The study of Thai stock market across the 2008 financial crisis
TThe study of Thai stock market across the 2008 financial crisis
K. Kanjamapornkul, ∗ Richard Pinˇc´ak,
2, 3, † and Erik Bartoˇs ‡ Department of Computer Engineering, Faculty of Engineering,Chulalongkorn University, Pathumwan, Bangkok 10330, Thailand Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 043 53 Koˇsice, Slovak Republic Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia Institute of Physics, Slovak Academy of Sciences,D´ubravsk´a cesta 9, 845 11 Bratislava, Slovak Republic
The cohomology theory for financial market can allow us to deform Kolmogorov space of timeseries data over time period with the explicit definition of eight market states in grand unifiedtheory. The anti-de Sitter space induced from a coupling behavior field among traders in case of afinancial market crash acts like gravitational field in financial market spacetime. Under this hybridmathematical superstructure, we redefine a behavior matrix by using Pauli matrix and modifiedWilson loop for time series data. We use it to detect the 2008 financial market crash by usinga degree of cohomology group of sphere over tensor field in correlation matrix over all possibledominated stocks underlying Thai SET50 Index Futures. The empirical analysis of financial tensornetwork was performed with the help of empirical mode decomposition and intrinsic time scaledecomposition of correlation matrix and the calculation of closeness centrality of planar graph.
Keywords: cohomology group, empirical mode decomposition, general equilibrium, time series, tensor net-work
I. INTRODUCTION
Recent studies [1–5] suggest that financial market is a complex dynamical system with underlying nonlinear andnonstationary financial time series data [6]. When market crashes, the time series data of price of stock will containwith systemic shift and display non-equilibrium entanglement state. The investigation of the 2008 market crash statein nonlinear and nonstationary financial time series data empirically [7, 8], is one of the central objectives in thestudy of a behavior of traders in financial market and it can be useful model to detect future market crash states.In a spacetime of differential geometry, it is possible to use a cohomology group to deform the Kolmogorov space intime series data to detect a dark state in financial market crash. Differential geometry of nonstationary and nonlinearpredictor and predictant states in mirror symmetry of time series data is one of the active researches. We assume thata nonstationary state can temporally deform the spacetime [9] into a homotopic class of supersymmetry breaking ofa supersymmetry of dark states [10]. The market state is decomposed into a coupling state between two linear andstationary states [11]. One of them is an equivalent of the path of a predictor state of time series of observation. Theother is a coincident path of an expectation path of dark state for predictant state of forecasting result evaluation inextradimensions of Kolomogorov space in time series data [12].In Walrasian microeconomics the existence of equilibrium paths in the dynamic economic system requires theequilibrium paths of the complete system to coincide with the equilibrium in economics [13] under linearly andstationary assumption of market state space model or dynamic stochastic general equilibrium [14] with nonstationarystate as a noise in financial market [15]. Under this assumption, the Brouwer’s fixed-point theorem is used to provethe existence of equilibrium price vector, in which it is not known that the algebraic topological structure underlyingmarket is based on an invariant structure of degree over the covering map between cohomology group of sphere [16].The economics is a source of definition of the Walrasian utility function in general equilibrium between supplyand demand [17] and the topology is concerned with global shape of space and, in particular, its finite or infiniteextension [18]. The result of interaction of these two theories in the study of time series in the nonstationary state is socalled cohomology theory for financial market [19]. A cohomology theory [20] is a mathematical branch of differentialgeometry and algebra used to explain anti-de Sitter (AdS) space, it is rich of powerful tools of Cartan calculus ofhyperbolic geometry and gauge theory [21, 22]. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ q -f i n . S T ] J un A general equilibrium price in financial market can be realized as cohomology sequence [23] of short exact sequencesof Kolmogorov space between Riemannian manifold. A generalization of complex plane of equivalent class of supplyand demand curve interacts with Wilson loop of Pauli matrix. It is a spinor field of time series data interpreted as abehavior matrix of traders. Every spinor field of non-orientation state of time series data can be written in the formof quantum triplet state [24] in framework of new cohomology theory for financial market.The equilibrium price of data recorded from stock market can provide a source of arbitrage opportunity derivedfrom market nonequilibrium state between supply and demand side. After all behavior traders found the opportunityto gain a profit from financial market, the chance will disappear according to efficient market hypothesis (EMH). Theorientation of stock Index Futures market is induced from the orientation of quaternionic field in time series data ofunderlying stocks, in which we can use the average correlation [25] as a standard tool for financial network analysis[26, 27]. The ultimate goal of the stock market microstructure prediction [28] is to find an arbitrage change to detectthe general equilibrium point of purchasing power parity [29]. It seems that precise definition of a nonstationary stateof time series by the projective approach of a hyperbolic space can help to analyze the non-equilibrium state or thecrash state in the financial market.A hyperstructure of non-Euclidean geometry of the behavior of a trader in a financial market [31] can induce aspin structure of principle bundle of a correlation matrix by usage of a cocycle over tangent of K¨ahler manifold.The underlying financial market of such introduced physical quantity of arbitrage opportunity can be realized as anew complicated topological structure related to the curvature of a spacetime of time series data inside an isometryof group action over tangent of higher topological space of market, such as Lie group of predictor and predictantappearing as an evolution feedback path of expectation market state in the Riemanian manifold. It can be generalisedto K¨ahler manifold or Calabi-Yau manifold, with the complex structure of a metric tensor in which it opens thebridge between Yang-Mill theory and the field of expectation induced by traders behaviour to the arbitrage amongtraders in a financial market. The cohomology sequence of Kolmogorov space of time series data can be inducedfrom a differential 2-form over tangent of K¨ahler manifold. It is an equivalent class of the path between predictorand predictant topological group. The Lie group structure of a correlation matrix induces its tangent manifold as Liealgebra as space of behavior of traders in financial market with a spinor field of double covering space of Kolmogorovspace of time series data. This structure can explain the fix point or equilibrium point in financial market network.In quantum physics, all measurement quantities are associated with Hermitian operator with its eigenvalues. Thehermitian property of a correlation matrix induces an asymmetric property of isometry or inertia frame of referencein space of time series data.The paper is organized as follows. In Section II we introduce the basic definition of a model of trader behaviorand new theoretical construction of de Rahm cohomology for financial market. We also define the another form ofcohomology group for physiology of time series data so called knot cohomology in time series data. At the end ofthis section we define an explicit form of eight market states in Kolmogorov space. In Section III we perform theempirical analysis of tensor correlation using Hilbert-Huang transform from nonstationary and non-orientation stateto stationary and orientation state in time series data before sending the result to the planar graph algorithm to builda financial tensor network. The result of tensor network is used to compute the closeness centrality of hyperbolicangle for the detection of a market crash state in time series data. In Section IV we show the plots of the results ofempirical analysis for daily close price of SET50 Index Futures with average of tensor network of correlation over 42stocks underlying Index Futures with goal to detect the non-equilibrium state, i. e., the 2008 financial market crash.We use the empirical analysis method of partial correlation matrix as a main tool. In Section V we discuss and makea common conclusion about the result of theoretical derivation of cohomology theory and the empirical result of adetection of 2008 financial market crash with tensor network analysis.
