Predicting critical crashes? A new restriction for the free variables

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Physica A 320 (2003) 539 – 547

www.elsevier.com/locate/physa

Predicting critical crashes? A new restriction for the free variables Hans-Christian Graf v. Bothmera;∗ , Christian Meisterb a Laboratoire

J. A. Dieudonne, Universite de Nice Sophia-Antipolis, Parc Valrose, F-06108 Nice, Cedex 2, France b Section Analysis, ITS/TWA, Delft, The Netherlands Received 12 September 2002

Abstract Several authors have noticed the signature of log-periodic oscillations prior to large stock market crashes. Unfortunately, the prediction of crashes with this method is di1cult, since good 2ts of the corresponding log-periodic equation to stock market prices are also observed in quiet times. We deduce a new inequality among the free variables in this equation, that must be satis2ed before a crash. This inequality alone yields crash predictions with a 25% success rate for the Dow Jones index. This compares to a 9% success rate obtained by previous approaches. Combining our method with previous ones we obtain a success rate of 54%. c 2002 Elsevier Science B.V. All rights reserved.  PACS: 89.65.Gh Keywords: Financial markets; Bubbles; Crash; Hazard rate; Log-periodicity

1. Introduction Several authors have noticed the signature of log-periodic oscillations prior to large stock market crashes [1–3,5]. Unfortunately, good 2ts of the corresponding equation to stock market prices are also observed in quiet times. To re2ne the method several approaches have been suggested: • Logarithmic divergence: Regard the limit where the critical exponent  converges to 0 [3]. ∗

Corresponding author. Supported by Marie Curie Fellowship HPMT-CT-2001-001238. E-mail addresses: [email protected] (H.-C. Graf v. Bothmer), [email protected] (C. Meister). URL: http://btm8x5.mat.uni-bayreuth.de/∼ bothmer/

c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/03/$ - see front matter
PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 1 5 3 5 - 2

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• Universality: De2ne typical ranges for the free parameters, by observing the best 2t for historic crashes [4]. We suggest a new approach. From the observation that the hazard-rate in Ref. [4] has to be a positive number, we get an inequality among the free variables of the equation for stock market prices. Checking 88 years of Dow Jones data for best 2ts, we 2nd that 25% of those that satisfy our inequality, occur less than 1 year before a crash. We compare this with other methods of crash prediction, i.e., the universality method of Johansen et al., which followed by a crash only in 9% of the cases. Combining the two approaches we obtain a method whose predictions are followed by crashes in 54% of the cases. 2. The hazard rate In Ref. [4] Johansen et al., suggest, that during a speculative bubble the crash hazard rate h(t), i.e., the probability per unit time that the crash will happen in the next instant if it has not happened yet, can be modeled by h(t) ≈ B0 (tc − t)− + B1 (tc − t)− cos(! log(tc − t) + ) : By assuming that the evolution of the price during a speculative bubble satis2es the martingale (no free lunch) condition, they obtain a diIerential equation for the price p(t) whose solution is 
 t p(t) = h(t  ) dt  ; log p(t0 ) t0 before the crash. Here denotes the expected size of the crash. This implies that the evolution of the logarithm of the price before the crash and before the critical date tc is given by (∗) log(p(t)) ≈ pc − B0 (tc − t) 

−

2

+ !2

B1 (tc − t) cos(! log(tc − t) + )

with  = 1 − , pc the price at the critical date, and a diIerent phase constant. Now the hazard rate is a probability and therefore positive. This leads to a necessary condition: 0 6 h(t) ⇔ 0 6 B0 (tc − t)− + B1 (tc − t)− cos(! log(tc − t) + ) ⇔ 0 6 B0 + B1 cos(! log(tc − t) + ) ; since t ¡ tc . At some times near the critical date cos(! log(tc − t) + ) takes on the values −1 and 1. This implies the necessary conditions 0 6 B0 ± B1 ⇔ |B1 | 6 B0 :

H.-C. Graf v. Bothmer, C. Meister / Physica A 320 (2003) 539 – 547

541

Drawdowns of the Dow Jones Index 1912 - 2000 0

-5

Drawdown [%]

-10

-15

-20

-25

-30

1900

1910

1920

1930

1940

1950 1960 Year

1970

1980

1990

2000

2010

Fig. 1. Drawdowns of the Dow Jones index from 1912 to 2000. There have been four drawdowns larger than 15% in the years 1929, 1932, 1933 and 1987.

