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Econophysics V: Market States and Subtleties of Correlations - Thomas Guhr

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Let’s face complexity September 4-8, 2017

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Econophysics V: Market States and Subtleties of Correlations - Thomas Guhr

  1. 1. Epps Effect Non–Gaussian Dependencies Market States Conclusions Fakult¨at f¨ur Physik Econophysics V: Market States and the Subtleties of Correlations Thomas Guhr Let’s Face Complexity, Como, 2017 Como, September 2017
  2. 2. Epps Effect Non–Gaussian Dependencies Market States Conclusions Outline mysterious vanishing of correlations: Epps effect Gaussian assumptions and correlations: copulae identification of market states time evolution of market states signatures of crisis Como, September 2017
  3. 3. Epps Effect Non–Gaussian Dependencies Market States Conclusions Epps Effect Como, September 2017
  4. 4. Epps Effect Non–Gaussian Dependencies Market States Conclusions Measurement of correlation coefficients stock prices: S(i)(t), i = 1, . . . , K returns r (i) ∆t = S(i)(t + ∆t) − S(i)(t) S(i)(t) depend on the chosen return interval Jan 2008 Feb 2008 Mar 2008 Apr 2008 May 2008 Jun 2008 Jul 2008 Aug 2008 Sep 2008 Oct 2008 Nov 2008 Dec 2008 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 S(t)/S(Jan2008) IBM MSFT Cij = corr(r (i) ∆t, r (j) ∆t) = r (i) ∆tr (j) ∆t − r (i) ∆t r (j) ∆t σ(i)σ(j) , u = 1 T T t=1 u(t) assume that T is long enough −→ no noise dressing ... but what is the dependence on the return interval ∆t ? Como, September 2017
  5. 5. Epps Effect Non–Gaussian Dependencies Market States Conclusions Empirical results measured correlations suppressed towards small return intervals ∆t −→ this is the Epps effect 0 5 10 15 20 25 30 35 40 ∆t [min] 0.70 0.75 0.80 0.85 0.90 0.95 1.00 corr(∆t)/corr(40min) ensemble of 50 stock pairs (normalized to saturation value) Como, September 2017
  6. 6. Epps Effect Non–Gaussian Dependencies Market States Conclusions Goal Variety of possible reasons discussed in finance, including highly speculative “emergence of correlations”. Existing studies mostly aim at schematically compensating the Epps effect. Being physicists, we ... look at the data — carefully, identify statistical causes, develop parameter free compensation, quantify what is left for other causes. Como, September 2017
  7. 7. Epps Effect Non–Gaussian Dependencies Market States Conclusions Asynchronity — formation of an overlap underlying fictitious time series actual time series γ: last trading time overlap ∆to(t)/∆t with synchronous information, outside random Como, September 2017
  8. 8. Epps Effect Non–Gaussian Dependencies Market States Conclusions Compensation of asynchronity g (i) ∆t(t) = r (i) ∆t(t) − r (i) ∆t(t) σ (i) ∆t corr(r (i) ∆t, r (j) ∆t) = g (i) ∆t(t)g (j) ∆t(t) corrasync(r (i) ∆t, r (j) ∆t) = g (i) ∆t(t)g (j) ∆t(t) ∆t ∆to(t) term–by–term compensation by multiplying with inverse overlap Como, September 2017
  9. 9. Epps Effect Non–Gaussian Dependencies Market States Conclusions Tick size and return distribution tick-Size q discretizes prices returns also affected clustering Como, September 2017
  10. 10. Epps Effect Non–Gaussian Dependencies Market States Conclusions Correlation coefficient for discretized data idea: discretization r(i)(t) → ¯r(i)(t) produces random errors ϑ(i)(t) r(i) (t) = ¯r(i) (t) + ϑ(i) (t) corrtick(r(i) , r(j) ) ≈ cov ¯r(i), ¯r(j) ˆσ(i)ˆσ(j) compensation by correcting with normalization estimation using average discretization error Como, September 2017
  11. 11. Epps Effect Non–Gaussian Dependencies Market States Conclusions Combined compensation take both, asynchronity and discretization, into account corr(r(i) , r(j) ) ≈ ¯r(i) ¯r(j) ∆t ∆to ˆσ(i)ˆσ(j) no interference, undisturbed superposition Como, September 2017
  12. 12. Epps Effect Non–Gaussian Dependencies Market States Conclusions Test with stochastic processes autoregressive GARCH(1,1) known to reproduce phenomenology of stock price time series and their distributions 0 5 10 15 20 25 30 ∆t/60 0.10 0.15 0.20 0.25 0.30 0.35 0.40 corr asynchrony correction tick−size correction combined correction full, parameter free compensation Como, September 2017
  13. 13. Epps Effect Non–Gaussian Dependencies Market States Conclusions Test with real data stocks from Standard & Poor’s 500, prices between $10.01–$20.00 0 5 10 15 20 25 30 ∆t [min] 0.6 0.7 0.8 0.9 1.0 corr(∆t)/corr(30min) corrected parameter free combined compensation −→ rest has other causes Como, September 2017