II. COHOMOLOGY THEORY IN TIME SERIES DATA
Let T n be the n -dimensional torus with ( S ) n = S × S × · · · S , the product topology of unit circle into T n agroup structure of space time series data. Let k -th homology group H k ( T n ) be a free abelian group of rank C nk . Wedefine the Poincare polynomial of a space of time series data X t as P X t .In classical analysis of time series x t ∈ X ⊂ R n , X is a compact subset, the Poincare polynomial is P X t = 1 . (1)The usage of Poincare polynomial in space of time series data is for measuring the deformation of distance betweenspace of time series and time scale of time series. For classical time series data we then get P X t = 1 = t − t = t − t = · · · . (2)If X t = S we get P X t = x , that means P X t = 1 + x, (3)then t − t = 1 + ( t − t ) . The distance in time series on sphere is not constant. But it is an equivalent class of complex projective space byusage of stereoprojection from real line to unit sphere, where ∞ is send to 1 in north pole in the sense of the contextof Poincare polynomial of time series data. There does not exist a cohomology group of space of time series data inwhich we can deform space over time period in time ordering like time series data. Our main goal is to create suchcohomology theory.Let X t be a Kolmogorov space of time series data. There exists the relationship between homotopy class [ S n , X t ]and cohomology group H n ( X t ) in algebraic topology[ S n , X t ] = H n ( X t ) := H n ( x t ; [ G, G ∗ ]) (4)and in this case we set [ G, G ∗ ] = Z / x → x → x → · · · x n , we induce a cohomology theory for financial market for timeseries data in which we can measure the invariance property of space of time series over time. We use the notationfor this new type of cohomology of time as HHH nt ( X ) HHH nt ( X t ) : H n ( x ; [ G, G ∗ ]) → H n ( x ; [ G, G ∗ ]) → H n ( x ; [ G, G ∗ ]) → · · · → H n ( x t ; [ G, G ∗ ]) ,HHH n − t − ( X t ) : H n − ( x ; [ G, G ∗ ]) → H n − ( x ; [ G, G ∗ ]) → H n − ( x ; [ G, G ∗ ]) → · · · → H n − ( x t − ; [ G, G ∗ ]) , · · · (5) HHH ( X t ) : H ( x ; [ G, G ∗ ]) → H ( x ; [ G, G ∗ ]) · · · with 0 → HHH nt ( X t ) → HHH n − t − ( X t ) → · · · → · · · → HHH ( X t ) → . (6)The cohomology in this context is used to measure the high dimensional invariance property of physiology of spacetime series data at the end point of time series measurement. Definition 1.
Let G be a Lie group of predictor, the path of result of prediction. Let G ∗ be a Lie group of predictant,the path of evaluation of result of prediction without of sample data. A time series x t is a path connected evolutionfeedback component with correct expectation path if and only if for every n, t > HHH nt ( X t ) = HHH nt ( x t ; [ G, G ∗ ]) = { [0] } (7) for every n -dimension of sphere to measure the invariant property of space over time scale. A spin group is a principle bundle with double covering and a following sequence is well known1 → Z / → Spin(2) = U(1) → SO(2) → Definition 2.
An equivalent class of up [ up ( x t )] ∈ Z / and down [ down ( x t )] ∈ Z / orientation state of time seriesdata is an equivalent class of Spin(2) over
SO(2) / Spin(2) = Z / in short exact sequence in Eq. (8). The exact sequence Eq. (8) can be applied to every dimensions, e. g., for n = 3 we have a high dimensional sphere S with Hopf fibration as a space underlying hidden dynamical system of financial market. If we have two cuttinglines L , L not perpendicular to each other in S , where S / Spin(3) (cid:39) H P , the supply S = L ( x, a , b ) ∈ H anddemand D = L ( y, a , b ) ∈ H of financial market intersects each other. This two lines will twist across each otherin equilibrium state like M¨obius operator in Markov trace. In equilibrium state we can transfer the measurementfrom one line to another line by using projective geometry, simply by interchange of coordinate system of supply S to demand side D of financial market by |S ( p ) | = |D ( p ) | (9) (a) (b) FIG. 1: (a) The knot model of time series data in knot cohomology model between D-brane and anti-D-brane of time seriesdata, (b) Wilson loop of time series data in the form of entanglement state coupling between predictor and predictant statesin end point of time series data. where p is a price. Let D = ln |D| and S = ln |S| so we have ln |S ( p ) | = ln |D ( p ) | . If D = ( x , x , x , · · · ) and S = ( y , y , y , · · · ) we induce a projective coordinate z = SD = xy = λx λy −→ yx = DS = 1 z . (10) Definition 3.
