Seeking Gold in Sand Financial applications of Random Matrix Theory in stock market data Mike S. Wang Faculty of Mathematics // Department of Chemistry University of Cambridge October, 2016 Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
Preview of the talk What is Random Matrix Theory? 1 Data Overview 2 Random Matrix Theory: The Marˇ cenko-Pastur Law 3 Mode & Clustering Analyses 4 A Multi-layer Structured Correlation Model & Its Predictions 5 Effect of Layer Division & an Excellent Match 6 Summary & Further Developments 7 Q&A Time 8 Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
1 What is Random Matrix Theory? Random Matrix Theory (RMT) is the study of matrices with random variable (r.v.) entries , e.g. X 11 X 12 · · · X 21 X 22 · · · . . . ... . . . . In particular, it concerns the emergent behaviours of random matrices in the asymptotic limit. Introduced by Wishart (1928), RMT gained prominence when Wigner (1950s) applied the theory to spacing of energy levels in nuclear physics. Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
1 What is Random Matrix Theory? Random Matrix Theory (RMT) is the study of matrices with random variable (r.v.) entries , e.g. X 11 X 12 · · · X 21 X 22 · · · . . . ... . . . . In particular, it concerns the emergent behaviours of random matrices in the asymptotic limit. Introduced by Wishart (1928), RMT gained prominence when Wigner (1950s) applied the theory to spacing of energy levels in nuclear physics. Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
1 What is Random Matrix Theory? Random Matrix Theory (RMT) is the study of matrices with random variable (r.v.) entries , e.g. X 11 X 12 · · · X 21 X 22 · · · . . . ... . . . . In particular, it concerns the emergent behaviours of random matrices in the asymptotic limit. Introduced by Wishart (1928), RMT gained prominence when Wigner (1950s) applied the theory to spacing of energy levels in nuclear physics. Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
2 Data overview: S&P 500 Procedure Price indices → logarithmic returns → de-meaned and normalised data. P stocks 126 . 8 30 . 5 · · · − 0 . 9 1 . 7 · · · 1 . 0 0 . 2 · · · 126 . 3 30 . 7 · · · 1 . 5 0 . 3 · · · 0 . 2 1 . 0 · · · �→ �→ T days . . ... . . . . ... ... . . . . . . . . . . . . � �� � � �� � � �� � raw data standardised data X correlation matrix E The log return is R i = log p i ( ≈ p i − p i − 1 ) , i > 1 p i − 1 p i − 1 where p i is the i -th trading day price index. The empirical correlation matrix is E = 1 T X t X . Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
3 Random Matrix Theory: the Marˇ cenko-Pastur law Covariance-correlation matrices are of fundamental importance to modern portfolio theory. They belong a class of random matrices called the Wishart ensemble. An important universality law for this ensemble in RMT: The Marˇ cenko-Pastur law If X : T × P has independently identically distributed (i.i.d.) r.v. entries with mean 0 and variance 1, then the limiting eigenvalue density function (e.d.f.) of matrix E = T − 1 X t X is � f ( λ ) = 1 ( λ + − λ )( λ − λ − ) 2 π r λ as P , T → ∞ and P / T → r ∈ (0 , 1), where λ ± = (1 ± √ r ) 2 . Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
3 Random Matrix Theory: the Marˇ cenko-Pastur law Covariance-correlation matrices are of fundamental importance to modern portfolio theory. They belong a class of random matrices called the Wishart ensemble. An important universality law for this ensemble in RMT: The Marˇ cenko-Pastur law If X : T × P has independently identically distributed (i.i.d.) r.v. entries with mean 0 and variance 1, then the limiting eigenvalue density function (e.d.f.) of matrix E = T − 1 X t X is � f ( λ ) = 1 ( λ + − λ )( λ − λ − ) 2 π r λ as P , T → ∞ and P / T → r ∈ (0 , 1), where λ ± = (1 ± √ r ) 2 . Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
3 The Marˇ cenko-Pastur law: a crude prediction underlying correlation matrix C predicted e.d.f. of E Mar ˇ cenko-Pastur distribution 0.9 √ ( λ + − x )( x − λ ) 0.8 1 f ( x ) = 2 π cx 0.7 c = P T 0.6 λ ± = (1 ± √ c ) 2 0.5 I P ⇒ f ( x ) 0.4 0.3 0.2 0.1 All noise!! 0 0 0.5 1 1.5 2 2.5 3 x histogram of observed eigenvalues 1.2 Mar ˇ cenko-Pastur distribution normalised eigenvalue density 1 0.015 0.01 0.8 market mode ? ⇒ 0.005 0.6 0 20 40 60 80 100 0.4 signals 0.2 0 0 1 2 3 4 5 6 7 8 eigenvalue Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
