Applications of RQA to Financial Time Series

1
Applications of RQA to Financial Time Series
F. Strozzi1, J. M. Zaldívar2 and J. P. Zbilut3
1Quantitative Methods Institute, Carlo Cattaneo
University, Castellanza (VA), Italy 2European
Commission, DG Joint Research Centre, IES, Ispra
(VA), Italy 3Department of Molecular Biophysics
and Physiology, Rush Medical College, Chicago, USA
2
Outline

• High frequency financial data sets
• Use of RQA ( Recurr) for detecting correlation
  between time series.
• Use of RQA ( Det, Lam) to distinguish financial
  time series from surrogate (linearly correlated
  noise).
• Use of RQA (Det, Lam) for measuring volatility
  in financial data sets
• Conclusions

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High Frequency Financial Data Sets
They are observations on financial variables
taken daily or at a finer time scale. They have
been widely used to study market microstructures.
They are also useful for studying the
statistical properties, volatility in particular,
of asset returns at lower frequencies. They are
irregularly spaced and they need a preliminary
treatment.
HFDF96, exchange rates --gtcorrelation Spot
electricity prices --gt surrogate, volatility
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Use of RQA for detecting correlation in exchange
rates
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High Frequency exchange rates

• HFDF96 Olsen Associates, high freqency, ½ h,
• Exchange rates between the US Dollar and 18
  other foreign currencies from 1.1.1996-31.12.1996

• Belgium Franc (BEF)
• Finnish Markka (FIM)
• German Mark (DEM)
• Spanish peseta (ESP)
• French Frank (FRF)
• Italian Lira (ITL)
• Dutch Guilder (NLG)
• ECU (XEU)
• Australian Dollar (AUD)
• Canadian Dollar (CAD)
• Swiss Frank (CHF)
• Danish Krone (DKK)
• British Pound (GBP)
• Malaysian Ringgit (MYR)
• Japanese Yen (JPY)
• Swedish Krona (SEK)
• Singapore Dollar (SGD)
• South African Rand (ZAR)

• Historical events in 1996
• 14 October FIM joined ERM
• 25 November ITL retunes in ERM
• ERMExchange Rate Mechanism

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exchange rates data treatment

• HFDF96

logarithmic medium price ym
normalised data 0-1
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Correlation in HFDF96 recurrence on epochs

• Time delay Dt (first minimum of the mutual
  information function) 232-283 i.e. 4.8-5.9 days.
• Embedding dimension dE using False Nearest
  Neighbour algorithm and E1E2 methods 7-14

Dt260
dE 11
Epochs of 336 points (1 week) shifted by 48
points (1 day). Linear correlation coefficient
CHF
XEU
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recurr correlation coefficients for the high
frequency currency exchange rates time series
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Stable Distribution
Is it possible to distinguish between real and
random stable distributed data?
A stable probability distribution is defined by
the Fourier transform of its characteristic
function
a?(0,2,
b?-1,1
g?0,?)
d?(-?,?)

• ? is the tail index, ß is a skewness parameter
• is a scale parameter, ? is a location parameter

Gaussian
Chauchy
Levy
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Fitting and generating stochastic data with
stable distributions

• Using STABLE for univariate data
  http//www.cas.american.edu/jpnolan
• We fitted the distribution of the first
  difference y(t1)-y(t) with a stable
  distribution.
• All zeros values were eliminated. After they were
  introduced at the same locations in the random
  time series.

Fitted density plot for the Japanese Yen (JPY)
exchange rate data
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Correlation coefficients for the stochastic time
series with the same stable distribution and zeros
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Comparison financial time series- stochastic
stable distribution time series using RQA
Sign test Null hypothesis the median of
correlation coefficient for real data is the same
that the median of correlation coefficient for
stable random data
the null hypothesis can be rejected at 5 level
of significance if
nmedian refers to the number of observations
lower than the median of random stable data. n is
the total number of observations.

• In 96 pairwise comparisons, over 153, on the
  currency exchange
• rate time series, the values of the correlation
  coefficients are
• higher than the higher value for the random time
  series, i.e.
• 0.646.

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Figure 100. Plot of the recurrence for the Euro
and the Finnish Markka. Blue values before the
entrance in the EMS, green values after the
entrance.
14

• Use of RQA to distinguish electricity spot prices
  from surrogate (linearly correlated noise).

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High Frequency spot electricity prices
Hourly spot prices in the Nordic electricity
market (Nord Pool) from January 1999 until
January 2007.(EUR/MWh)
Hourly spot prices in the Nordic electricity
market (Nord Pool) from May 1992 until December
1998. (NOK/MWh)
Norway Statnett Market
Sweden Nord Pool
Finland
W Denmark
E Denmark
Kontek
1996- EU Electricity Directive starts to have
impact EU countries open their electricity
markets to competition ( high consumers can
choose their provider). 1993- Nord Pool (Nordic
Electricity Market) was created by
Norway. 2005-KT area. (Kontek cable connection
Zealand-Germany). A competition starts between
Nord Pool and European Energy Exchange (EEX)
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Spot electricity price dependencies

• The variation of the prices in the Nord Pool
  system is well correlated with the variations in
  precipitations because of its dependence from
  hydropower generation.
• In the dry periods the price and its
  volatility increase due to the dependence from
  other source of energy (petrol)

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Surrogate data Null hypothesis
Linearly correlated noise. The null hypothesis
the time series are originated by a linear random
process with the same autocorrelation function
or, equivalently, with the same Fourier Power
Spectrum.