On the other hand, these conditions are also su1cient for h(t) ¿ 0 because of the fact that cos(! log(tc − t) + ) is always between −1 and 1. To summarize, if the assumptions of Johansen et al., are valid, we must have |B1 | 6 B0 prior to a critical crash. 3. Eighty eight years of Dow Jones To check this model of speculative bubbles, we have investigated the Dow Jones index from 1912 to 2000. This period contains 23,668 trading days. Johansen et al., de2ne a crash as a continuous drawdown (several consecutive days of negative index performance) larger than 15%. Fig. 1 shows the drawdowns of the Dow Jones index from 1912 to 2000. Observe that there have been four drawdowns larger than 15% namely in the years 1929, 1932, 1933 and 1987. As our basic data set, we have calculated numerically the best 2t of Eq. (∗) to a sliding window of 750 trading days, every 5 trading days. This yields 4761 best 2ts. The complete data set is available at http://btm8x5.mat.uni-bayreuth.de/∼ bothmer.

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Mean Square Errors Dow Jones 1912-2000 500 all errors errors before crashs 400

300

200

100

0 0

1

2 3 mean square error

4

5

Fig. 2. Distribution mean square errors of best 2ts to the Dow Jones index. The best 2ts do not occur within 1 year before a large crash.

In what follows, we will call a crash prediction successful, if it was issued at most 1 year before a crash. With this de2nition there are 229 best 2ts that could possibly give a successful crash prediction. By predicting crashes randomly, one would obtain a successful prediction in 4.8% of the cases. 3.1. Mean square errors  The 2rst approach to detecting speculative bubbles is to look for good 2ts of (∗) to the Dow Jones index. If the mean square error  of the 2t is su1ciently small one issues a crash prediction. Fig. 2 shows the mean square errors of all our best 2ts compared with the best 2ts before a crash. Unfortunately, small errors also occur in quiet times. If one issues a crash prediction if the mean square error is smaller than 0.75 one obtains only 72=(72 + 1732) ≈ 3:9% successfull predictions (Table 1). Since this is worse than issuing random predictions, we conclude that one cannot predict a crash by looking only at the mean square error. This observation has also been made by Sornette and Johansen.

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Table 1 Crash predictions using the mean square error of best 2ts to the Dow Jones index

(2

No crash prediction ¿ 0:75) Crash prediction (2 ¡ 0:75)

Before crash

Not before crash

175 72

2799 1732

Table 2 Crash predictions using the critical time of best 2ts to the Dow Jones index

No crash prediction (tc ¿ today + 1 year) Crash prediction (tc ¡ today + 1 year)

Before crash

Not before crash

93 136

2652 1879

3.2. Critical times tc If the model of Johansen et al., is correct one should expect that a crash occurs close to the critical date tc . Using this, one can issue a crash prediction when the critical date tc is less than 1 year away. In our data set 136=(136 + 1879) ≈ 6:7% of these predictions are successful (Table 2). This is slightly better than random predictions, but still not very good. 3.3. Universality Johansen et al., suggest that speculative bubbles exhibit universal behavior. This would imply that  and ! take on roughly the same values for each speculative bubble. Fig. 3 shows the distribution of ! before crashes compared with the distribution during other times. One can clearly observe an unexpected peak around ! = 9 before the crashes. If we issue a crash prediction in the range 7 ¡ ! ¡ 13 we obtain a success rate of 79=(79 + 803) ≈ 8:9% (Table 3). Fig. 4 shows the distribution of  before crashes and not before crashes. Here we observe a tendency toward lower values of , but no clear peak. We interpret this as evidence, that one should look for logarithmic divergence, i.e., the limit of  tending to 0, as suggested by Vandewalle et al. [3]. We will investigate this approach in a later paper. 3.4. Positive hazard rate In Section 2 we have explained that in the model of Johansen et al. the hazard rate h(t) must be positive. We proved that this is equivalent to |B1 | 6 B0 :

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H.-C. Graf v. Bothmer, C. Meister / Physica A 320 (2003) 539 – 547

Distribution of Omega Dow Jones 1912-2000

best fits before crashs other best fits

0.08

0.06

0.04

0.02

0 0

10

20

omega

30

40

50

Fig. 3. Up to 1 year before a large crash the best 2ts of Eq. (∗) to the Dow Jones index use unusual values of !.

