How do financial correlations grow? How do financial correlations grow? C. Borghesi Borghesi (Paris), S. (Paris), S. Micciche Micciche’ (Palermo), MM (Trieste) ’ (Palermo), MM (Trieste) C. � Financial correlations Financial correlations � � Noise and structure Noise and structure � � Dynamics: evolution and formation Dynamics: evolution and formation � � How fast is information aggregated across stocks? How fast is information aggregated across stocks? � � Early results: Market forms as an embryo Early results: Market forms as an embryo � � Here: HF data from NYSE, LSE, PB Here: HF data from NYSE, LSE, PB � ∆ t ∆ t market develops as a baby ( ∆ > ∆ internal dynamics → → market develops as a baby ( � internal dynamics t > t 0 ) 0 ) � � Structure of overnight returns markedly different Structure of overnight returns markedly different � � Summary of results Summary of results �
Financial correlations Financial correlations x PFE x GE T = window size t 0 = initial time ∆ t = time scale τ = time shift
The correlation matrix The correlation matrix Log-returns: + T t [ ] 1 ∑ 0 ∆ = − ∆ ∆ = ∆ = t t t K ( ) log ( ) ( ) , ( ) 1 , , x t p t p t t x x t i N i i i i i T = t t 0 Pearson coefficient/sample correlation: [ ][ ] ∑ + ∆ ∆ ∆ ∆ t T − + τ − t t t t 0 ( ) ( ) x t x x t x = i i k k τ ∆ = = t t K ( , , , ) , , 1 , , C t T t 0 i k N [ ] [ ] , 0 i k ∑ ∑ + + 2 2 t T ∆ ∆ t T ∆ ∆ − − t t t t 0 0 ( ) ( ) x t x x t x = i i = k k t t t t 0 0 Questions: Structure: i,k Noise dressing: T Evolution: t 0 Efficiency τ Growth: ∆ t
Noise or real correlations? Noise or real correlations? � Eigenvalue Eigenvalue distribution distribution � and random matrix theory and random matrix theory ( (Laloux Laloux et al./ et al./ Gopikrishnan Gopikrishnan et al. …) et al. …) Λ � The bulk of The bulk of eigenvalue eigenvalue � distribution is dominated by distribution is dominated by sampling effects (noise) sampling effects (noise) � One large One large eigenvalue eigenvalue � (market mode) (market mode) � Few Few eigenvalues eigenvalues with with � (N/T) 1/2 “economic” meaning “economic” meaning � “Localized” small “Localized” small � eigenvalues eigenvalues
Structure Structure ∆ t ( ∆ t= 1 day, N~ T~ 500) = 1 day, N~ T~ 500) ( � Eigenvectors analysis Eigenvectors analysis � ( (Gopikrishnan Gopikrishnan et al. ) et al. ) � Minimal spanning trees Minimal spanning trees � ( (Mantegna Mantegna et al. ) et al. ) � There is significant non There is significant non- - � trivial structure in C trivial structure in C
Cluster structure Cluster structure Idea: = 1 if i= j C i,j = = C s if s i = s j = C 0 else Model: s i = sector to which = µ + η + − − ε 2 2 stock i belongs ( ) ( ) ( ) 1 ( ) x t a t g t a g t i i s s i s i i i i Maximize likelihood: ⎡ ⎤ − 2 1 { } ( ) c n c ∑ = − + − s s s log data | structure ⎢ log 1 log ⎥ P n − s 2 2 ⎣ ⎦ n n n > s : n 0 s s s s
S&P500 clusters ~ sectors n s Group: size/c/g/e 18 115.202408 0.465534151 -4.64141703 Computers Group: size/c/g/e 24 190.345795 0.431334049 -6.17717028 Oil & Gas AMAT 247 Applied Materials Equipment (Semiconductor) ENE 710 Enron Corp. Natural Gas TXN 235 Texas Instruments Electronics (Semiconductors) SLB 395 Schlumberger Ltd. Oil & Gas (Drilling & Equipment) NSM 235 National Semiconductor Electronics (Semiconductors) RDC 395 Rowan Cos. Oil & Gas (Drilling & Equipment) INTC 235 Intel Corp. Electronics (Semiconductors) HAL 395 Halliburton Co. Oil & Gas (Drilling & Equipment) BHI 395 Baker Hughes Oil & Gas (Drilling & Equipment) AMD 235 Advanced Micro Devices Electronics (Semiconductors) TX 390 Texaco Inc. Oil (International Integrated) SUNW 190 Sun Microsystems Computers (Hardware) RD 390 Royal Dutch Petroleum Oil (International Integrated) IBM 190 International Bus. Machines Computers (Hardware) Group: size/c/g/e 8 29.0933895 0.604280651 -2.01765895 CHV 390 Chevron Corp. Oil (International Integrated) HWP 190 Hewlett-Packard Computers (Hardware) SGP 285 Schering-Plough Health Care (Drugs-Major Pharmacs) P 385 Phillips Petroleum Oil (Domestic