• et is un uncorrelated Gaussian noise of unit
  variance
• s is chosen so that the variance of the
  surrogates matches with the one of original data
• ak contain information on Correlation function

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Surrogate data discriminating statistic Q
The dynamic is chaotic? Q
Correlation dimension Q Lyapunov exponent Q
Forecasting error
Are there differencies in the RP structures? Q
RQA measures
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Electricity spot prices Recurrence Plots
EUR/KMh t 13, dE 10, e10
NOK/MWh t15, dE10, e40

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Recurrence Plots Surrogate linearly correlated
NOK/MWh t15, dE10, e40
EUR/KMh t 13, dE 10, e10
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Surrogate linear surrogate (NOK/MWh)
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Surrogate linear surrogate (EUR/MWh)
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Use of RQA for measuring volatility in
electricity spot prices
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Volatility
Higher determinism and laminarity mean that the
states of the system stay closer in time for
longer periods forming diagonal or vertical
segments in RP. Hypothesis higher determinism
or laminarity implies smaller volatility

• There are three main types of volatility
• Realized volatility, also called hystorical
  volatility determined by past observation.
  Standard deviation of the change in value of a
  financial instrument with a specific time horizon
  .
• Model volatility a virtual variable in a
  theoretical model such as GARCH or stochastic
  volatility.
• Implied volatility a volatility forecast
  computed from market prices of derivatives such
  as options, based on a model of underlying
  process (such as log-normal random-walk).

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Volatility
Inverse of standard deviation and determinism
(top) and laminarity (bottom) for EUR/MWh
Inverse of standard deviation and determinism
(top) and laminarity (bottom) for NOK/MWh
720 point window (one month), data are shifted
720 points
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Volatility NOK/MWh
Nonlinear metrics of the Nord Pool spot prices
time series in NOK/MWh Values are computed from
a 720 point window (one month), data are shifted
720 points. RQA parameters t 15, dE10,
distance cutoff max. distance between points/10,
line definition 100 points (4 days). Vertical
lines correspond to the following dates 1st
January 1993, 1st January 1996, 29th December
1997 and 1st July 1999 (see historical
background).
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Volatility EUR/MWh
RQA measures of EUR/MWh Values are computed from
a 720 point window (one month), shifted of 720
points. RQA parameters t 13, dE10, distance
cutoff max. distance between points/10, line
definition 100 points (4 days). Vertical lines
correspond to the following dates 1st October
2000, 5th October 2005 (see historical
background).
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Conclusions

• Correlation (exchange rates)
• A method to assess the correlation between time
  series was developed based on recurrence.
• Series in EUR zone (BEF, FIM, DEM, ESP, FRF, ITL,
  NLG) were highly correlated (R0.9) in 1996 but
  also JPY-CAD-GPB.
• HFDF96 time series are more correlated than
  stable distribution random time series.
• Surrogate (Electricity spot prices)
• Using RQA it is possible to distinguish between
  spot electricity prices and linearly correlated
  noise.
• Volatility (Electricity spot price)
• determinism laminarity could provide a new
  measure of volatility in financial time series.

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Volatility r2

• NOK/MWh
• Det-1/Std r20.4740
• Lam-1/Std r20.5809
• Lam-Det r20.8804

• EUR/MWh
• Det-1/Std r20.4683
• Lam-1/Std r20.4512
• Lam-Det r20.8870

Det-1/Std, Lam-1/Std not very linearly
correlated
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Recurrence Quantification Analysis (RQA)
The concept of recurrence plot (RP) was
introduced by Eckmann et al. (1987) as
follows Let
be the reconstructed delayed vector.
Then, it is possible to define the distance
matrix D as If this Euclidean distance falls
within a defined radius, r, the two vectors are
considered to be recurrent and graphically this
can be indicated by a dot.
Finnish Markka-US dollar (HFDF96)
Malaysian Ringgit-US dollar (HFDF96)
Swiss Franc-US dollar (HFDF96)
http//homepages.luc.edu/cwebber/
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Introduction Recurrence Quantification Analysis
(RQA)

• To extend the original concept and made it more
  quantitative Zbilut and Webber (1992) developed a
  methodology called Recurrence Quantification
  Analysis. They defined several variables to
  quantify recurrence plots
• recurrence percentage of colored pixels in the
  RP. Quantifies the amount of cyclic behaviour.
• determinism Percentage of recurrent points
  which form lines parallel
• to the main diagonal. In the case of a
  deterministic system, these parallel lines
• are an indication of the trajectories being close
  in phase space for time scales
• that are equal to the length of these lines.
• laminarity measures the percentage of vertical
  lines which indicate the occurrence of laminar
  states i.e., periods of tranquility or slowly
  drifting dynamics.
• entropy Shanon entropy of line segments
  distributions. Quantifies the richness of
  deterministic structure.
• trend measure of the recurrence points away
  from the central diagonal. It is a measure of non
  stationarity.
• trap time The average length of all vertical
  lines, indicating an average time the system is
  trapped" into a laminar state as defined above
• 1/linemax reciprocal of the longest diagonal
  line segment, related to largest positive local
  Lyapunov exponent (Trulla et al., 1996)

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Volatility
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Surrogate temporally uncorrelated
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Recurrence Plots Surrogate Temporally
uncorrelated
NOK/MWh t15, dE10, e40
EUR/KMh t 13, dE 10, e10
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Use of RQA for measuring volatility in
electricity spot prices
720 point window (one month), data are shifted
720 points
NOK/MWh
EUR/MWh
Finland
W Denmark
Norway
E Denmark
Sweden
Kontek