Oscillations analysis, economics, business, finance, and Financial Markets theory.

1. “Log- Periodic Analysis of Critical
Crashes in the Portuguese Stock
Market”
Master Thesis
MASTER PROGRAMME IN CORPORATE SCIENCES
Supervisors:
Mestre Pedro Nuno Rino Carreira
Doutor Pedro Luis Pereira Verga Matos
Candidate:
Jorge Victor Quiñones Borda
December 22, 2015

2. Presentation Outline
1) Motivation of the Research
2) Introduction to the Theory of self-similar oscillatory finite-time
singularities in Finance
3) Brief Literature Review: Rationality, Bubbles, Herding and Imitation
4) The “Linear” Log-Periodic Formula
5) International Context: 1998, 2007 and 2015
6) Methodology, Results and Discussion
7) Conclusions
8) Recommendations
9) Suggested Future Research

3. 1) Motivation
Dragon Kings: “events which are even beyond
the extrapolation of the fat tail distribution of
the rest of the population” (Sornette, 2007)
Black Swans are events of low predictability
and high impact

4. Graph taken from Grossman (2013)
Dragon Kings appear much more
frequently than what could be expected
if all returns came from the same
probability distribution function.
Effects can be devastating for most
participants.
Maybe it could be possible to see them
coming (at least some of them).
Dragon Kings vs Black Swans

5. 2) Introduction to the Theory of self-similar
oscillatory finite-time singularities in
Finance
• Stock Markets are “complex evolutionary adaptive systems populated by bounded
rational agents interacting with each other” (Sornette, 2002)
• The threshold nature of decision making leads to the emergence of irregular cycles and
to critical* behavior.
• Critical situations tend to be rule, rather than the exception.
• Due to the repetitive interactions of market participants, coherent large-scale collective
behavior could appear and…
“The whole turns out to be much more than the sum of its parts”
(Sornette, 2003)
* In the context of complex dynamical systems, critical points are defined as points where a
normally well-behaved quantity explodes to infinity (Johansen, et al., 2000).

6. What causes the crashes?
The process leading to crashes include the
building up of correlations (Johansen, et al.,
2000)
Markets eventually land into a crash because
those correlations lead to a global cooperative
behaviour (Kaizoji, et al., 2008).
The log-periodic critical theory can be applied
both to bubbles ending in a crash and to those
that land smoothly (Johansen, et al., 1999).
There are shared characteristics among
crashes along history, despite technological
advances (most financial crashes have been
preceeded by log-periodic precurssors).
What about the applicability?
Representation of a) An equilibrated
market on a 256x256 plane, b) A market in
a Critical State, c) A bubble.
Taken from (Sornette, 2003)
Investors “are still essentially driven by at least a grain of greed and fear in their quest for a
better well-being” (Sornette, 2003)

7. 3) Brief Literature Review:
Rationality, Bubbles, Herding and
Imitation
In a rational market “prices are set as if all investors are rational” (Rubinstein, 2000), in
those situations is not assumed that all investors are rational.
It has been shown that some the main characteristics of financial market (volatility
clustering and heavy tails) can appear even when taking a ´zero-intelligence´ approach.
(Thompson, et al., 2013)
“Institutions strongly shape our behaviour, so that some
of the properties of markets may depend more on the
structure of institutions than on the rationality of
individuals.” (Farmer, et al., 2005)

8. Bubbles
Their own existence are matter of debate,
because:
• We can never observe the fundamentals in
a direct way (Youssefmir, et al., 1998).
• Bubbles can be re-interpreted as
fundamentals not observed by the
researcher (Sornette, et al., 2002).
During a bubble, the prices of assets not only
reflect fundamentals, but also an extra that may
arise “because market participants believe that
they will be able to sell the asset in the future to
other participants at a higher price above and
beyond what they actually think it is worth”
(Youssefmir, et al., 1998).
Risk/Reward rations are constantly changing (Khandani, et al., 2007),

9. Herding and Imitation
Threshold models try to predict “from the
initial distribution of thresholds, the ultimate
number or proportion making each of the two
decisions” (Granovetter, 1978):
• In the case of the Stock Market, this is a
difficult task since every investor has its
own personal threshold.
There could be a point where
investors stop worrying about
fundamentals and only try to get
information about the actions of
other market participants.
“As the imitation spreads, it reinforces
itself in that individuals show an
increasing tendency to imitate. In other
words, the attractive power of an
opinion increases with the number of
individuals who share it, so that those
who have up to a point remained
indifferent to the influence of a model
may finally give in”
(Orlean, 1989)
Cartoon from The New Yorker