A Wilson loop of time series data, denoted as W x t ( σ i ) is a twistor in complex projective space C P . Itturn one side of Euclidean plane twist into another side behind the plane with a help of modified M¨obius map z = z . Therefore, we have W x t ( λSλD ) = DS . The equivalent class of supply and demand S ∈ [ s ], D ∈ [ d ] is induced fromthe gluing process of 0 to ∞ by modified M¨obius map z = z , such that [ s ] ∼ [ d ] if and only if there exist λ i where x i = λ i y i . The projective coordinate is a local coordinate on Minkowski space. Let g ij be Jacobian matrix of supplyand demand, we have a shallow to hidden state of transformation by using Wigner ray transform g ij (cid:55)→ λg ij , λ ∈ U (1) (cid:39) Spin(2) (11)where λ = σ x and σ x are Pauli matrices in Spin(2) group. Definition 4.
Let S be supply linear compact operator in Banach space and D be demand operator. A Wigner ray oftime series data at general equilibrium point in financial time series data is a point of price x t ( S, D ) such that thereexist a ray of unitary operator λ ∈ SU(2) (cid:39)
Spin(3) W x t < S, D > := < λS, λD > = λ < S, D > = < S, D > . (12)We introduce three types of Killing vector field for market potential field of behavior of traders in market. A ± , ( f ) = ∂∂f : SO(2) → Spin(2) : M → T x MA ± , ( σ ) = ∂∂σ : SO(2) → Spin(2) : M → T x M (13) A ± , ( ω ) = ∂∂ω : SO(2) → Spin(2) : M → T x M where “+” means optimistic behavior of trader, “ − ” means pessimistic trader, f is a fundamentalist trader, σ is anoise trader and ω is a bias trader. A is an agent field of adult behavior of trader as fundamentalist, A is an agentfield of teenager behavior of trader as herding behavior or noise trader, A is an agent field of child behavior of traderor bias trader. The behavior states are spin up or down as the expected states in market communication layer oftransactional analysis framework [32]. The strategy of fundamentalist trader is to expect price in which period priceis in minimum point s and maximum point s (crash and bubble price). Then the optimistic fundamentalist f + willbuy at minimum point s of price below fundamental value and the pessimistic fundamentalist f − will sell productor short position at maximum point of price if the maximum point is over a fundamental value. The strategy of noisetrader σ ± is different from fundamentalist. The up state is signified as optimistic market expected state and downstate as pessimistic as market crash state of intuition state of forward looking trader [33]. An equivalent class ofsupply induced from interaction of behavior of fundamentalist and noise trader in mirror symmetry model of expectedon arbitrage opportunity over physiology of time series data in entanglement states between s , s and s ∗ , s ∗ is shownin Fig. 1a (the definition of s , s , s ∗ and s ∗ can be found in [12]).We can use Pauli matrix σ i ( t ) to define a strategy of all agents in stock market as basis of quaternionic field spanby noise trader and fundamentalist. Let +1 stands for a buy at once and − i be inert tobuy and − i be inert to sell. The row of Pauli matrix represent the position of predictor and the position in columnrepresent the predictant state. Definition 5.
For noise trader, we use finite state machine for physiology of time series to accepted pattern definedby A := σ ( t ) = σ y = s ∗ ( t ) s ∗ ( t ) s ( t ) 0 − is ( t ) i = s ∗ ( t ) s ∗ ( t ) s ( t ) 0 σ + s ( t ) σ − (14) where σ y is a Pauli spin matrix. Definition 6.
For fundamentalist trader, we use finite state machine for physiology of time series to accepted patterndefined by A := f ± = − W ( σ z ) = s ∗ s ∗ s ( t ) 0 − s ( t ) 1 0 = s ∗ s ∗ s ( t ) 0 f − s ( t ) f + (15)Let ω be a bias behavior A of market micropotential field. From [ σ y , σ z ] = 2 iσ x follows[ f, − W − ( σ )] = 2 iω, (16)where W is a Wilson loop (knot state between predictor and predictant) for time series data and W − is inverse ofWilson loop (unknot state between predictor and predictant). We have an entanglement state (Fig. 1b) induced fromstrategy of noise trader as herding behavior explained by s − i → s ∗ , s i → s ∗ (17)with W x t (cid:18) (cid:20) s s ∗ (cid:21) (cid:19) = (cid:20) s ∗ s (cid:21) (18) A. de Rahm coholomogy for financial market
In this section we explain the source of second cohomology group of spinor field in financial time series data. Itcomes from the entanglement state of field of behavior of trader A i with respect to an equivalent class of error ofexpectation of out coming of physiology of time series data ∂ [ s ] = [ s ] ∗ − [ s ] and ∂ [ s ] = [ s ] ∗ − [ s ].The mathematical structure in this section is related to Chern-Simon theory so called gravitational field in threeforms of connection F (cid:53) ∈ Ω ( T x M ⊗ T ∗ x M ) over three vector fields in financial market, Ψ i ∈ M a market state field, A i is an agent behavior field and [ s i ] is a field of physiology of financial time series data over Kolmogorov space. Let A = ( A , A , A ) be market micro potential field of agent or behavior of trader in market. Let Ψ ( S , D ) i ([ s i ] , [ s j ]) be afield of market 8 states of equivalent class of physiology of time series [ s i ] , [ s j ].In financial market microstructure [34] we define a new quantity in microeconomics induced from the interactionof order submission from supply and demand side of the orderbook, so called market micro vector potential [28] twist FIG. 2: The order submission as time series data in complex projective space. field (Fig. 2) or connection in Chern-Simon theory for financial time series A ( x, t ; g ij , Ψ ( (cid:126) S , (cid:126) D ) i , s i ( x t )), where Ψ ( (cid:126) S , (cid:126) D ) i isa state function for supply (cid:126) S and demand (cid:126) D of financial market. Let Ψ ( (cid:126) S , (cid:126) D ) i ( F i ) be a scalar field induced from thenews of market factor F i , i = 1 , , ,
4. Let x t ( A , x ∗ , t ∗ ) be a financial time series induced from the field of marketbehavior of trader, where x ∗ and t ∗ are hidden dimensions of time series data in Kolmogorov space. Definition 7.
