Final project: Exploring the structure of correlation Forrest White, Jason Wei Joachim Edery, Kevin Hsu Yoan Hassid MS&E /02/2010

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

Epps effect empirical correlations virtually disappear at high frequency trading asynchronous Epps effect observed but data still significant Stylized facts Factor model Copula Conclusion 3

Memory effect and fractal analysis Stylized facts Factor model Copula Conclusion 4 Time series IC & AIC (instantaneous correlation) Average Instantaneous correlation : Detrented Fluctual Analysis : 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)

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

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

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

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

Factor model Stylized facts Factor model Copula Conclusion 9 Compute the scores/loadings with a PCA Model values : X i (t) ≈ β i V 1 (t)+ γ i V 2 (t) + δ i V 3 (t) … Correlation : ρ ij ≈ ρ iV1 ρ jV1 + ρ iV2 ρ jV2 +…

Distribution of correlation Stylized facts Factor model Copula Conclusion 10 empirical distribution : t-distribution fits better 1 factor model : normal distribution closer normal fit when time scale of returns increases

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

Factor model Stylized facts Factor model Copula Conclusion 12 Interpretation

Factor model Stylized facts Factor model Copula Conclusion 13 Selection Healthxxx Utilitiesxxx Financexxxx consumer dxxx consumer sxxx industrialsxxx info techxxxx materialsxxxx telecomxxx energyxxxxxx

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

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

Stylized facts Factor model Copula Conclusion 16 Copula Market absolute log-returnsML Gaussian copulaML T copuladfRelative difference < 0.30% (0-20% quantile) % < 0.58% (0-40% quantile % < 1.00% (0-60% quantile) % < 1.65% (0-80% quantile) % all % Gaussian is ok for low returns T-distribution  T-copula ?

Conclusion Stylized facts Factor model Copula Conclusion 17 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

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

Memory effect and fractal analysis Stylized facts Factor model Copula Conclusion 19 Time series IC & AIC (instantaneous correlation) normalized returns : Instantaneous correlation : Average Instantaneous correlation :

Memory effect and fractal analysis Stylized facts Factor model Copula Conclusion 20 Detrented Fluctual Analysis, with A=IC or A= AIC DFA functions : qth order of detrended function : power law behavior : 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)

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

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