10. 4) The “Linear” Log-Periodic Model
Every trader is inserted inside a network:
• They receive information from other traders
(while also generating their own
information).
• Traders tend to imitate their closest peers
(Johansen, et al., 2000).
• The normal state of the market is when
most trades cancel out.
Representation of a trader as part of a network
Something that can´t be forgotten:
Social Pressures exist.
“Macro-level coordination can arise from micro-level imitation” (Johansen, et al., 2000).

11. Price dynamics
Mainly, we have 2 type of investors:
1. Fundamental value investors* that buy/sell
stocks when the market values differ in a
significant amount from the fundamental
value of the stock.
2. Trend followers that buy when they detect
a rise in the price, causing a further rise.
They sell when they see falling prices,
deepening crises.
The interaction between fundamental value
investors and the trend followers could lead to
log-periodic oscillations, which could be the
signature of an upcoming crisis.
*Contrarian behaviour has been grouped on its effects with
fundamental value investors, since they provide negative
feedbacks, also inertia (the fact that markets require time to
adjust), have alaso been considered (Zhou, et al., 2009).
Most chartist techniques are
not linear since they under-
react to small price changes
and over-react to large ones
(Ide, et al., 2002)

12. Price Dynamics
Source: Prepared by the Author

13. Price dynamics
We could identify two types of crashes by its causes:
Exogenous Endogenous
“Natural deaths of self-organized self-
reinforcing speculative bubbles giving rise to
specific precursory signatures in the form of
log-periodic power laws accelerating super-
exponentially” (Johansen, et al., 2010).
Extraordinary events
Crashes are “constructed progressively by the
market as a whole”, and that will reflect a
´systemic instability´ (Sornette, et al., 1997) .
“Fragility emerges due to increasing structural
correlation (…) Fragility is hidden from us
because it is emergent” (Crutchfield, 2009)
We can observe that
markets are usually
not at point 0, and
they tend to go far
from it.

14. The “Linear” Log Periodic Formula
log 𝑝 𝑡 = 𝐴 + 𝐵 𝑡 𝑐 − 𝑡 𝛽{1 + 𝐶 cos 𝜔 log 𝑡 𝑐 − 𝑡 + 𝜙
tc is determined by initial conditions at some point in time and gives the
estimated time of termination of bubbles, “which can occur approximately
two times out of three in the form of a significant correction or a crash”
(Zhou, et al., 2009).
That a crash is not a certainty but only a probability is a
necessary condition to ensure the coherence of the model,
since otherwise, investors could leave the market and
avoid crashes (Johansen, et al., 1999).
Where:
tc = critical time
ω = log-periodic angular frequency
ϕ = phase
β = exponent
𝜆 ≡ 𝑒
2𝜋
𝜔 : represents
the ratio of
consecutive time
intervals.

15. Restrictions generally applied during log-periodic analysis:
β → 0.2 – 0.8 (the exponent needs to be
between 0 and 1, in order to accelerate and to
remain finite, but we will use a more stringent
suggested range)
ω → 5 – 15 (this corresponds to 1.5 < λ < 3.5)
tc → tc > tlast
φ → No restriction
Ranges recommended in Sornette (2004).
𝑁𝑜𝑠𝑐 =
𝜔
2𝜋
ln
𝑡 𝑐 − 𝑡𝑓𝑖𝑟𝑠𝑡
𝑡 𝑐 − 𝑡𝑙𝑎𝑠𝑡
The most probable value for Nosc ≈ 1.5, and if
Nosc≥ 3 we can reject with 95% of confidence
that the log-periodicity observed comes from
noise (Zhou, et al., 2009).
Number of oscilations:

16. 5) International Context:
1998, 2007 and 2015
East Asian crisis: found its cause in the rapid forced
liberalization of financial markets and the easy access to
international credits without adequate supervision (Radelet, et
al., 1998).
Russian Crash which could have been “triggered by the Asian
crises, but it was to a large extent fuelled by the collapse of a
banking system, which in the course of the bubble had created
an outstanding debt of $19.2 billion” (Johansen, et al., 1999).
1998
2007-2008 For the US, it´s origin can be traced to the Housing Market.
2015 China crashes, government takes measures to reduce volatility.