  14. 14. Epps Effect Non–Gaussian Dependencies Market States Conclusions Results purely statistical causes have strong impact on Epps effect identified what is left for other causes (e.g. lags, etc) parameter free compensation significant better precision when estimating correlations can easily be applied Como, September 2017
  15. 15. Epps Effect Non–Gaussian Dependencies Market States Conclusions Non–Gaussian Dependencies Como, September 2017
  16. 16. Epps Effect Non–Gaussian Dependencies Market States Conclusions Correlation coefficient and joint probability density function correlation coefficient reduces complex statistical dependence to a single number only meaningful if dependence is multivariate Gaussian, e.g. bivariate fX,Y (x, y) = 1 2π √ 1 − c2 exp − 1 2 x2 − 2cxy + y2 1 − c2 if not, have to retrieve better information from full joint probability density function fX,Y (x, y) which contains all information −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 0.05 0.10 0.15 Como, September 2017
  17. 17. Epps Effect Non–Gaussian Dependencies Market States Conclusions Tools and definitions marginal distribution: fX (x) = +∞ −∞ fX,Y (x, y)dy comulative distribution: FX (x) = x −∞ fX (x )dx u quantile: left of x = F−1 X (u) are u percent of events joint probability: FX,Y (x, y) = x −∞ dx y −∞ dy fX,Y (x , y ) Como, September 2017
  18. 18. Epps Effect Non–Gaussian Dependencies Market States Conclusions Copulae fX,Y (x, y) fX (x), fY (y) copX,Y (u, v) separate statistical dependencies and marginal distributions CopX,Y (u, v) = FX,Y F−1 X (u), F−1 Y (v) copX,Y (u, v) = ∂2 ∂u∂v CopX,Y (u, v) . (similar to “moving frame” or “unfolding” in quantum chaos) Como, September 2017
  19. 19. Epps Effect Non–Gaussian Dependencies Market States Conclusions Comparison true versus Gaussian copulae K return time series r(i)(t), i = 1, . . . , K calculate standard Pearson correlation coefficients Cij for each pair (i, j) uniquely determines bivariate Gaussian distribution for pair (i, j) fi,j (x, y) = 1 2π 1 − C2 ij exp − 1 2 x2 − 2Cij xy + y2 1 − C2 ij evaluate corresponding Gaussian copula cop (G) i,j (u, v) Como, September 2017
  20. 20. Epps Effect Non–Gaussian Dependencies Market States Conclusions Comparison true versus Gaussian copulae — continued analyze true copula copi,j (u, v) calculate distance d(u, v) = 1 K(K − 1)/2 i<j copi,j (u, v) − cop (G) i,j (u, v) Como, September 2017
  21. 21. Epps Effect Non–Gaussian Dependencies Market States Conclusions Empirical study TAQ data 2007–2010, S&P 500, more than 12 billion transactions Quantile 1 0.0 0.2 0.4 0.6 0.8 Quantile 2 0.0 0.2 0.4 0.6 0.8 0.00 0.05 0.10 0.15 0.20 0.25 ∆t = 2h Quantile 1 0.0 0.2 0.4 0.6 0.8 Quantile 2 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 ∆t = 30min structure of copula stable when varying return interval bivariate Gaussian assumption drastically underestimates simultaneous extreme events Como, September 2017
  22. 22. Epps Effect Non–Gaussian Dependencies Market States Conclusions Results standard Pearson correlation coefficient problematic for data which are not bivariate Gaussian copulae provide alternative by extracting statistical dependencies independent of marginal distributions risk of simultaneous extreme events in real data much higher than usually assumed Como, September 2017
  23. 23. Epps Effect Non–Gaussian Dependencies Market States Conclusions Identifying Market States and their Dynamics Como, September 2017
  24. 24. Epps Effect Non–Gaussian Dependencies Market States Conclusions States of financial markets market is non–stationary different states before, during and after a crisis market can function in different modes qualitative/empirical — also quantitative ? S&P 500 Aug 2008 Sep 2008 Oct 2008 Nov 2008 Dec 2008 Jan 2009 700 800 900 1000 1100 1200 1300 1400 points Como, September 2017
  25. 25. Epps Effect Non–Gaussian Dependencies Market States Conclusions Similarity measure correlations provide detailed information about the market introduce distance of two correlation matrices ζ(T) (t1, t2) = C (T) ij (t1) − C (T) ij (t2) ij i, j running index of risk element or company t1, t2 times at which the two correlation matrices calculated T sampling time backwards −→ distances ζ(T)(t1, t2) array or matrix in points (t1, t2) Como, September 2017
  26. 26. Epps Effect Non–Gaussian Dependencies Market States Conclusions Empirical results T = 2 months Como, September 2017
  27. 27. Epps Effect Non–Gaussian Dependencies Market States Conclusions Empirical results T = 2 months T = 1 week Como, September 2017