Integrated) CPQ 190 COMPAQ Computer Computers (Hardware) PFE 285 Pfizer, Inc. Health Care (Drugs-Major Pharmacs) OXY 385 Occidental Petroleum Oil (Domestic Integrated) AAPL 190 Apple Computer Computers (Hardware) MRK 285 Merck & Co. Health Care (Drugs-Major Pharmacs) AHC 385 Amerada Hess Oil (Domestic Integrated) ORCL 185 Oracle Corp. Computers (Software & Services) LLY 285 Lilly (Eli) & Co. Health Care (Drugs-Major Pharmacs) UCL 380 Unocal Corp. Oil & Gas (Exploration & Productn) NOVL 185 Novell Inc. Computers (Software & Services) KMG 380 Kerr-McGee Oil & Gas (Exploration & Productn) JNJ 280 Johnson & Johnson Health Care (Diversified) Group: size/c/g/e 5 20.0271244 3.02181792 -4.17928696 MSFT 185 Microsoft Corp. Computers (Software & Services) BR 380 Burlington Resources Oil & Gas (Exploration & Productn) BMY 280 Bristol-Myers Squibb Health Care (Diversified) PDG 265 Placer Dome Inc. Gold & Precious Metals Mining XON 0 EXXON CORP CA 185 Computer Associates Intl. Computers (Software & Services) AHP 280 American Home Products Health Care (Diversified) NEM 265 Newmont Mining Gold & Precious Metals Mining SNT 0 SONAT INC MOT 180 Motorola Inc. Communications Equipment ABT 280 Abbott Labs Health Care (Diversified) HM 265 Homestake Mining Gold & Precious Metals Mining PZL 0 PENNZOIL CO DIGI 0 DSC COMM CORP ABX 265 Barrick Gold Corp. Gold & Precious Metals Mining c s ORX 0 ORYX ENERGY CO DEC 0 DIGITAL EQUIPMEN ECO 0 ECHO BAY MINES Gold & Precious Metals Mining MOB 0 MOBIL CORP ACAD 0 AUTODESK INC LLX 0 LOUISIANA LAND HP 0 HELMERICH & PAYN DI 0 DRESSER INDUS ARC 0 ATL RICHFIELD CO AN 0 AMOCO CORP n s Data from R.N. Mantegna
Hierarchical clustering of assets Hierarchical clustering of assets (N= 2000 NYSE 90- -98): 98): (N= 2000 NYSE 90 • Statistically significant • Zipf’s law • no orthogonality “noise level” Data from R.N. Mantegna Note: totally unsupervised method Number of clusters not predefined!
Time evolution (t 0 ): Persistence Time evolution (t 0 ): Persistence δ t 1500 days, δ (Onnela Onnela et al. 2003, T~ 500 et al. 2003, T~ 500÷ ÷ 1500 days, t 0 = 21 days) ( 0 = 21 days) More than 80% of the links of the MST are conserved from one time window to the next The fraction of conserved links has algebraic decay with time lag between windows (no finite memory)
Time evolution (t 0 ): Recurrence Time evolution (t 0 ): Recurrence (MM ‘02) (MM ‘02) Daily return day t assets IBM, AAPL,HWP,…. GM,F,…. PDG, NEM,ABX,…. SGP,PFE,MRK,… { Computers,… cars,… gold/mining,… health,… Do blue days, red days, … exist or is market activity following a continuum
Market states: Clustering days Market states: Clustering days = ⎧ 1 ' t t ⎪ 1 N ∑ = = = = ≡ ⎨ Idea : ( , ' ) ( ) ( ' ) C t t x t x t g s s s ' i i s t t N ⎪ = 1 ≠ i ⎩ 0 s s ' t t maximize likelihood on the class of matrices of this form Results: log likelihood/asset • Market states do exist • scale free frequency distribution for most frequent clusters of days • Identify market states ω ={s ω,1 ,s ω,2 ,…} days
Conditional dynamics Dynamics after crashes <r| ω > = average return in state ω
τ ) equal time correlations ( τ Non- -equal time correlations ( ) Non (B. Toth Toth + J. + J. Kertesz Kertesz 2005) 2005) (B. C(big today , small tomorrow ) Correlation lag τ → market is getting more and more efficient
Growth Growth ∆ t How does market structure forms as ∆ t grows? grows? How does market structure forms as Early results Early results
The Epps effect The Epps effect ∆ t correlation grows with ∆ t correlation grows with Information is aggregated faster today than in the past in bigger companies (J. Kwapien, S. Drozdz, J. Speth 2003)
Market as an embryo: Market as an embryo: growth and differentiation growth and differentiation ∆ t= 1/20 day ∆ t= 1/10 day ∆ t= 1 day (Bonanno Bonanno et al. 2004, et al. 2004, ( Tumminello et al. 2006) et al. 2006) Tumminello ∆ t= 1/5 day ∆ t= 1/2 day + number of eigenvalues out of the noise band increases with ∆ t → structure forms as ∆ t increases
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