Let S be a supply potential field defined by rate of change of supply side of market potential fields ofbehaviour trader in induced field of behavior of agent A in stock market. (cid:126) S = − ∂ A ∂ [ s ] , (cid:126) N S = − i ∂ A ∂ [ s ] , (cid:126) D = ∂ A ∂ [ s ] (cid:126) N D = + i ∂ A ∂ [ s ] (19) with A = ( A ( f ) , A ( σ ) , A ( ω )) . (20) Definition 8.
Let (cid:126) D be a demand potential field (analogy to magentic field). It is defined by rate of change of marketpotential fields in induced field of (cid:126) S , (cid:126) D = (cid:53) × (cid:126) S . (21) Definition 9.
Let F µν be a stress tensor for market microstructure with component A µ of market potential field (theanalogy with connection one form in Chern-Simon theory) defined by the rank 2 antisymmetric tensor field strength F µν = ∂ µ A ν − ∂ ν A µ , d A d [ s ] = S ν = F µν Ψ ν , d A d [ s ] = D µ = (cid:15) κλµν F κλ Ψ ν . (22)These three vector fields of a behavior of trader play role of 3-form in tangent of complex manifold. It is an elementof section Γ of manifold of market A ∈ Γ( ∧ T ∗ x M ⊗ H ) = Ω ( M ) . Let consider 3-differential form of behavior of trader as market potential field A d A = (cid:88) ijk =1 , , F (cid:53) [ si ] ijk dA i ∧ dA j ∧ dA k (23) · · · ←−−−− Ω ( M ) d ←−−−− Ω ( M ) d ←−−−− Ω ( M ) d ←−−−− Ω ( M ) d ←−−−− (cid:121) (cid:121) (cid:121) (cid:121) · · · ←−−−− C ( X t ) ∂ −−−−→ C ( X t ) ∂ −−−−→ C ( X t ) ∂ −−−−→ C ( X t ) ∂ −−−−→ FIG. 3: The cohomology map between predictor and predictant states. with a market state Ψ ( S,D ) ∈ C ( M ). de Rahm cohomology for financial market is defined by using the equivalentclass over this differential form. The first class is so called market cocycle Z n ( M ) = { f ∗ : C n ( M ) → C n +1 ( M ) } withcommutative diagram between covariant and contravariant functor · · · ←−−−− H ( M ) d ←−−−− H ( M ) d ←−−−− H ( M ) d ←−−−− H ( M ) d ←−−−− (cid:121) ξ (cid:121) ξ (cid:121) ξ (cid:121) ξ (cid:121) ξ · · · ←−−−− H ( X t ) ∂ −−−−→ H ( X t ) ∂ −−−−→ H ( X t ) ∂ −−−−→ H ( X t ) ∂ −−−−→ (cid:121) ν (cid:121) ν (cid:121) ν (cid:121) ν (cid:121) ν · · · ←−−−− H ( x t : [ G, G ∗ ]) d ←−−−− H ( x t : [ G, G ∗ ]) d ←−−−− H ( x t : [ G, G ∗ ]) d ←−−−− H ( x t : [ G, G ∗ ]) d ←−−−− d : Ω ( M ) → Ω ( M )) , d : [ s i ]( S, D ) (cid:55)→ A ( A , A , A ) , d [ s i ]( S, D ) = 0 = A , (26)where [ s i ]( S, D ) is an equivalent class of physiology of time series data. The boundary map of cochain of marketΩ ( M ) is a second equivalent class used for modulo state,Im( d : Ω ( M ) → Ω ( M ) , d : Ψ ( S,D ) (cid:55)→ [ s i ]( S, D ) . (27) Definition 10.
The de Rahm cohomology for financial time series is an equivalent class of second cochain of market(Fig. 4) H DR ( M ) = Ker( d : Ω ( M ) → Ω ( M )) / Im( d : Ω ( M ) → Ω ( M )) . (28)The meaning of a new defined mathematical object is in use in measurement of a market equilibrium in algebraictopology approach by AdS Yang-Mills field in finance. From the definition above we get a communication behaviorbetween two sides of market by the twist field of behavior trader of differential 3-form in financial market withunderlying Kolmogorov space X t by ∗ F (cid:53) [ si ] = (cid:73) H DR ( M ) Ψ ( S , D ) k F kµν A ν = F (cid:53) [ s ∗ i ] . (29)The section of manifold induces a connection of differential form. The connection is used typically for gravitationalfield, in econophysics, the connection is used for measurement of arbitrage opportunity, we have (cid:88) ij g ij ∂A i ∂ [ s ] ∧ ∂A j ∂ [ s ] ∈ H DR ( X t ) . (30) FIG. 4: Figure shows D-brane and anti D-brane of Calabi-Yau hypersurface of eight market states in time series model.The Euclidean plane cannot visualize the physiology of market state in this model because all of eight states lay in theextradimensions of Kolmogorov space in time series data.