17. 6) Methodology, Results and
Discussion
Data: Prices of the index of the Portugal PSI-20 Index retrieved from
Datastream, starting in 31-12-1993.
Methodology
log 𝑝 𝑡 = 𝐴 + 𝐵 𝑡 𝑐 − 𝑡 𝛽{1 + 𝐶 cos 𝜔 log 𝑡 𝑐 − 𝑡 + 𝜙Formula to fit:
Crashes to study: 1998, 2007 and 2015.
tfirst (the first data point in the sample) will be located at the beginning of the
new trend while tlast (the last data point in the sample) will be changed after
every fitting procedure, as a way to analyze the sensitivity of the results.
The fitting was done by minimizing the sum of root squared deviations (SRSD)
and since there could be many local minimums due to the multiple variables, 400
random starting points were tested for each variable.

18. 98.31, 9.57
98.55, 9.52
98.76, 9.00
8
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10
96.01
96.08
96.16
96.24
96.33
96.41
96.49
96.57
96.64
96.73
96.81
96.89
96.97
97.05
97.13
97.21
97.29
97.37
97.45
97.53
97.61
97.69
97.77
97.85
97.94
98.02
98.10
98.18
98.25
98.34
98.42
98.50
98.58
98.66
98.74
98.82
98.90
98.98
Portuguese Stock Index - 20
LN(PSI20)
Results and Discussion
Analysis of the 1998 Crash
1996-1998

19. F
o
r
e
c
a
s
t
s
o
b
t
a
i
n
e
d
tlast: 98,25 tflast: 98,34
tlast: 98,50 tlast: 98,59

20. 96 A B C β φ tc ω λ SRSD
SRSD
/ NDP tlast tlast natural Nosc NDP
1 9,41 -0,68 0,14 0,51 199,91 98,18 7,35 2,35 12,57 0,02 98,08 29-01-1998 3,60 542
2 9,41 -0,68 0,14 0,51 199,91 98,18 7,35 2,35 12,74 0,02 98,12 13-02-1998 4,23 553
3 9,41 -0,69 0,14 0,51 199,91 98,18 7,42 2,33 12,88 0,02 98,17 02-03-1998 6,05 564
4 9,69 -0,93 0,09 0,42 199,42 98,28 8,91 2,02 13,53 0,02 98,21 17-03-1998 4,84 575
5 11,78 -2,86 -0,02 0,20 74,86 98,64 14,09 1,56 13,69 0,02 98,25 01-04-1998 4,27 586
6 11,79 -2,86 -0,02 0,20 74,91 98,64 13,96 1,57 14,09 0,02 98,29 16-04-1998 4,49 597
7 10,94 -1,99 0,03 0,30 109,37 98,66 15,00 1,52 14,83 0,02 98,33 01-05-1998 4,95 608
8 10,94 -1,99 0,03 0,30 109,36 98,66 15,00 1,52 15,36 0,02 98,38 18-05-1998 5,33 619
9 10,87 -1,92 0,03 0,30 109,35 98,66 15,00 1,52 16,12 0,03 98,42 02-06-1998 5,70 630
10 10,61 -1,65 0,04 0,35 109,26 98,68 15,00 1,52 17,93 0,03 98,46 17-06-1998 5,97 641
11 10,57 -1,60 0,04 0,36 109,13 98,70 15,00 1,52 20,63 0,03 98,50 02-07-1998 6,21 652
12 9,61 -0,71 -0,08 0,74 194,99 98,55 15,00 1,52 22,95 0,03 98,54 17-07-1998 12,93 663
13 9,61 -0,68 -0,08 0,77 194,77 98,60 15,00 1,52 24,27 0,04 98,59 03-08-1998 13,05 674
Summary Table for the 1998 Crisis
tfirst: 96,01

21. 98.10
98.20
98.30
98.40
98.50
98.60
98.70
98.80
98.00 98.10 98.20 98.30 98.40 98.50 98.60 98.70
tc
tlast
1998 Crisis - PSI20
Pea
k 1
Pea
k 2
Lowest
Point
8.9
9
9.1
9.2
9.3
9.4
9.5
9.6
98 98.2 98.4 98.6 98.8 99
Portuguese Stock Index - 20
LN(PSI20)
98,76;
9,00
98,31;
9,57 98,55;
9,52
We can observe that
predictions for tc started to
converge between Peak 2
and the Lowest Point.

