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This poster shows empirical information of some characteristics obtained after evaluating

the dynamical behavior of 13 important stock market indices and 10 foreign exchange

markets by using fractal and multifractal tools such as spectral analysis, Hurst exponents

(R/S, generalized and wavelets), Holder and Lyapunov exponents.

OBJECTIVE: To show how an alternative approach like the multifractalism might be

very helpful on further understanding and describing financial markets behavior and how

it is a helpful tool in asset and specially risk management, based on the results here

obtained.

DATA ANALYZED:

According to the efficient market hypothesis point of view, only relevant negative

information could cause a financial crisis (Samuelson 1964 and Fama 1970, 1991); however, until now the lineal paradigm of the structure of markets might not explain

exactly the frequent cracks and lows of the financial markets, and precisely, this is an

interesting academic discussion.

Markets are similar to dynamical complex systems, then the idea of analyzing financial

markets during crisis is based on the scientific evidence in physics that some complex

systems reveal better their inner properties under stressed conditions than in normal

situations.

Johansen, Sornette y Ledoit (2000) showed that under the Efficient Market Hypothesis,

other theories and standard statistic methodologies, a financial crack like the one that occurred in October 1987 or 2008, have a probability of occurrence of 1/1035 times and th

a 5% fall in the Dow Jones Industrial Average Index has a frequence of occurrence once

each 1000 years, in theory. But through history and in the reality these statistics are not

achieved because the frequency of occurrence is much more higher.

Methods

Conclusions

APPLICATIONS ON FINANCE

STEPHANIE RENDON DE LA TORRE UNIVERSIDAD NACIONAL AUTONOMA DE MEXICO – UNAM

Figure #1

Spectral Analysis: Brasilian Real/USD

Bibliography

.

Wolf, A., Swift, J.B., Swinney, H.L., y Vastano J.A. (1985). Determining Lyapunov exponents from a time series. Physica. 16D, pp. 285-317.

Sornette, D. A., Johansen J. y Bouchaud, P. (1996). Stock Market Crasher, Precursors and Replicas. Journal of Physics in France. 6, pp. 167-175.

Samuelson P.A. (1964). Proof that Properly Anticipated Prices Fluctuate Randomly. Industrial Management Review, Spring, 6 (2).

Qian, B., Rasheed y Khaled (2004). Hurst exponent and financial market predictability. Department of Computer Science, University of Georgia, USA.

Peters, E.E. (1991). Chaos, order in the capital markets. New York: John Wiley & Sons, Inc.

Mandelbrot, B.B., (1997). Fractals and Scaling in finance. Springer. New York: USA.

Mantenga, R. N., Palágyi, Z. y Peters, H. E. (1999). Applications of statistical mechanics to Finanance. Physica A. 274. pp. 216-221.

Lyapunov, A. M. (1966). Stability of motion. Academic Press. New York: USA.

Johansen, A., Sornette D. y Ledoit O. (2000). Crashes as Critical Points. International Journal for Theoretical & Applied Finance. 3(2), pp. 219-255.

Jaffard, S. (2004). Wavelet techniques in multifractal analysis. In: Fractal Geometry and Applictions: A Jubilee of Benoit Mandelbrot. Proceedings of Symposia in Pure Mathematics.

72, pp. 91-151.

Hurst, H.E. (1951). Long-term- Storage of Reservoirs. Transactions of the American Society of Civil Engineers. p. 116.

Iacobucci, A. (2003). Spectral Analysis for Economic Time Series. Documents de Travail de l'OFCE. Observatoire Francais des Conjonctures Economiques (OFCE),

Fama E.F. (1970). Efficient Capital Market: Review of Theory and Empirical Work. The Journal of Finance, 25.

Di Matteo, T., Aste T. y Dacorogna, M. M. (2005). Long term memories of developed and emerging markets: using the scaling analysis to characterize their stage of development.

Journal of Banking and Finance. 29(4), pp. 827-851.

Cornelis, A. y Rossitsa, Y. (2004). Multifractal Spectral Analysis of the 1987 Stock Market Crash, Kent State University, recuperado de

http://65.54.113.26/Publication/58551073/multifractal-spectral-analysis-of-the-1987-stock market-crash

Agaev, I.A., Kuperin Yu. A. (2005). Multifractal Analysis and Local Hölder Exponents Approach to Detecting Stock Market Crashes.

Abry, P., Veitch, D. (1998). Wavelet Analysis of Long-Range-Dependent Traffic. IEEE Transactions on Information Theory, 44 (1), pp. 2–15.

1) The analyzed time series are not well represented by normal distributions. They show Levy-Pareto distributions.There might be a serious

problem that the belief of the normal distribution drives to underestimate volatility, then the financial losses could result catastrophic.-

None of the markets analyzed showed linear distributions.

2) The time series showed long term dependencies in general. The Hurst coefficients are found to be different from those of the well-studied

random walk with H = 0.5. These exponents are located in the persistence region. However, during some periods the values were close to

0.5. This contradiction makes sense because it depends on the local behavior of H(t) in the interval of the studied time and H value cannot

be static, otherwise it will mean that the investors always look to the same window of past tense to take investment decisions through time.

