lines are reconstructed and extrapolated from the fits to a six-term zero-phase Weierstrass-type function. We also present the two previous fits (red lines) performed on Aug. 24, 2002 (shown in Fig. 1) for comparison. The blue dots show the daily price evolution from Aug. 9, 2000 to Oct. 17, 2003. The large (respectively small) ticks in the abscissa correspond to January 1st (respectively to the first day of each quarter) of each year.

The development observed in the update on June, 19, 2003 is further confirmed: the fit obtained here is very close to that on June 19, 2003. The 'zero-phase' Weierstrass-type function, which up to May 18, 2003 selected a series of downward critical crashes, is now selecting as the dominant critical points the bullish accelerations. The formula is thus deciphering the coexistence of two sets of critical points: (i) the crashes previously recognized which have punctuated the descent in the last three years and (ii) the bursts of upward accelerating rallies. This formula is however not rich enough in its present version to capture these two sets simultaneously and has to choose between the two, as a result of their relative strengths. This new twist does not change fundamentally our prediction of a drastic turn towards a systematic downward trajectory till the summer of 2004.

Fig. 4 extends Figs. 1 and 2 by performing a sensitivity analysis on the simple log-periodic formula (continuous lines in Figs. 1 and 2), in order to assess the reliability and range of uncertainty of the prediction. Using the fit shown in black solid lines in Fig. 2, we have generated 10 realizations of an artificial S&P 500 by adding GARCH noise to the black solid line. GARCH means "generalized auto-regressive conditional heteroskedasticity". It is a process often taken as a benchmark in the financial industry and describes the fact that volatility is persistent. 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 to ensure the agreement between the statistical properties of these synthetic time series and the known properties of the empirical distribution of returns. The fits are shown as the bundle of 10 curves in magenta. This bundle of predictions is coherent and suggests a good robustness of the prediction. The typical width of the blue dots give 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. 1 and 2 by performing a sensitivity analysis on the log-periodic formula with a second log-periodic harmonic (dashed lines in Figs. 1 and 2), in order to assess the reliability and range of uncertainty of the prediction. Using the fit shown in dashed solid lines in Fig. 2, 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. This test shows that the log-periodic formula with a second log-periodic harmonic (dashed lines in figures 1 and 2) is also providing stable scenarios: the precise timing of the highs and lows remain robust with respect to the realization of the noise. The real S&P 500 price trajectory is shown as the red wiggly line.

Fig. 6 extends Fig. 3 by performing a sensitivity analysis on the 'zero-phase' Weierstrass-type function, 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 A 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 by the black continuous curve. We have then fitted each of these 10 synthetic noisy clones of the S&P 500 (shown as the blue dots) by our 'zero-phase' Weierstrass-type function. This yields the narrow bundle of 10 curves shown here in magenta. This bundle of predictions is very coherent and suggests a good robustness of the prediction. The typical width of the blue dots give 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. 7 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. 4-6: (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. Strikingly, we first observe that our log-periodic formula is able to account quite well for the behavior of the VIX index, strengthening the evidence that the market is presently in a strong herding (anti-bubble) phase. Note also the rather good stability of the predictions, suggesting a reasonable reliability.

Fig. 8 compares the fit with the simple log-periodic formula shown as the continuous red line (which is the same as the continuous black line in Fig.2) with the fit using the log-periodic formula derived from a second-order Landau expansion shown as the red dashed line. In our paper appeared in the December issue of Quantitative Finance in 2002, we stated that the simple log-periodic formula is enough to fit the S&P 500 antibubble and we thus concluded that the S&P 500 index had not yet entered into the second phase in which the angular log-frequency may start its shift to another value, as did the 1990 Nikkei antibubble after about 2.5 years. This figure confirms our announcement in our last update of September 17, 2003 that we now detect the occurrence of such a change of regime. The statistical tests summarized at the end of our last update of September 17, 2003 and given in details in "Testing the Stability of the 2000-2003 US Stock Market Antibubble" give the probability (shown as the full black dots and the right vertical scale) to reject the hypothesis that the market has entered the second phase in which the angular log-frequency is shifting to another value. These results here open very seriously the possibility that, indeed, we have entered a cross-over regime in log-frequency shift. The improved second-order log-periodic formula shown as the red dashed line suggests that there will be a delay in the expected drop, which rather than occurring now, may wait until November-December 2003. Another possibility discussed in our recent paper "Testing the Stability of the 2000-2003 US Stock Market Antibubble" is that the antibubble may have ended and the market is in a new regime. Our analysis does not select this scenario as the prefered one.

The construction of this last figure was suggested to us by several readers of our webpage. The idea is that it may be informative to view the antibubble from the perspective of a european investor or more generally a foreigner invested in euros. This idea is interesting because it connects with the fueling impact of foreign investments on US markets, as we document in a recent publication D. Sornette and Wei-Xing Zhou, Evidence of Fueling of the 2000 New Economy Bubble by Foreign Capital Inflow: Implications for the Future of the US Economy and its Stock Market, Physica A (http://arXiv.org/abs/cond-mat/0306496). This last figure shows the S&P 500 expressed in US dollars and in euro, together with their fit. The fit in magenta of the S&P 500 in US dollars is the same as the continuous black line in Figure 2. Not surprisingly, the two fits are similar, but the amplitude of logperiodicity is larger when the S&P 500 is expressed in euro. Note also the lag consistent with Figure 8. However, the drop predicted to occur soon could be due in part to a fall of the dollar.

**Cautionary note:**

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 the theory and equations used above 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.