We simplify the model of financial market in unified theory of E × E ∗ (Fig. 4) under mathematical structure ofKolmogorov space underlying time series data. Let M ∈ H P / Spin(3) be the market with supply and demand sidesand M ∗ : → H P / Spin(3) → H P be dual market of behavior of noise trader σ and fundamentalist f . Then we provean existence of general equilibrium point in the market by using the equivalent classes of supply [ s ] and demand [ d ]in the complex projective space [ s ] , [ d ] ∈ C P . Kolmogorov space S = H P (cid:39) E can be used as explicit state offinancial market and its dual H P ∗ /spin (3) (cid:39) E ∗ as dark state in financial market. It is a ground field of first knotcohomology group tensor with first de Rahm cohomology group H ( x t ; [ G, G ∗ ]; H P / Spin(3)) ⊗ H ( M ; H P ∗ / Spin(3)).We induce a free duality maps ξ : H ( x t ; [ G, G ∗ ]; H P / Spin(3)) → H ( M ; H P ∗ / Spin(3)) as market state E × E ∗ ingrand unified theory model of financial market. For simplicity we let a piece of surface cut out from Riemann spherewith constant curvature g ij define as a parameter of D-brane field basis with the same orientation as a defintion ofsupply and demand in complex plane (flat Riemann surface). Now we blend complex plane to hyperbolic coordinatein eight hidden dimensions of Ψ = (Ψ , · · · Ψ ∗ ) ∈ S . Definition 11.
Let [ G, G ∗ ] be homotopy class between predictor and predictant Lie group, G ⊂ S and G ∗ ⊂ S (Fig. 5a). The physiology shape of eight states is shown in Fig. 5b. We assume that the ground state of marketpotential field basis over hidden state of time series data (not D-brane E × E ∗ , just projection of boundary of D-brane). Let C P (cid:39) [ G, G ∗ ] (cid:51) Ψ i , i = 1 , . . . , defined by Ψ ( (cid:126) S , (cid:126) D ) i ∈ [ G, G ∗ ] ⊂ C P (cid:39) S (cid:39) S /S → S → S → H P . (31) State 1. induced from coupling between supply side of market and supply side in dual market Ψ ( S , D )1 ([ s i ] , [ s j ] ∗ ) = ( S, N S ) ⊂ → ([ U ] , [ N U ]) = (Ψ ( σ + ) , Ψ ∗ ( σ + )) = ([e i θ ] , [e i( − π + γ ) ]) , < θ < π , < γ < π State 2. induced from coupling between Ψ ( (cid:126) S , (cid:126) D )2 ([ s i ] , [ s j ] ∗ ) = ( S, N S ∗ ) ⊂ → ([ U ] , [ N U ∗ ]) = (Ψ ( σ + ) , Ψ ∗ ( f − )) = ([e i θ ] , [e i( − π + γ ) ]) , < θ < π , π < γ < π (U,NU)(cid:13)(U,NU*)(cid:13)(U*,NU)(cid:13)(U*,NU*)(cid:13) (D,ND)(cid:13)(D,ND*)(cid:13)(D*,ND)(cid:13)(D*,ND*)(cid:13) (a) U(cid:13)D(cid:13) U*(cid:13) D*(cid:13) NU(cid:13)ND(cid:13)NU*(cid:13)ND*(cid:13)
State 1(cid:13)State 2(cid:13)State 3(cid:13) State 4(cid:13)State 5(cid:13)State 6(cid:13)State 7(cid:13)State 8(cid:13)Agent(cid:13) Dual Agent(cid:13)Positive noise trader(cid:13) Negative noise trader(cid:13)Positive fundamentalist(cid:13)Negative Fundamentalist(cid:13) Positive noise trader(cid:13)Negative noise trader(cid:13) Positive fundamentalist(cid:13)Negative Fundamentalist(cid:13) (b)
FIG. 5: (a) Picture shown type I of market field. It is originally denoted as market 8 states. The first block in this diagramis predictor Lie group G . The second block is predictant Lie group G ∗ as dark state. A market field is an evolution feedbackpath communicate between these 2 manifolds (b) Picture shown a physiology of shape time series data in complex plane of ourdefinition of 8 states of coupling between behavior of traders in financial market . State 3. induced from coupling between Ψ ( (cid:126) S , (cid:126) D )3 ([ s i ] , [ s j ] ∗ ) = ( D, N D ) ⊂ → ([ D ] , [ N D ]) = (Ψ ( σ − ) , Ψ ∗ ( σ + )) = ([e i θ ] , [e i( − π + γ ) ]) , π < θ < π , π < γ < π State 4. induced from coupling between Ψ ( (cid:126) S , (cid:126) D )4 ([ s i ] , [ s j ] ∗ ) = ( D, N D ∗ ) ⊂ → ([ D ] , [ N D ∗ ]) = (Ψ ( f + ) , Ψ ∗ ( σ − )) = ([e i θ ] , [e i( − π + γ ) ]) , π < θ < π , π < γ < π State 5. induced from coupling between Ψ ( (cid:126) S , (cid:126) D )5 ([ s i ] , [ s j ] ∗ ) = ( S ∗ , N S ) ⊂ → ([ U ∗ ] , [ N U ]) = (Ψ ( σ − ) , Ψ ∗ ( f + )) = ([e i θ ] , [e i( − π + γ ) ]) , π < θ < π, < γ < π State 6. induced from coupling between Ψ ( (cid:126) S , (cid:126) D )6 ([ s i ] , [ s j ] ∗ ) = ( S ∗ , N S ∗ ) ⊂ → ([ U ∗ ] , [ N U ∗ ]) = (Ψ ( f + ) , Ψ ∗ ( f − )) = ([e i θ ] , [e i( − π + γ ) ]) , π < θ < π, π < γ < π State 7. induced from coupling between Ψ ( (cid:126) S , (cid:126) D )7 ([ s i ] , [ s j ] ∗ ) = ( D ∗ , N D ) ⊂ → ([ D ∗ ] , [ N D ]) = (Ψ ( f − ) , Ψ ∗ ( σ + )) = ([e i θ ] , [e i( − π + γ ) ]) , π < θ < π, π < γ < π State 8. induced from coupling between Ψ ( (cid:126) S , (cid:126) D )8 ([ s i ] , [ s j ] ∗ ) = ( D ∗ , N D ∗ ) ⊂ → ([ D ∗ ] , [ N D ∗ ]) = (Ψ ( f − ) , Ψ ∗ ( f + )) = ([e i θ ] , [e i( − π + γ ) ]) , π < θ < π, π < γ < π. A market cyclic cohomology is induced from boundary operator between open string correlator ∂C φM (Ψ ( (cid:126) S , (cid:126) D )1 , Ψ ( (cid:126) S , (cid:126) D )2 , · · · Ψ ( (cid:126) S , (cid:126) D )8 ) = ∂C φM (Ψ ( (cid:126) S , (cid:126) D )2 , · · · Ψ ( (cid:126) S , (cid:126) D )8 , Ψ ( (cid:126) S , (cid:126) D )1 ) (32)with cyclic symmetry C φM (Ψ ( (cid:126) S , (cid:126) D )2 , · · · Ψ ( (cid:126) S , (cid:126) D ) M , Ψ ( (cid:126) S , (cid:126) D )1 ) = (1) M − ( (cid:126) S , (cid:126) D )1 (Ψ ( (cid:126) S , (cid:126) D )2 + ··· +Ψ ( (cid:126) S , (cid:126) D ) M ) C φM (Ψ ( (cid:126) S , (cid:126) D )1 , Ψ ( (cid:126) S , (cid:126) D )2 , · · · Ψ ( (cid:126) S , (cid:126) D ) M ) (33)0 III. METHODOLOGYA. Empirical analysis of financial tensor network for non-equilibrium state