This suggests a multifractal feature.

3) The time series show certain chaotic and scalable behavior.

4) Hölder exponents (pointwise and local) modeled correctly the behavior of crisis moments and this could be a helpful tool in risk

management by detecting patterns not only a posteriori but a priori.

5) In general emerging markets show higher Hurst coefficients than developed financial markets. Hurst coefficients seem to decrease and

approximate to 0.5 as their markets become more evolved.

6) There are some cases where H was lower than 0.5. This can be explained by the loss of confidence in the markets feeded by uncertainty on

volatility and this is why H diminishes under 0.50. The mean reversion might be explained by the short sells in stock markets during

bearish periods. The more aggressive the new information is, the quicker and longer the markets’ movements will be.

7) Spectral analysis showed betas in all cases different from 2 and power laws S(f)~(f -β). This confirms scalating behavior.

8) The scalating behavior of markets could be a useful tool for identifying or measuring the stability level of a market (from less mature –

emerging to mature- developed market).

9) Two groups could be seggregated (divided by the development level of their stock markets) the ones that show higher levels of H and the

ones that show levels around 0.50. Mature markets (levels close to 0.5) CAC, NIKKEI, FTSE, INDU, DAX y SPX. Emerging markets

(levels close to 1): HSCEI, TA25, KOSPI, IPC, XU100 y JCI (after analyzing several periods of time, 1 year, 4 years, 10 years, 50 years).

In currencies: systematically all H were higher than in the stock indices. Higher H (mature markets): CAD/USD, EUR/USD, YEN/USD,

GBP/USD, AUD/USD. Less developed (values closer to H): CHF/USD, MXN/USD, BRL/USD, RUB/USD, CNY/USD.

10) There is a correspondence between the crisis patterns detected by the local and pointwise Hölder exponents and the periods when the

highest values of H were produced. Both tools could be useful to model crisis periods.

12) All Lyapunov exponents were bigger than 0; in theory, this is an important evidence of deterministic chaotic behavior, however different

values of the exponents were produced after changing the embedded dimensions. The iterative periods (at any scale) are higher in stock

indices than in currencies, then the cycles of memory in stock markets are longer than in the fx markets analyzed.

Introduction Results

Efficient Market Hypothesis Fractal Market Hypothesis

Gaussian statistics- Independence Non Gaussian statistics: Dependence

Geometric Brownian motion Fractionary motion

Stationary processes Non-Stationary processes

No historical correlations Historical correlations

There is no memory process. Past events

do not influence present or future events

There is memory (events are linked). Investors are

influenced by what has happened in the past. Their

expectations are based on previous experiences.

Patterns are not repeated at any scale

Patterns are repeated in all scales of time (minutes, days,

years).

Continuous and stable at all sclaes

Discontinuities occur at any scale. (black swan events)

Leptokurtosis

All the information is reflected in prices

today.

Each individual sees information with different

perspectives and in different

Rationality and risk aversion “Herd instinct” -

FX MARKET STOCK INDEX Country FX Períod Country IA Name Period

Mexico MXN/USD 1993-2013 Mexico IPC Índices de Precios y Cotizaciones 1994-2013

Canada CAD/USD 1993-2013 Indonesia JCI Jakarta Composite Index 1983-2013

Russia RUB/USD 1993-2013 South Korea KOSPI Korea Composite Stock Price Index 1980-2013

China CNY/USD 1993-2013 China HSCEI Hong Kong Hang Seng Index 1993-2013

Japan YEN/USD 1993-2013 Japan NIKKEI Nikkei 225 1970-2013

UK GBP/USD 1993-2013 UK FTSE FTSE 100 1984-2013

Europa EUR/USD 1993-2013 Germany DAX Deutsche Borse AG German Stock Index DAX 1970-2013

Australia AUD/USD 1993-2013 France CAC CAC 40 (París) 1988-2013

Brasil RLS/USD 1993-2013 Brasil BOVESPA IBOVESPA Brasil Sao Paolo Stock Exchange Index 1989-2013

Switzerland CFH/USD 1993-2013 Turkey XU100 Borsa Istanbul Stock Exchange National 100 Index 1988-2013