22. 107.54, 9.53
107.74, 9.38
108.06, 9.28
8.4
8.6
8.8
9
9.2
9.4
9.6
103.15
103.26
103.37
103.48
103.59
103.71
103.81
103.93
104.04
104.15
104.26
104.37
104.48
104.59
104.70
104.82
104.92
105.04
105.15
105.26
105.37
105.48
105.59
105.70
105.82
105.93
106.04
106.15
106.26
106.37
106.48
106.59
106.70
106.82
106.93
107.04
107.15
107.26
107.37
107.48
107.60
107.70
107.82
107.93
108.04
Portuguese Stock Index - 20
LN(PSI20)
Analysis of the 2007 Crash
2003-2008

23. F
o
r
e
c
a
s
t
s
o
b
t
a
i
n
e
d
tlast: 107,04 tlast: 107,13
tlast: 107,21 tlast: 107,29

24. 103 A B C β φ tc ω λ SRSD
SRSD
/ NDP tlast tlast natural Nosc NDP
1 9,33 -0,25 0,16 0,76 -20,35 107,02 11,59 1,72 23,00 0,02 106,96 15-12-2006 7,65 994
2 9,33 -0,25 0,16 0,75 -20,29 107,01 11,50 1,73 23,09 0,02 107,00 01-01-2007 11,11 1005
3 9,35 -0,27 0,16 0,72 -20,64 107,05 12,23 1,67 23,32 0,02 107,04 16-01-2007 12,55 1016
4 9,37 -0,28 0,15 0,70 -20,90 107,10 12,57 1,65 23,87 0,02 107,09 01-02-2007 11,54 1028
5 9,52 -0,35 0,11 0,64 -22,52 107,44 14,48 1,54 25,43 0,02 107,13 15-02-2007 6,02 1038
6 9,54 -0,36 0,11 0,63 -22,71 107,47 14,72 1,53 25,79 0,02 107,16 01-03-2007 6,18 1048
7 9,54 -0,36 0,11 0,63 -22,72 107,47 14,75 1,53 25,89 0,02 107,21 16-03-2007 6,52 1059
8 9,54 -0,35 0,11 0,63 -22,88 107,50 14,97 1,52 26,00 0,02 107,25 02-04-2007 6,85 1070
9 9,54 -0,35 0,11 0,64 -22,92 107,51 15,00 1,52 26,25 0,02 107,29 17-04-2007 7,20 1081
10 9,54 -0,35 0,11 0,64 -22,95 107,51 15,00 1,52 26,33 0,02 107,33 02-05-2007 7,62 1092
11 9,53 -0,35 0,11 0,64 -22,94 107,51 15,00 1,52 26,47 0,02 107,38 17-05-2007 8,32 1103
Summary Table for the 2007 Crisis
tfirst: 103,15

25. 106.90
107.00
107.10
107.20
107.30
107.40
107.50
107.60
106.90 107.00 107.10 107.20 107.30 107.40
tc
tlast
2007 Crisis - PSI20
Peak
1
9.25
9.3
9.35
9.4
9.45
9.5
9.55
107 107.2 107.4 107.6 107.8 108
Portuguese Stock Index - 20
LN(PSI20)
107,54;
9,53
Predictions for tc started to
converge before Peak 1.
What is usually seen, is
to get tc values
preceeding the real time
of the crash, as in this
case.