If we consider two time series of close price of stocks r i ( t ), r j ( t ) we can break a mirror symmetry of financial marketby Wigner ray SU(2) transformation act on scalar product (see Def. 4). In Euclidean geometry we measure a 2-formof angle, price r j by typical correlation formula. We visualize the process of financial tensor network [35] by empiricalanalysis of a skeleton of time series data, i. e., (ITD − IMF)chain(1 , n ) (see [12]), with hyperbolic angle over financialtensor network [36]. The result is powerful to detect non-orientation state of market crash by closeness centralitymeasure over planar graph [37] over distance of partial correlation matrix [38]. The schema of our methodology ispresented in Fig. 6.Let V R =: V R ( M ) be a vector of real value as a function of ground field of underlying financial market network M .We can extend ground field to C , H and H P . We define their dual vector fields of 42 stock as eigenvector of vectorspace V by V = << v , v , · · · , v >> (the full list of 42 stocks is shown in Tab. I). We consider V (cid:39) T p M, dim V = 42 , V ∗ R : V → R , (34)where T p M is a tangent of manifold M . In behavior model we extend a field from R to H so we get one form ofground field with 8 states by V ∗ H P : V → H P , V ∗ H P = ∧ V H p . (35)Let r i , r j , i, j = 1 , cdots,
42. The correlation matrix in financial network classically induces from linear map betweenvector space of market. Let r , r be returns of stocks stock , stock in financial market stock , stock ∈ M . We makean assumption that there exist dual market M ∗ , hidden financial market with hidden stocks stock ∗ , stock ∗ ∈ M ∗ . Let r be correlation of return of stock and stock , r = Corr( r , r ), it induces a hidden correlation between hiddenstocks r ∗ = Corr( r , r ) ∗ and the correlation matrix between two markets M ⊗ M ∗ = Corr( r , r ) ⊗ Corr( r , r ) ∗ . The precised geometrical definition between two stocks in financial market M is defined byCorr( M ) = stock ∗ ( t ) stock ∗ ( t ) stock ( t ) r r stock ( t ) r r ∈ SO(2) (cid:39) S ← Spin(2) (cid:39) S , (36)where (cid:39) is a homotopy equivalent. Because the value of correlation is from the interval [ − , ⊂ R , the equilibriumproperty of market is coming from isometry property of the determinant of correlation matrix.First we consider determinant of correlation matrix between two stocks for zero value,det Corr( M ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) stock ∗ ( t ) stock ∗ ( t ) stock ( t ) r r stock ( t ) r r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (37)Since r = r = 1, r = r , we get r r = 1 . (38)If we generalized to hermitian product of correlation in complex form, we get r r = < r , r > = 1 . (39)Thus, the isometry property of market induces an inertia frame of reference from the Hermitian property of correlationin complex conjugate r † = r ∈ C . We have r = 1 so r = ±
1. In the state with r = 1 the stock is in the samedirection of stock . For r = −
1, the stock is in the opposite direction of stock . These two states are the explicitstates of financial market.In complex projective space, we can extend the value of correlation to general form in Riemannian manifold ( M, g )by extend r , r ∈ C P , r = r † , where † is Hermitain product in quantum mechanics. We define metric tensor g by using a differential 2-form over vector space M of correlation between two stocks (correlation in complex projectivespace (K¨ahler manifold) in this sense).The determinant of correlation matrix or a wedge product of column of correlation of all returns of stocks in stockmarket can be minus one and it induces a complex structure as a spin structure in principle bundle of time series data.We found that when det(Corr( M )) = 1, where Corr( M ), M ∈ SL(2 , C ) is a correlation matrix of financial network1 FIG. 6: The flowchart of empirical analysis of (ITD − IMF)chain(1 , n ) of close price in financial tensor network with closenesscentrality. The algorithm in this flowchart use to detect market crash state in year 2008 from time series of close price of42 stocks in 2000 days. The output of this algorithm is a financial tensor network as planar graph [37] and a time series ofcloseness centrality of planar graph . with M¨obius map M [1, 39]. This space of time series data is in equilibrium with two equilibrium points. One isreal and explicit form, the other is hidden and in complex structure like dark matter or invisible hand in economicsconcept. If det(Corr( M )) = 0 a space of time series data is out of equilibrium and can induces market crash withmore herding behavior of noise trader. This result is related to definition of log return and arbitrage opportunity ineconophysics.We can also compute a hyperbolic angle by using Hilbert–Huang transform of financial time series with the instan-taneous frequency IMF1 of return r i θ it = arg[HHT( r it )] , θ jt = arg[HHT( r jt )] (40)2which induce an instantaneous correlation as instantaneous frequency bycos( θ t ) = Corr t ( r i , r j ) . (41)If we interchange between complex plane to hyperbolic plane by break a symmetry, the cosine function can be replacedby the hyperbolic cosine cosh(Ψ t ) = Corr t ( r i , r j ) . (42)We have θ t = arccos(Corr t ( r i , r j )) , Ψ t = arccosh(Corr t ( r i , r j )) (43)with help of e i θ + e − i θ r i , r j ) . (44)since − i = e − i π and i = e i π , the Pauli matrix σ x := f and the Wilson loop for time series data of Pauli matrix to be σ = W ( σ x ) defined by e Ψ π − e − Ψ π (cid:20) − π Ψ e π Ψ (cid:21) = Corr( r i , r j ) . (45) B. Closeness centrality of planar graph in tensor network
