Applications of RQA to Financial Time Series

F. Strozzi1, J. M. Zaldívar2 and J. P. Zbilut3

1Quantitative Methods Institute, Carlo Cattaneo University, Castellanza (VA), Italy2European Commission, DG Joint Research Centre, IES, Ispra (VA), Italy3Department of Molecular Biophysics and Physiology, Rush Medical College, Chicago, USA

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

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 -->correlationSpot electricity prices --> surrogate, volatility

Use of RQA for detecting correlation in 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)

High Frequency exchange rates

Historical events in 1996:

• 14 October FIM joined ERM• 25 November ITL retunes in ERM

ERM=Exchange Rate Mechanism

logarithmic medium price ym 2

)log()log( askbidm

ppy

minmax

min

mm

mm

yy

yyy

normalised data [0-1]

• HFDF96

exchange rates: data treatment

• Time delay t (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 E1&E2 methods: 7-14

t=260 dE =11

XEU

CHF

Epochs of 336 points (1 week) shifted by 48 points (1 day). Linear correlation coefficient

Correlation in HFDF96: %recurrence on epochs

AUD BEF CAD CHF DEM DKK ESP FIM FRF GPB ITL JPY MYR NLG SEK SGD XEU ZAR AUD 0.835 0.602 0.615 0.840 0.798 0.840 0.817 0.859 0.512 0.482 0.686 0.488 0.879 0.731 0.508 0.782 0.533 BEF 0.689 0.465 0.936 0.953 0.890 0.810 0.924 0.554 0.751 0.667 0.604 0.927 0.815 0.719 0.930 0.596 CAD 0.325 0.726 0.788 0.767 0.633 0.750 0.841 0.726 0.905 0.801 0.697 0.751 0.560 0.818 0.868 CHF 0.584 0.405 0.441 0.363 0.590 0.562 0.158 0.551 0.010 0.554 0.144 0.001 0.367 0.163 DEM 0.945 0.859 0.834 0.935 0.654 0.742 0.728 0.617 0.903 0.756 0.686 0.906 0.574 DKK 0.899 0.818 0.931 0.642 0.842 0.725 0.731 0.885 0.856 0.754 0.965 0.683 ESP 0.865 0.929 0.617 0.617 0.763 0.740 0.927 0.894 0.586 0.936 0.769 FIM 0.815 0.464 0.513 0.639 0.708 0.847 0.856 0.580 0.814 0.619 FRF 0.674 0.681 0.781 0.616 0.938 0.807 0.616 0.923 0.685 GPB 0.633 0.845 0.626 0.552 0.491 0.337 0.644 0.658 ITL 0.568 0.679 0.570 0.636 0.782 0.794 0.574 JPY 0.629 0.730 0.683 0.401 0.740 0.773 MYR 0.564 0.815 0.607 0.750 0.778 NLG 0.829 0.616 0.911 0.676 SEK 0.672 0.884 0.782 SGD 0.740 0.490 XEU 0.780 ZAR

%recurr: correlation coefficients for the high frequency currency exchange rates time series

Stable Distribution

0,2

1 ,2/10 ,1

A stable probability distribution is defined by the Fourier transform of its characteristic function t

:

dtetxf itx

2

1),,,;(

)sgn(1||exp tititt

1 t

1

)log()/2(

)2/tan(

Gaussian22 2

Chauchy Levy

(0,2], [-1,1] [0,) (-,)

is the tail index, β is a skewness parameter is a scale parameter, is a location parameter

Is it possible to distinguish between real and random stable distributed data?

• Using STABLE for univariate data http://www.cas.american.edu/~jpnolan

• We fitted the distribution of the first difference y(t+1)-y(t) with a stable distribution.

• All zeros values were eliminated. After they were introduced at the same locations in the random time series.

Data set Data set

AUD 1.57 -0.004 0.343 -0.556 GBP 1.54 -0.018 0.262 -0.386

BEF 1.46 0.005 0.405 0.939 ITL 1.45 0.018 0.583 -1.018

CAD 1.53 -0.038 0.508 1.845 JPY 1.61 0.010 0.386 0.535

CHF 1.56 0.060 0.296 0.029 MYR 1.42 0.005 0.368 -0.136

DEM 1.55 0.006 0.395 0.899 NLG 1.54 0.015 0.440 0.651

DKK 1.53 -0.018 0.518 1.555 SEK 1.44 0.010 0.746 0.138

ESP 1.32 0.004 0.492 0.998 SGD 1.50 0.016 0.628 -0.455

FIM 1.43 0.003 0.434 0.452 XEU 1.55 0.004 0.794 0.724

FRF 1.50 0.015 0.456 0.583 ZAR 1.26 0.044 0.173 0.050

Fitted density plot for the Japanese Yen (JPY) exchange rate data

Fitting and generating stochastic data with stable distributions

AUDsd BEFsd CADsd CHFsd DEMsd DKKsd ESPsd FIMsd FRFsd GPBsd ITLsd JPYsd MYRsd NLGsd SEKsd SGDsd XEUsd ZARsd