Israel TA25 Tel Aviv 25 Index 1992-2013

EEUU SPX Standard & Poor 500 1928-2013

EEUU INDU Dow Jones Industrial Average 1928-2013

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

H Wavelet

H R/S

H Generalized

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

H Wavelet

H R/S

H Generalized

1.7

1.8

1.9

2

2.1

2.2

2.3

β

Stock

Index Embedd

ed

dimensi

on

N λi Máx

duration

of

iterations

Iteratio

ns

averag

e

BOVES

PA 3 3 0.067 15 18

3 5 0.052 19

4 4 0.051 20

IPC 3 3 0.042 24 23

3 4 0.047 21

4 4 0.04 25

INDU 3 4 0.015 67 50

3 4 0.016 63

4 5 0.051 20

SPX 3 4 0.017 59 72

4 4 0.015 67

5 5 0.011 91

TA-25 3 3 0.069 14 18

4 4 0.057 18

5 4 0.048 21

DAX 3 4 0.049 20 24

3 4 0.043 23

4 5 0.035 29

CAC 2 3 0.069 14 15

3 3 0.073 14

4 4 0.062 16

NIKKEI 4 3 0.031 32 33

4 4 0.031 32

5 5 0.028 36

JCI 3 3 0.016 63 54

3 5 0.018 56

3 4 0.023 43

XU100 3 3 0.046 22 23

3 5 0.042 24

4 4 0.042 24

KOSPI 3 3 0.052 19 23

3 4 0.055 18

4 4 0.032 31

FTSE 3 3 0.073 14 17

4 4 0.055 18

4 5 0.051 20

HSCEI 3 3 0.076 13 16

3 5 0.062 16

4 4 0.054 19

FX

exchange

Embe

dded

dimension

n Λi Max # of iterations

Average iterations

MXN/U

SD

3 3 0.047 21 23

3 3 0.042 24

4 4 0.042 24

EUR/U

SD

3 3 0.119 8 10

4 3 0.096 10

4 5 0.085 12

CAD/U

SD

3 3 0.094 11 12

3 4 0.091 11

4 4 0.073 14

AUD/U

SD

3 3 0.087 11 12

4 4 0.075 13

3 4 0.09 11

JPY/US

D

3 3 0.117 9 10

3 5 0.099 10

4 3 0.096 10

CNY/U

SD

5 3 0.026 38 43

5 4 0.023 43

4 5 0.021 48

RLS/U

SD

3 3 0.036 28 34

4 4 0.026 38

5 4 0.029 34

GBP/U

SD

3 3 0.146 7 8

4 3 0.131 8

4 4 0.124 8

RUB/U

SD

3 3 0.024 42 41

4 4 0.024 42

3 4 0.026 38

CHF/U

SD

3 3 0.091 11 13

4 4 0.077 13

5 5 0.068 15

For determining Hurst coefficient by R/S and H generalized: First of all, let us consider a price time series of length n given

by: {p(t1 ), p(t2 ), ..., p(tn )}, and the price τ -returns r(τ ) having time scale τ and length n that is represented in terms of r(τ ) =

{r1 (τ ), r2 (τ ), ..., rn (τ )}, with ri (τ ) = ln p(ti + τ ) − ln p(ti ). After dividing the time series or returns into n subseries of

length M , we label each subseries EM,d (τ ) = {r1,d (τ ), r2,d (τ ), ..., rM,d (τ )}, with d = 1, 2, ..., N . Then, the deviation DM,d

(τ ) can be defined directly from the mean of returns r¯M,d (τ ) as The hierachical average value (R/S)M

(τ ) that stands for the rescaled/normalized relation between

RM,d (τ ) and SM,d (τ ) become where the subseries RM,d (τ ) and the standard deviation SM,d (τ )

are, respectively, given by and

Here, H (τ ) is called the Hurst exponent, and the relationship between the fractal dimension Df and the Hurst exponent H (τ ) can

be written as Df = 2 − H (τ ). As is well known, we can dynamically evolve the Hurst exponent in the following way: (1) The

time series is persistent if H (τ ) ∈ (0.5, 1.0]. This means that it is characterized by long-run memory affecting all time scales.

One has increasing persistence as H (τ ) approaches 1.0. The persistence process means that the chances will continue to be up or

down in the future if the price is up or down. (2) When H (τ ) = 0.5, the time series is uncorrelated, and this case really reduces

to a Gaussian or a gamma white-noise process. The stochastic process with H (τ ) = 0.5 is also referred to as a fractional Brownian

motion. (3) One has antipersistence if H (τ ) ∈ (0, 0.5].

The q-th price-price correlation function Fq (τ ) that depends only on the time lag τ takes the form

where Hq is the generalized qth-order Hurst exponent and the angular brackets denote a statistical average over time. In reality, it

would be expected that a nontrivial multi-affine spectrum can be obtained as Hq varies with q. This has been exploited in the

multifractal method, and the large fluctuation effects in the dynamical behav ior of the price can be explored.

Spectral analysis: The spectral density function: The notion of power is used as energy: the measure of energy by the

unit of time to obtain a finite spectral density in order to represent the whole stochastic process. Now, the definition of the power

spectral density for stochastic processes is: The spectral density of the log of a stock price is described by the

functional form:

S(f)~ 1/(f 2) this is a power law and it is the prediction for the spectral density in a random walk. The power spectrum is a second

order statistic measure and knowing its slope helps on validating a particular scaling model.

Hölder exponents: pointwise and local determination based on oscillations by (Agaev y Kuperin 2005)

Lyapunov exponents: Wolf (1985) algorhythm. The formal equation for the ith Lyapunov exponent λi, for the ith dimension (pi (t))

is: λ i= lim (1/t) ln ((pi (t) / pi (0)) where the exponents are ordered by the dominant to the weakest. The largest might be approximated by: t→∞

Lyapunov exponents results

FX Exchanges Betas results Evolution of FX RUB/USD

Hurst coefficients FX Markets Hurst coefficients Stock Markets

Hurst coefficients FX Markets Hurst coefficients Stock Markets