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Exploring Correlation Structures: Empirical Studies & Modeling Analysis

Delve into the structure of correlation through empirical data analysis and Copula Factor modeling. Uncover stylized facts on memory effects, fractal analysis, and asymmetry in correlations. Understand the relationship between betas and correlations, and evaluate factor models to interpret and select suitable models. Explore the distribution of correlation and the effectiveness of one-factor models. Discover the implications of Gaussian and T-distribution copulas, trading strategies, and the potential for implied versus historical correlation. This comprehensive study presents conclusions and suggests avenues for further research in financial correlation analysis.

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Exploring Correlation Structures: Empirical Studies & Modeling Analysis

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  1. Final project:Exploring the structure of correlation Forrest White, Jason Wei Joachim Edery, Kevin Hsu Yoan Hassid MS&E 444 - 06/02/2010 MS&E 352 - 2/25/2010

  2. Stylized facts Conclusion • Verification of empirical facts on correlation • Data : 15min closing prices from Jan 2007 to Jan 2009 of the S&P 500 Copula Factor model Stylized facts 2

  3. Eppseffect Conclusion • empirical correlations virtually disappear at high frequency • tradingasynchronous • Eppseffectobserved but data still significant Copula Factor model Stylized facts 3

  4. Memory effect and fractal analysis Conclusion • Time series IC & AIC (instantaneouscorrelation) • AverageInstantaneouscorrelation : • DetrentedFluctualAnalysis : • interpretation of H2 as Hurst exponent: • 0.5<H2<1 : long-range memory • 0<H2<0.5 : mean-reverting • H2 = 0.5 : no memory (Brownian motion) Copula Factor model Stylized facts 4

  5. Memory effect and fractal analysis Conclusion Copula • long-range memory for correlation on average • behavior close to gaussian for pairwise Factor model • Multi-fractal behavior • Asymmetricshape Stylized facts 5

  6. Correlations vs absolute returns Conclusion • Expect big correlation for extreme return periods Copula Factor model Stylized facts 6

  7. Asymmetry in Correlations Conclusion • Expect asymmetry for extreme negative return periods vs extreme positive return periods Copula Factor model • Time period may be too short Stylized facts 7

  8. Beta vs Correlations Conclusion • Stocks with the same betas show higher correlation Copula • High Beta Mid Beta Low Beta Factor model • Low Beta Mid Beta High Beta Stylized facts 8

  9. Factor model Conclusion • Compute the scores/loadings with a PCA • Model values : Xi(t) ≈ βiV1(t)+ γiV2(t) + δiV3(t) … • Correlation : ρij ≈ ρiV1 ρjV1 + ρiV2 ρjV2 +… Copula Factor model Stylized facts 9

  10. Distribution of correlation Conclusion Copula Factor model • empirical distribution : t-distribution fitsbetter • 1 factor model : normal distribution • closer normal fit when time scale of returnsincreases Stylized facts 10

  11. One factor model Conclusion • The one factor model works, on average! • It tends to underestimate correlation for stocks of the same nature (sectors, betas…) Copula Factor model Stylized facts 11

  12. Factor model Conclusion • Interpretation Copula Factor model Stylized facts 12

  13. Factor model Conclusion • Selection Copula Factor model Stylized facts 13

  14. Factor model Conclusion • Results • Consumer d. Energy Materials Copula • Materials Energy Consumer d Factor model Stylized facts 14

  15. Copula Conclusion • Marginals + copula  Joint distribution • Sklar’s theorem, other properties • Gaussian copula : Copula • Easy but bad tail fitting • Empirical ρ : 45%Optimal ρ : 60% Factor model Stylized facts 15

  16. Copula Conclusion • Gaussian is ok for low returns • T-distribution  T-copula ? Copula Factor model Stylized facts 16

  17. Conclusion Conclusion • Some empirical facts in correlation can be captured with a low dimension model • The Gaussian copula is very limited • Trading strategies exist to take advantage of patterns • Further studies • Implied correlation vs historical correlation? • Different time periods • Higher frequencies Copula Factor model Stylized facts 17

  18. Q&A Conclusion • Thank you Copula Factor model Stylized facts 18

  19. Memory effect and fractal analysis Conclusion • Time series IC & AIC (instantaneouscorrelation) • normalizedreturns : • Instantaneouscorrelation : • AverageInstantaneouscorrelation : Copula Factor model Stylized facts 19

  20. Memory effect and fractal analysis • DetrentedFluctualAnalysis • , with A=IC or A= AIC • DFA functions : • qthorder of detrendedfunction : • power lawbehavior : • interpretation of H2 as Hurst exponent: • 0.5<H2<1 : long-range memory • 0<H2<0.5 : anti-persistent • H2 = 0.5 : no memory (Brownian motion) Conclusion Copula Factor model Stylized facts 20

  21. Memory effect and fractal analysis Conclusion Copula Factor model • long-range memory for correlation on average : persisentbehavior, possible predictability • behavior close to gaussian for pairwisecorrelation Stylized facts 21

  22. Memory effect and fractal analysis • Hq non constant : multifractality of signal • Signal complex and turbulent withinhomogeneities in properties • Spectrum of singularities : • Asymmetry in spectrum => Conclusion Copula Factor model Stylized facts 22

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