When market crashes the stocks in financial market will fall down in the same direction. Therefore the correlationbetween all stocks will be very high, it will induce approaching of all vertices to the others (on average). The closenesscentrality is the most suitable to measure market crash of tensor financial network of planar graph. The time varyingdistance is defined by d ijt = (cid:112) − Corr( r i , r j )). Let G t be random variable of graph with G t = ( V t , E t ), where V t is a time series of vertex V t ∈ X t of a 1-dim CW complex, i. e., connected graph, of realization of Kolmogorov spacefor time series data X t and E t is time series of edge. The time varying closeness centrality is defined by C t ( k ) = 1 (cid:80) h ∈ G t d G t ( h, k ) (46)Since the covering space of correlation matrix is a spin group of Pauli matrix. The map between them, we define inthis work as modified Wilson loop for time series data. The algorithm to do minimum spanning tree [40] and planargraph [41] is analogy to discrete dynamic programming approach to optimized an one dimensional CW complexof financial tensor network with θ as degree (sum of edge in vertex), as maximized parameter in financial tensornetwork. That mean the special case of tensor field of correlation matrix, Corr( M ) ⊗ Corr( M ) ∗ can use the formularcosh(Corr( M )) := Corr ∗ ((Corr( M ))) and Corr(Corr( M )) := cos(Corr( M )) for some special case of some type oftensor field. We do tensor in quantum information theory by using Kronecker product of matrix of correlation of 42stocks. We get 42 ×
42 result of matrix of correlation in tensor field. After that we use average correlation to findan average of matrix and plot into time series of correlation in tensor field. The tensor matrix of correlation can becompute by hand as followings,
Corr(IMF i ( r ) , IMF i ( r )) Corr(IMF i ( r ) , IMF i ( r )) · · · Corr(IMF i ( r ) , IMF i ( r n ))Corr(IMF i ( r ) , IMF i ( r )) Corr(IMF i ( r ) , IMF i ( r )) · · · Corr(IMF i ( r ) , IMF i ( r n )) · · · · · · · · · · · · Corr(IMF i ( r n ) , IMF i ( r )) Corr(IMF i ( r n ) , IMF i ( r )) · · · Corr(IMF i ( r n ) , IMF i ( r n )) × , i =1 , ··· n ⊗ Corr(IMF i ( r ) , IMF i ( r )) Corr(IMF i ( r ) , IMF i ( r )) · · · Corr(IMF i ( r ) , IMF i ( r n ))Corr(IMF i ( r ) , IMF i ( r )) Corr(IMF i ( r ) , IMF i ( r )) · · · Corr(IMF i ( r ) , IMF i ( r n )) · · · · · · · · · · · · Corr(IMF i ( r n ) , IMF i ( r )) Corr(IMF i ( r n ) , IMF i ( r )) · · · Corr(IMF i ( r n ) , IMF i ( r n )) × , i =1 , ··· n (47)3 Time(Days) S t o ck P r i c e ( B a t h )
42 stock priceSET INDEX2008 marketcrisis
FIG. 7: The plot of close prices for 42 stocks of SET50 Index Futures during 2000 days including 2008 market crash. The crashdate is located under 272 point of the plot with all 42 stocks falling in the same direction down. = Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r n ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r n ))) · · ·· · · · · · · · · · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r n ) , c i ( r ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r n ) , c i ( r n ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r n ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r n ))) · · ·· · · · · · · · · · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r n ) , c i ( r ))) · · · Corr(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r n ) , c i ( r n ))) · · ·· · · · · · · · · · · · × = cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r n ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r n ))) · · ·· · · · · · · · · · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r n ) , c i ( r ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r n ) , c i ( r n ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r n ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r ) , c i ( r n ))) · · ·· · · · · · · · · · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r n ) , c i ( r ))) · · · cosh(Corr( c i ( r ) , c i ( r )) , Corr( c i ( r n ) , c i ( r n ))) · · ·· · · · · · · · · · · · × IV. RESULTS
We used close prices of 42 stocks in the Thai stock market SET50 Index Futures (Fig. 7), starting from 23/08/2007to 29/10/2015 with 2000 data points. The results show the calculations for 2000 days interval with window size of 20days. September 16th, 2008 as the crash point of financial market is located at date number 272. The details of all42 stocks are listed in Table I (8 stocks of SET50 Index Futures need to be cut out since the time duration of pricesis insufficient for the calculation, the prices of other 42 stocks pass through 2008 financial market crisis [26]).In Fig. 8 we show the average correlation of instantaneous frequency IMF7 compared with the average correlationof instantaneous frequency of (ITD − IMF)chain(1 , n ) for all 42 stocks together. The peak of market crash at 272 isrelatively high but not highest for overall investigated period.Next we have computed the closeness centrality of IMF1, (ITD − IMF)chain(1 , n ) and IMF7 which are displayedin Fig. 9. The end of time axes represents the market crisis date. One can see that the (ITD − IMF)chain(1 , n ) of 42stocks has a very high peak and clearer result than IMF1 and IMF7.4 A v e r age C o rr e l a t i on o f I n s t an t eneou s F r equen cy o f I M F A v e r age c o rr e l a t i on o f i n s t an t eneou s f r equen cy o f ( i m f − i t d ) c ha i n ( ) FIG. 8: The plot on left shows the average correlation of instantaneous frequency of IMF7. The right plot shows the averagecorrelation of instantaneous frequency of (ITD − IMF)chain(1 , n ) of all 42 stocks in 2000 days interval with window size of 20days. The peak of 2008 market crash is located at the date 272.