4 Mode analysis 0.08 Materials 0.07 Utilities Financials 0.4 Energy Telecom Consumer Staples 0.06 Consumer Discretionary 0.2 0.05 0.04 0 0.03 -0.2 Healthcare 0.02 Industrials IT Financials Materials -0.4 Energy 0.01 IT Telecom Consumer Staples Industrials Utilities Consumer Discretionary Healthcare 0 -0.6 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 450 The market mode (L: IPR 3 . 98 × 10 − 5 ) and the lowest mode (R: IPR 0.149). Localisation The inverse participation ratio is defined by IPR ( v ) = � P v i | 4 where ˜ i =1 | ˜ v is the vector v demeaned and normalised. Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
4 Clustering analysis Market mode removal: E ′ = E − λ 1 v 1 v t 1 ( λ 1 , v 1 largest eigenvalue pair); Dissimilarity distance: d ij = 1 − corr ( i , j ); � 1 Average linkage: D IJ = i ∈ I , j ∈ J d ij . | I || J | 1 ROST RSH 50 BBBY JWN 100 HD BBY 0.5 COST 150 strong clustering JCP 200 KSS WMT TGT 250 FDO LTD 0 300 TJX GPS 350 ANF WAG 400 HCBK -0.5 CVS 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 450 100 200 300 400 Dendrogram (L) Heatmap Minimum spanning tree (R) Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
5 A multi-layered correlation model: preview visualisation 1 histogram of observed eigenvalues 50 0.9 1.2 Mar ˇ cenko-Pastur distribution 0.8 normalised eigenvalue density 100 simulated analytic prediction 1 0.7 150 0.6 0.8 200 0.5 250 0.6 0.4 300 0.4 0.3 350 0.2 0.2 400 0.1 450 0 100 200 300 400 0 5 10 15 eigenvalue Heatmap (L) and analytic prediction (R) for the empirical e.d.f of a multi-layered model. Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
5 A multi-layered correlation model: analytic prediction Model : ν ( λ ) = P − 1 � P i =1 δ ( λ − λ i ), the e.d.f. of underlying correlation matrix C . Prediction : f ( λ ), the limiting e.d.f. of empirical correlation matrix E . The Stieltjes transform pair � ∞ d λ f ( λ ) G ( z ) = λ − z , f ( λ ) = lim ǫ → 0 Im { G ( λ + i ǫ ) } −∞ The Marˇ cenko-Pastur equation � ∞ 1 λν ( λ ) − G ( z ) = z − r d λ 1 + λ G ( z ) −∞ A polynomial equation P P P [1 + λ i G ( z )] = 1 � � � [1 + zG ( z )] λ i G ( z ) [1 + λ j G ( z )] . T i =1 i =1 j � = i Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
5 A multi-layered correlation model: analytic prediction Model : ν ( λ ) = P − 1 � P i =1 δ ( λ − λ i ), the e.d.f. of underlying correlation matrix C . Prediction : f ( λ ), the limiting e.d.f. of empirical correlation matrix E . The Stieltjes transform pair � ∞ d λ f ( λ ) G ( z ) = λ − z , f ( λ ) = lim ǫ → 0 Im { G ( λ + i ǫ ) } −∞ The Marˇ cenko-Pastur equation � ∞ 1 λν ( λ ) − G ( z ) = z − r d λ 1 + λ G ( z ) −∞ A polynomial equation P P P [1 + λ i G ( z )] = 1 � � � [1 + zG ( z )] λ i G ( z ) [1 + λ j G ( z )] . T i =1 i =1 j � = i Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
5 A multi-layered correlation model: analytic prediction Model : ν ( λ ) = P − 1 � P i =1 δ ( λ − λ i ), the e.d.f. of underlying correlation matrix C . Prediction : f ( λ ), the limiting e.d.f. of empirical correlation matrix E . The Stieltjes transform pair � ∞ d λ f ( λ ) G ( z ) = λ − z , f ( λ ) = lim ǫ → 0 Im { G ( λ + i ǫ ) } −∞ The Marˇ cenko-Pastur equation � ∞ 1 λν ( λ ) − G ( z ) = z − r d λ 1 + λ G ( z ) −∞ A polynomial equation P P P [1 + λ i G ( z )] = 1 � � � [1 + zG ( z )] λ i G ( z ) [1 + λ j G ( z )] . T i =1 i =1 j � = i Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
5 A multi-layered correlation model: analytic prediction Model : ν ( λ ) = P − 1 � P i =1 δ ( λ − λ i ), the e.d.f. of underlying correlation matrix C . Prediction : f ( λ ), the limiting e.d.f. of empirical correlation matrix E . The Stieltjes transform pair � ∞ d λ f ( λ ) G ( z ) = λ − z , f ( λ ) = lim ǫ → 0 Im { G ( λ + i ǫ ) } −∞ The Marˇ cenko-Pastur equation � ∞ 1 λν ( λ ) − G ( z ) = z − r d λ 1 + λ G ( z ) −∞ A polynomial equation P P P [1 + λ i G ( z )] = 1 � � � [1 + zG ( z )] λ i G ( z ) [1 + λ j G ( z )] . T i =1 i =1 j � = i Mike S. Wang Seeking Gold in Sand with Random Matrix Theory
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