# The geometry of the market

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1 The geometry of the market

2 Contents 1 The Market Geometry Market metric Principal directions Effective dimensions A measure of structure New structure after Market networks Networks in and outside crises Connection intensities 3 Market shape: precursor of endogeneous crisis and return to normality Geometrical volume Skewness and kurtosis

3 1 - The Market Geometry r: return The metric r ( k ) = log( p ( k )) log( p ( )) p: price t k t 1 (From the Euclidean distance of the time series) d ij = 2(1 cij) (Mantegna & Stanley, 1999) Market space 1. The N stock coordinates are computed from the distances. Embedding space dimension = N-1 2. Find the center of mass R=Σ m(k) x(k) / Σ m(k) 3. Find coordinates in the center of mass system y(k) = x(k) - R 4. Construct the inertial tensor T ij =Σ m(k) y i (k) y j (k) 5. Find the eigenvectors and eigenvalues of T : (λ( i,e i ). Compare with the same construction for permuted data IBM d GE

4

5 Geometria Effective do Mercado dimensions Financeiro What is the number of dimensions of the manifold containing the systematic components of all stocks? S&P5 and Dow Jones, daily data 35 stocks, 1 years 249 stocks, 33 years 253 stocks, 22 years 7 stocks, 1 years 253 stocks, 35 years 424 stocks, 1 years In all cases: dimensions are enough! IBM The systematic component is contained in f dimensions << N Dimensions are not necessarily sectors. Impact on the choice of portfolios d GE

6 Geometria Effective do Mercado dimensions Financeiro Ordered and normalized eigenvalues λ+(1-λ*) Stocks Stocks λ : actual λ* : random λ + (1- λ*)..5 λ + (1- λ*)

7 Redes Shapede andmercado crises Shape changes in the market space Spherical market Typical in all projections for surrogate data or in business-as-usual periods.5 Jul.1998 to Jul.2 During crises there is distortion and reduction of volume Uti IT Fin Heal Cons Ind Ene

8 Redes Shapede andmercado crises Surrogate Data 253 stocks stocks.5.5 e e e1 e e e stocks stocks.5.5 Mar.1998 to Mar.28 e3 e e e e Uti IT Fin.5 Heal -.5 Cons Ind Ene

9 Geometria A structure do Mercado index Financeiro 5 23 stocks Structure index λ : actual λ : random S S t = i = 1 λ t '( i) λt ( i) 1 After 1997 deviations from sphericity become more frequent. What happened? S 1 17/4/71 7/1/7 3/3/82 2/9/87 12/3/93 2/9/98 23/2/4 15/8/ stocks A new regime after S>5

10 Redes Marketde networks Mercado Market networks: From a fully connected network to a sparse one 1. minimal spanning tree 2. L D = smallest distance in R that guarantees the network connectivity 3. hierarchical clustering Aug2 Industrial 73 Health Energy 28 Consumer 114 Financial 44 7 d d i, i, j j > L D 2 L D 2 b b i, j i, j = 1 = Technology Utilities 5 39 Few connections in normal periods

11 Redes Market de networks Mercado Market networks Jan28 Energy The economic agents are much more correlated during market crises Industrial 73 Health 28 Consumer 114 Financial 44 7 Technology Utilities 5 39 An increased number of connections, mostly inside sectors

12 Redes Market de networks Mercado Sep21 Energy 28 Industrial 73 Consumer 114 Health 44 Financial 7 Technology Utilities 5 39

13 Redes de Mercado Market networks Another structure index Another structure index Ratio of weak and strong links stocks From Jan.1998 to Mar.28 R R>.5 > = ), ( ), ( ), ( ), ( D t D t L j i d t L j i d t t j i d j i d R

14 Synchronization s s i i t = 1 d ( i, j) = otherwise L Redes Synchronization de Mercado D 2 Mar1998 The state depends on the existence of a strong link inside or outside the sector Sep21 Uti IT Fin Heal Cons Ind Ene

15 Redes Synchronization Topologia de Mercado Aug2 Dec27 Sep2 Jan28 Feb28 Uti IT Fin Heal Cons Ind Ene

16 Market shape as a precursor Redes of crises de Mercado and indicator of return to normality - Characterizing shape: - Volume, asymmetry and tails - Mardia s Skewness K 3 and Kurtosis K 4 K 3 = 1 n 2 { n n ( ) ( )} ' 1 X i X S X j X i= 1 j= 1 3 K 4 = 1 n { n ( ) ( )} ' 1 X X S X X i i i= The volume is the product of the first eigenvaues - K 3 and K 4 are computed after the dimensions are renormalized to have the same volume

17 Market shape as a precursor Redes of crises de Mercado and indicator of return to normality 1 2 w=8 2nd Black Monday: 27/1/97 w=8 2nd Black Monday: 27/1/97 V 1-3 K /9/97 17/9/97 27/9/97 7/1/97 17/1/97 27/1/97 /11/ /8/97 7/9/97 17/9/97 27/9/97 7/1/97 17/1/97 27/1/97 /11/97 1/11/97 w=8 2nd Black Monday: 27/1/ K 4 P S&P /9/97 17/9/97 27/9/97 7/1/97 17/1/97 27/1/97 /11/97 7/9/97 17/9/97 27/9/97 7/1/97 17/1/97 27/1/97 /11/97

18 Market shape as a precursor Redes of crises de Mercado and indicator of return to normality 1 2 w=8 Black Monday: 19/1/ w=8 Black Monday: 19/1/ K V /9/87 1/1/87 2/1/87 3/1/87 9/11/87 19/11/87 29/11/87 9/12/87 19/12/ /9/87 1/1/87 2/1/87 3/1/87 9/11/87 19/11/87 29/11/87 9/12/87 19/12/87 P S&P w=8 Black Monday: 19/1/ /9/87 1/1/87 2/1/87 3/1/87 9/11/87 19/11/87 29/11/87 9/12/87 19/12/87 K /9/87 1/1/87 2/1/87 3/1/87 9/11/87 19/11/87 29/11/87 9/12/87 19/12/87

19 Market shape as a precursor Redes of crises de Mercado and indicator of return to normality - From some of the cases studied so far it looks as if the shape of the market might provide some precursors of impending crises and would also be an indicator of when normality has returned. - However this is just work in progress. More to come.

20 Redes References de Mercado - RVM., T. Araújo and F. Louçã; Reconstructing an Economic Space from a Market Metric, Physica A 323 (23) T. Araújo and F. Louçã; The Geometry of Crashes: A Measure of the Dynamics of Stock Market Crises, Quantitative Finance 7 (27) T. Araújo and F. Louçã; The Seismography of Crashes in Financial Markets, Physics Letters A 372 (28) T. Araújo and F. Louçã; Tribes under Threat - The Collective Behavior of Firms During Stock Market Crises, Working paper ISEG 28/28, to appear in Int. Journ. of Industrial and Corporate Change - T. Araújo and F. Louçã; Trouble Ahead - The Subprime Crisis as Evidence of a New Regime in the Stock Market, Working paper ISEG 28/2, Economics Letters

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