Table 3 Crash predictions using the log frequency of best 2ts to the Dow Jones index

No crash prediction Crash prediction (7 ¡ ! ¡ 13)

Before crash

Not before crash

150 79

3728 803

From our best 2ts we can calculate the value (B0 − |B1 |) ; which should also be positive during a speculative bubble, since is a positive number. Consequently, we can issue a crash warning if (B0 − |B1 |) is positive. In our data set 96=(96 + 276) ≈ 25:8% of the predictions are successful (Table 4). This is already a practical success rate, but we can do even better, if we combine this with universality.

H.-C. Graf v. Bothmer, C. Meister / Physica A 320 (2003) 539 – 547

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Distribution of Beta Dow Jones 1912-2000

best fits before crashes other best ifits

0.25

0.2

0.15

0.1

0.05

0 0

0.2

0.4

0.6

0.8

1

beta Fig. 4. Up to 1 year before a large crash the best 2ts of Eq. (∗) to the Dow Jones index use lower values of .

Table 4 Crash predictions using best 2ts to the Dow Jones index with positive hazard rate

No crash prediction ( (B0 − |B1 |) 6 0) Crash prediction ( (B0 − |B1 |) ¿ 0)

Before crash

Not before crash

133 96

4255 276

3.5. Positive hazard rate and universality Combining the last two approaches we issue a crash prediction, if the hazard rate is everywhere positive and ! is in the range of Section 3.3. In our data set 65=(65+55) ≈ 54:1% of these crash predictions are successfull (Table 5). Fig. 5 shows when these crash predictions were issued. For every trading day we have plotted the number of crash predictions during the past year and the drawdown of the Dow Jones index. Notice that the crashes of 1929 and 1987 have been predicted well in advance. The crashes of 1932 and 1933 have not been directly predicted, but we argue that they are

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Table 5 Crash predictions using best 2ts to the Dow Jones index with positive hazard rate and universality for !

No crash prediction  Crash prediction ( (B0 − |B1 |) ¿ 0 and 7 ¡ ! ¡ 13)

Before crash

Not before crash

164

4476

65

55

Detection of Speculative Bubbles of the Dow Jones Index 0

-5

-5

-10

-10

-15

-15

-20

-20

-25

-25 -30

-30 Detections during past year Drawdowns

-35 1900

1910

1920

1930

1940

1950 1960 Year

Number of detections during past year

Drawdown [%]

Positive hazard rate and 7
-35 1970

1980

1990

2000

2010

Fig. 5. Best 2ts of Eq. (∗) with positive hazard rate and 7 ¡ ! ¡ 13 usually occur before large crashes.

in the aftermath of 1929 and represent the bursting of the same speculative bubble. The crash predictions of 1997 were followed by two small crashes in October 1997 and 1998, which did not quite reach 15%. One could argue that they represent a crash in two steps. 4. Summary We have derived a new restriction of the free variables in the model of Johansen et al. [4] for stock market prices during a speculative bubble. This restriction alone yields crash predictions with a 25% success rate for the Dow Jones index. This is an improvement over the 9% success rate obtained by using universality. Combining our approach and the universality method we obtain a success rate of 54%.

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We think that these results represent strong evidence for the model of Johansen et al. describing speculative bubbles in the stock market. References [1] D. Sornette, A. Johansen, J.-P. Bouchaud, Stock market crashes, precursors and replicas, J. Phys. I France 6 (1996) 167–175, cond-mat/9510036. [2] J.A. Feigenbaum, P.G.O. Freud, Discrete scale invariance in stock markets before crashes, Int. J. Mod. Phys. 10 (1996) 3737–3745, cond-mat/9509033. [3] N. Vandewalle, Ph. Boveroux, A. Minguet, M. Ausloos, The krach of October 1987 seen as a phase transition: amplitude and universality, Physica A 255 (1–2) (1998) 201–210. [4] A. Johansen, O. Ledoit, D. Sornette, Crashes as critical points, Int. J. Theoret. Appl. Finance 3 (2) (2000) 219–255, cond-mat/9810071. [5] A. Johansen, D. Sornette, The Nasdaq crash of April 2000: yet another example of log-periodicity in a speculative bubble ending in a crash, Eur. Phys. J. B 17 (2000) 319–328, cond-mat/0004263.