  28. 28. Epps Effect Non–Gaussian Dependencies Market States Conclusions Identification of market states array ζ(T)(t1, t2) reveals changes in market structures over long time horizons identify states by cluster analysis ensemble of correlation matrices C (T) ij (t), t = t(a), . . . , t(b) find two disjunct clusters where distance ζ(T) from average within each cluster is smallest repeat that for these two clusters, and so on stop when distances within groups comparable to distances between grop Como, September 2017
  29. 29. Epps Effect Non–Gaussian Dependencies Market States Conclusions States of US financial market 1992–2010 1 2 4 5 6 7 8 3 bold face numbers label market states if no threshold −→ division process continued until all correlation matrices are identified small numbers label year and two–months period Como, September 2017
  30. 30. Epps Effect Non–Gaussian Dependencies Market States Conclusions Market states in basis of industrial sectors overall average correlation matrix E M I CD CS H F IT C U E M I CD CS H F IT C U −1.0 −0.5 0.0 0.5 1.0 Como, September 2017
  31. 31. Epps Effect Non–Gaussian Dependencies Market States Conclusions Market states in basis of industrial sectors overall average correlation matrix E M I CD CS H F IT C U E M I CD CS H F IT C U −1.0 −0.5 0.0 0.5 1.0 the eight states: 1 E M I CD CS H F IT C U E M I CD CS H F IT C U 2 E M I CD CS H F IT C U E M I CD CS H F IT C U 3 E M I CD CS H F IT C U E M I CD CS H F IT C U 4 E M I CD CS H F IT C U E M I CD CS H F IT C U 5 E M I CD CS H F IT C U E M I CD CS H F IT C U 6 E M I CD CS H F IT C U E M I CD CS H F IT C U 7 E M I CD CS H F IT C U E M I CD CS H F IT C U 8 E M I CD CS H F IT C U E M I CD CS H F IT C U Como, September 2017
  32. 32. Epps Effect Non–Gaussian Dependencies Market States Conclusions Time evolution of US market states subsequent formation of US market states 1992–2010 market jumps between different states old states die out −→ states have a lifetime how does this lifetime relate to other time scales (e.g. times between crashes) ? Como, September 2017
  33. 33. Epps Effect Non–Gaussian Dependencies Market States Conclusions Time evolution of correlation coefficients distribution time resolved analysis of distribution p(Cij ) during the 2008 crisis surface plot black: September 2008, red: December 2008 distribution became broader before the crisis started in October 2008, partly due to decoupling of energy sector very narrow during the crisis with large mean value ←→ panic Como, September 2017
  34. 34. Epps Effect Non–Gaussian Dependencies Market States Conclusions Stable period within 2008–2009 crisis correlations during crisis October 15th, 2008, to April 1st, 2009 three–week stable period January 1st, 2009, to January 21st, 2009 E M I CD CS H F IT C U E M I CD CS H F IT C U 0.00 0.15 0.30 0.45 0.60 0.75 0.90 excluding stable period E M I CD CS H F IT C U E M I CD CS H F IT C U −0.15 0.00 0.15 0.30 0.45 0.60 0.75 0.90 stable period stable period very similar to market state number 7 Como, September 2017
  35. 35. Epps Effect Non–Gaussian Dependencies Market States Conclusions Results usually stock market data analyzed “as if they were thrown on the floor” similarity measure reveals structural changes clear identification of market states time evolution of states followed −→ dynamical information changes during 2008–2009 crisis: correlation matrix and distributions early warning system ? Como, September 2017
  36. 36. Epps Effect Non–Gaussian Dependencies Market States Conclusions Conclusions Epps effect: it helps to look at the data copulae: the world of finance is not Gaussian market states: they exist, have time evolution and lifetime Como, September 2017
  37. 37. Epps Effect Non–Gaussian Dependencies Market States Conclusions M.C. M¨unnix, R. Sch¨afer and T. Guhr, Compensating Asynchrony Effects in the Calculation of Financial Correlations, Physica A389 (2010) 767 Impact of the Tick–size on Financial Returns and Correlations, Physica A389 (2010) 4828 Statistical Causes for the Epps Effect in Microstructure Noise, Int. J. Theoretical and Applied Finance 14 (2011) 1231 M.C. M¨unnix and R. Sch¨afer, A Copula Approach on the Dynamics of Statistical Dependencies in the US Stock Market, Physica A (2011) 4251 M.C. M¨unnix, T. Shimada, R. Sch¨afer, F. Leyvraz, T.H. Seligman, T. Guhr and H.E. Stanley, Identifying States of a Financial Market, Scientific Reports 2 (2012) 644 Como, September 2017

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