Final project: Exploring the structure of correlation

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