We have constructed the tensor network of tensor correlation to find the population of noise trader in financialmarket as demonstrated in our theory. We have used the hyperbolic cosine of (ITD − IMF)chain(1 , n ). The result ofcloseness centrality together with graphical representation of tensor network of 42 stocks of time series at the date2000 is shown in Fig. 10. One can see that the value of closeness centrality of 2008 market crash at the date 272 issignificantly higher in comparison with neighborhood values. The figure also demonstrates the network separationinto two clusters and some isolated nodes. The tensor network in our analysis is varying over time period in realtime. When market crashes the network changes its structure topology and stocks are closer to each other, they areforming group.We have computed tensor networks from IMF1 to IMF7 to select which IMF is the best tool for a detection of 2008market crash. We have found that the plot for closeness centrality of algorithm for IMF7 is the best tool to detectmarket crash, as is shown in Fig. 11. The first peak of closeness centrality is very thick with wider band than otherpeaks in the plot, it is a capture of 2008 market crash state. The tensor network above the plot demonstrates thesituation at the date 2000, it is separated into two clusters and isolated nodes.
V. DISCUSSION AND CONCLUSION
Recently, econophysicists introduced a partial correlation matrix approach to analyze market structure empirically[38]. The first and second eigen vectors of correlation matrix of stocks underlying financial market are denoted aseigen vectors of market in eigenportfolio model of market [42]. Using this mathematical construction, the existence ofa cohomology theory for financial market was taken into account in this work to be justified as the first cohomologytheory in econophysics. The cohomology theory can be used to deform Kolmogorov space of time series data.The meaning of correlation matrix of assets in financial market is deeply related to a risk analysis of portfoliomanagement [43] so called time-varying beta risk [44]. The geometric structure of correlation matrix is in a relationto the Killing vector field and the covariant derivative of geometrical property of financial market.The evolution of average edges of all 42 stocks underlying market is very high in the market crash state. Thecohomology theory is used to construct 1-dim CW complex as planar graph in financial tensor network and to detectthe topological defect in the market (the market crash state).In this work, we have explained the existence of cohomology group over Kolmogorov space of time series data. Thegeneral equilibrium in financial market exists when the sequence of market cocycle is an exact sequence. In the caseof a complex projective space or K¨ahler manifold of market, the equilibrium point is in non-orientation state.We have developed a new theory so called cohomology theory for financial market with new series of definitionof three types of market potential field with orientation and non-orientation state of entanglement state in modifiedWilson loop of time series data to detect the entanglement state of market crash for Index Futures market. AllIndex Futures have underlying asset, for example SET50 Index Futures have 42 selected stocks from 42 underlyingcompanies which are registered in Thailand stock market (SET) as based space and with SET50 Index Futures pricesas tangent space. We have used the closeness centrality of a tensor field of partial correlation and the planar graph of5
Time(Days) C en t r a li t y o f i m f . . . . . . Time(Days) C en t r a li t y o f ( i m f − i t d ) c ha i n ( ) Time(Days) C en t r a li t y o f i m f FIG. 9: The plots show a closeness centrality of IMF1 in above panel, (ITD − IMF)chain(1 , n ) in the middle and IMF7 in thelower panel. The right limit of time axes (the date 272) represents the date of 2008 financial market crisis.FIG. 10: The plot of closeness centrality of algorithm for hyperbolic cosine of (ITD − IMF)chain(1 , n ) for 42 stocks over theperiod of 2000 days with window size of 20 days. A tensor network of 42 stocks at the date 2000 is depicted above the plot. FIG. 11: The plot of closeness centrality of algorithm for IMF7 for 42 stocks over the period of 2000 days with windows sizeof 20 days. The first peak is a capture of 2008 market crash state. A tensor network of all stocks at the date 2000 is depictedabove the plot, it demonstrates the network separation into two clusters and some isolated nodes.
Hilbert-Huang transform with the hyperbolic spectrum of instantaneous frequency as the main tools for a detection ofmarket crash over time series data. The result of analysis shows that the closeness centrality of hyperbolic spectrumof IMF7 and (ITD − IMF)chain(1 , n ) can be also used to detect the next market crashes.
Acknowledgments
K. Kanjamapornkul is supported by Scholarship from the 100th Anniversary Chulalongkorn University Fund forDoctoral Scholarship. This research is supported by 90th Anniversary of Chulalongkorn University, RachadapisekSompote Fund. The work was partly supported by VEGA Grant No. 2/0009/16 and APVV-0463-12. R. Pinˇc´akwould like to thank the TH division at CERN for hospitality. We would like to express our gratitude to Librade( ) for providing access to their flexible platform, team and community. Their professional insightshave been extremely helpful during the development, simulation and verification of the algorithms. [1] W.-Q. Duan and H. E. Stanley. Cross-correlation and the predictability of financial return series.
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