AUDsd 0.444 0.212 0.136 0.397 0.230 0.625 0.274 0235 0.045 0.055 0.463 0.430 0.109 0.191 0.227 0.207 0.021 BEFsd 0.387 0.234 0.480 0.568 0.639 0.388 0.120 0.334 0.186 0.073 0.390 0.166 4e-4 0.100 0.224 0.066 CADsd 0.228 0.192 0.597 0.272 0.415 0.062 0.167 0.223 0.050 0.329 0.132 0.125 0.349 0.128 0.135 CHFsd 0.288 0.406 0.086 0.222 0.008 0.039 0.369 0.064 0.001 0.413 0.083 0.035 0.033 0.069 DEMsd 0.115 0.240 0.199 0.046 0.139 0.283 0.121 0.344 0.200 0.169 0.145 0.163 0.170 DKKsd 0.439 0.468 0.075 0.177 0.064 0.158 0.078 0.148 0.230 0.113 0.300 0.270 ESPsd 0.638 0.331 0.464 0.020 0.152 0.259 0.242 0.090 0.232 0.157 0.290 FIMsd 0.164 0.448 0.144 0.210 0.068 0.340 0.255 0.523 0.132 0.613 FRFsd 0.368 0.115 0.004 0.026 0.391 0.646 0.024 0.335 0.024 GPBsd 0.011 0.359 0.094 0.517 0.099 0.004 0.487 0.254 ITLsd 0.045 0.196 0.337 0.132 0.239 0.187 0.244 JPYsd 0.181 0.328 0.135 0.077 0.342 0.113 MYRsd 0.141 0.190 0.201 0.464 0.368 NLGsd 0.144 0.114 0.506 0.371 SEKsd 0.224 0.093 0.454 SGDsd 0.195 0.426 XEUsd 0.184 ZARsd

Correlation coefficients for the stochastic time series with the same stable distribution and zeros

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.

Comparison financial time series- stochastic stable distribution time series using RQA

nnnmedian /12/1/

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.

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.

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

Hourly spot prices in the Nordic electricity market (Nord Pool) from May 1992 until December 1998. (NOK/MWh)

Hourly spot prices in the Nordic electricity market (Nord Pool) from January 1999 until January 2007.(EUR/MWh)

SwedenNord Pool

Finland W Denmark E Denmark KontekNorway

Statnett Market

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)

High Frequency spot electricity prices

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)

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.

t

q

kktkt

exax

1

et is un uncorrelated Gaussian noise of unit varianceis chosen so that the variance of the surrogates matches with the one of original dataak contain information on Correlation function

Surrogate data: discriminating statistic Q

The dynamic is chaotic? Q = Correlation dimensionQ = Lyapunov exponentQ = Forecasting error

Are there differencies in the RP structures?Q = RQA measures

Electricity spot prices: Recurrence Plots

EUR/KMh =13, dE =10, =10 NOK/MWh =15, dE=10, =40

Recurrence Plots: Surrogate linearly correlated

EUR/KMh =13, dE =10, =10 NOK/MWh =15, dE=10, =40

Data set %recur %deter maxline entropy trend % laminar TrapTime

Bnok 16.095 67.13 3545 8.593 -8.687 69.994 308.044

Surr001 8.150 6.129 4808 6.740 2.306 1.796 123.511

Surr002 1.926 4.521 1355 4.913 -0.142 0.000 -1

Surr003 2.807 8.026 4808 6.028 -1.616 0.000 -1

Surr004 30.218 36.309 4808 7.994 -3.360 35.521 214.805

Surr005 1.735 13.216 1844 6.117 -0.983 0.055 110

Surr006 1.007 32.018 1178 6.287 -0.752 16.980 166.134

Surr007 4.785 13.279 2674 6.895 -0.802 7.533 153.511

Surr008 14.122 17.880 4350 7.357 -4.479 9.293 154.815

Surr009 5.934 13.528 3130 7.195 -2.458 6.301 159.498

Surr010 1.193 5.900 1064 4.696 -0.677 0.347 119.500

Surr011 4.860 51.638 4808 7.918 -1.542 52.444 266.162

Surr012 31.899 52.675 4808 8.415 12.407 54.522 218.168

Surr013 4.795 9.417 4808 6.880 0.524 0.724 143.393

Surr014 5.725 9.169 4154 6.783 -2.980 1.774 144.963

Surr015 4.972 6.340 2370 6.606 -2.341 1.720 114.988

Surr016 18.050 23.404 4808 7.678 -4.183 12.845 161.470

Surr017 10.846 43.188 4614 8.799 -7.222 38.298 338.899

Surr018 4.956 8.523 4808 6.596 -2.654 3.484 141.553

Surr019 6.323 4.462 4808 6.184 -1.989 0.375 114.208

Surrogate: linear surrogate (NOK/MWh)