26. 115.02, 8.44
115.27, 8.75
115.52, 8.57
115.54, 8.67
115.65, 8.51
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9
112.45
112.53
112.60
112.68
112.76
112.83
112.91
112.99
113.06
113.14
113.22
113.29
113.37
113.45
113.52
113.60
113.68
113.75
113.83
113.91
113.98
114.06
114.14
114.21
114.29
114.37
114.44
114.52
114.60
114.67
114.75
114.83
114.90
114.98
115.07
115.15
115.22
115.31
115.39
115.46
115.54
115.62
Portuguese Stock Index - 20
LN(PSI20)
Analysis of the 2015 Crash
2012-2015

27. F
o
r
e
c
a
s
t
s
o
b
t
a
i
n
e
d
tlast: 115,17 tlast: 115,25
tlast: 115,34
tlast: 115,42

28. 112 A B C β φ tc ω λ SRSD
SRSD
/ NDP tlast tlast natural Nosc NDP
1 8,37 0,31 0,56 0,20 31,84 115,20 5,64 3,05 34,15 0,05 115,09 02-02-2015 2,88 685
2 8,37 0,31 0,55 0,20 31,67 115,25 5,92 2,89 34,61 0,05 115,13 16-02-2015 2,99 695
3 8,37 0,31 0,55 0,20 31,54 115,28 6,19 2,76 34,90 0,05 115,17 02-03-2015 3,13 705
4 8,40 0,28 0,60 0,25 0,17 115,28 6,23 2,74 35,05 0,05 115,21 16-03-2015 3,63 715
5 8,63 0,01 -21,10 0,23 53,27 115,29 5,25 3,31 36,56 0,05 115,25 02-04-2015 3,57 728
6 8,30 0,39 0,40 0,20 -13,07 115,47 8,41 2,11 36,86 0,05 115,29 16-04-2015 3,81 736
7 8,65 0,01 -20,07 0,23 52,95 115,39 6,10 2,80 37,54 0,05 115,34 04-05-2015 3,95 747
8 8,42 0,26 0,57 0,20 -13,45 115,54 8,53 2,09 37,86 0,05 115,38 18-05-2015 4,03 757
9 8,42 0,27 0,53 0,20 30,04 115,65 9,72 1,91 38,77 0,05 115,42 02-06-2015 4,09 768
10 8,45 0,24 0,59 0,20 -13,96 115,65 9,55 1,93 39,31 0,05 115,46 16-06-2015 4,29 778
11 8,66 0,01 -19,07 0,27 52,50 115,51 7,21 2,39 40,29 0,05 115,50 02-07-2015 6,70 790
Summary Table for the 2015 Crisis
tfirst: 112,45

29. 115.15
115.20
115.25
115.30
115.35
115.40
115.45
115.50
115.55
115.60
115.65
115.70
115.00 115.10 115.20 115.30 115.40 115.50 115.60
tc
tlast
2015 Crisis - PSI20
Peak
1
Peak
2
Lowest
Point
8.4
8.45
8.5
8.55
8.6
8.65
8.7
8.75
8.8
115 115.1 115.2 115.3 115.4 115.5 115.6
Portuguese Stock Index - 20
LN(PSI20)
115,27;
115,54;
8,67
115,65;
8,61
Predictions for tc started to
converge around Peak 1.
Since the trend has changed
after Peak 1 a different model
would be needed in order to
analyze the points afterwards.

30. 7) Conclusions
• The analysis of log-periodicity could have given investors clues about
upcoming crashes:
1998: There is a convergence of points before the lowest point.
2007: The investors could have sell before the peak.
2015: A convergence appears around the first peak. This specific model is not
appropriate for the new trend that develops after the first peak.
However:
• It could be better to wait for a convergence of critical points after successive
last points, in order to consider that we have a high probability of a crash.
It´s also worth mentioning, that the number of oscillations also needs to be
considered, as a way to distinguish real log-periodic oscillations from noise.

31. • The use of this methodology could be helpful in order to prevent major
losses in diversified stocks portfolios, whenever we are at a bubble
situation. Given that Log-Periodic Analysis can also be applied to the GNP, a
joint analysis could be done.
• If this technique is going to be applied to individual stocks instead of
indexes, noisier series are to be expected.
• Different methods could be used to verify that the markets are in fact in a
bubble in conjunction to the log-periodic analysis of prices, as a way to
increase the reliability of our forecasts.
• This kind of analysis can be useful for risk management at different levels,
not only for investors, but also for policy makers.
8) Recommendations

32. 9) Suggested Future Research
• Study of anti-bubbles for the Portuguese Stock Market.
• Analysis of individual stocks.
• Analysis using the so-called “Non-Linear” Log-Periodic Formula for longer periods.

33. Thank You
“Stock market crashes are often unforeseen for most people,
especially economists” (Sornette, 2003)