Data set %recur %deter maxline entropy trend % laminar TrapTime

Beur 7.12 35.33 2094 7.658 -4.587 33.94 263.525

Surr001 12.524 3.665 3340 6.355 -6.259 2.539 149.367

Surr002 1.643 5.894 2238 5.270 -1.100 1.872 119.367

Surr003 3.840 1.397 2150 4.533 -0.998 0.000 -1

Surr004 4.377 1.105 1324 3.970 -0.286 0.000 -1.000

Surr005 10.677 1.825 4187 5.730 -5.483 1.527 126.613

Surr006 8.658 18.813 4826 7.538 -5.638 9.854 146.364

Surr007 0.491 3.888 690 2.807 -0.346 0.000 -1.000

Surr008 23.790 11.105 4826 7.509 -7.639 9.252 162.159

Surr009 30.269 10.831 4826 7.393 -1.830 7.108 151.053

Surr010 20.536 4.700 4826 6.845 -7.466 6.416 150.611

Surr011 2.336 3.777 1888 5.094 -1.160 1.529 134.161

Surr012 3.715 1.475 3517 4.059 -1.627 0.108 117.250

Surr013 4.994 3.736 3721 5.972 -3.343 1.886 135.457

Surr014 21.649 9.020 4826 7.162 -2.900 9.664 154.810

Surr015 20.052 8.142 2669 7.247 -4.243 4.171 146.484

Surr016 6.811 5.384 3998 6.574 -4.098 0.758 125.312

Surr017 3.161 4.113 1964 5.641 -2.076 0.715 131.650

Surr018 7.809 3.369 2429 6.204 -0.473 2.766 132.437

Surr019 12.185 1.330 4826 5.429 1.503 0.088 125.600

Surrogate: linear surrogate (EUR/MWh)

Use of RQA for measuring volatility in electricity spot prices

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).

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

Volatility

Inverse of standard deviation and %determinism (top) and %laminarity (bottom) for NOK/MWh

Inverse of standard deviation and %determinism (top) and %laminarity (bottom) for EUR/MWh

720 point window (one month), data are shifted 720 points

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, dE=10, 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).

Volatility: EUR/MWh

RQA measures of EUR/MWh: Values are computed from a 720 point window (one month), shifted of 720 points. RQA parameters: =13, dE=10, distance cutoff: max. distance between points/10, line

definition: 100 points (~4 days). Vertical lines correspond to the following dates: 1 st October 2000, 5th October 2005 (see historical background).

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 (R~0.9) 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.

• NOK/MWh

%Det-1/Std r2=0.4740

%Lam-1/Std r2=0.5809

%Lam-%Det r2=0.8804

Volatility: r2

• EUR/MWh

%Det-1/Std r2=0.4683

%Lam-1/Std r2=0.4512

%Lam-%Det r2=0.8870

%Det-1/Std, %Lam-1/Std

not very linearly correlated

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 asIf 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.

jiij SSd ),....}2(),(),({)( ttsttststS

http://homepages.luc.edu/~cwebber/

Finnish Markka-US dollar (HFDF96) Malaysian Ringgit-US dollar (HFDF96) Swiss Franc-US dollar (HFDF96)

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 parallelto the main diagonal. In the case of a deterministic system, these parallel linesare an indication of the trajectories being close in phase space for time scalesthat 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)

Introduction: Recurrence Quantification Analysis (RQA)

Volatility

Data set %recur %deter maxline entropy trend % laminar TrapTime

Bnok 16.095 67.13 3545 8.593 -8.687 69.687 308.044

Surr001 0 0 -1 -1 0 0 -1

Surr002 0 0 -1 -1 0 0 -1

Surr003 0 0 -1 -1 0 0 -1

Surr004

Surr005

Surr006

Surr007

Surr008

Surr009

Surr010

Surr011

Surr012

Surr013

Surr014

Surr015

Surr016

Surr017

Surr018

Surr019

Surrogate: temporally uncorrelated

Recurrence Plots: Surrogate Temporally uncorrelated

EUR/KMh =13, dE =10, =10 NOK/MWh =15, dE=10, =40

Norway Sweden

Finland W Denmark E Denmark Kontek

NOK/MWh EUR/MWh

720 point window (one month), data are shifted 720 points

Use of RQA for measuring volatility in electricity spot prices