Theory and Applications of Financial Chaos Index
TTheory and Applications of Financial Chaos Index
Masoud Ataei Shengyuan Chen Zijiang Yang M.Reza Peyghami
Abstract
We develop a new stock market index that captures the chaos existing in the market by measuring themutual changes of asset prices. This new index relies on a tensor-based embedding of the stock marketinformation, which in turn frees it from the restrictive value- or capitalization-weighting assumptions thatcommonly underlie other various popular indexes. We show that our index is a robust estimator of themarket volatility which enables us to characterize the market by performing the task of segmentation witha high degree of reliability. In addition, we analyze the dynamics and kinematics of the realized marketvolatility as compared to the implied volatility by introducing a time-dependent dynamical system model.Our computational results which pertain to the time period from January 1990 to December 2019 implythat there exist a bidirectional causal relation between the processes underlying the realized and impliedvolatility of the stock market within the given time period, where it is shown that the later has a strongercausal effect on the former as compared to the opposite. This result connotes that the implied volatility ofthe market plays a key role in characterization of the market’s realized volatility.
Keywords.
Pairwise Comparisons, Tensor Decompositions, Transfer Entropy, VIX, Stock Market Seg-mentation
In this paper, we set out to analyze the stock markets by resorting to the theory and applications of tensors.We develop a special class of spatio-temporal tensors and show that our created algebraic object is a suitablestructure to embed the collective judgment of agents who are present in the market. It is then the useof such effective embedding of the stock market information that allows us to take advantage of differentmathematical and statistical properties of the constructed tensors, in order to define an index which capturesthe chaos existing in the market by measuring the mutual changes of asset prices. The proposed financialchaos index is shown to be indeed a robust estimator of market volatility, which in turn enables us toreliably utilize it for various purposes including analysis of market regimes, performing the task of marketsegmentation and testing the efficient market hypothesis.
M. Ataei: Department of Mathematics and Statistics, York University, Ontario, Canada;e-mail: [email protected]. Chen: Department of Mathematics and Statistics, York University, Ontario, Canada;e-mail: [email protected]. Yang: School of Information Technology, York University, Ontario, Canada;e-mail: [email protected] Peyghami: Department of Mathematics and Statistics, York University, Ontario, Canada;e-mail: [email protected] a r X i v : . [ q -f i n . S T ] J a n Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
To begin with, we formulate our tensor-based model of the stock market on the basis of a rich body ofliterature that has been conducted by numerous scholars during the past decades who have developed thetheory of pairwise comparative judgments to its near maturity stage. This theory thus far has mainly reliedon mathematical aspects of the so-called pairwise comparison matrices and their special subclass widelyknown as the reciprocal pairwise comparison matrices, which finds various application in different areasof science including the analytic hierarchy processes, e.g., see (Brunelli, 2014; Ishizaka and Labib, 2009;Ishizaka and Labib, 2011; Mu and Pereyra-Rojas, 2016; Mu and Pereyra-Rojas, 2017a; Mu and Pereyra-Rojas, 2017b; Saaty and Vargas, 2012) for review of the foundations and recent developments. Our primarycontribution in this work is to extend the theory of pairwise comparative judgments to a tensor domain,using the invaluable findings available on the pairwise comparison matrices and their various properties.To achieve the above-mentioned goal, we first axiomitize the pairwise comparative judgments, and in lightof the defined axiom, we state and re-interpret and expand on the most important results pertaining to mathe-matical and statistical properties of the pairwise comparison matrices and their special subclass of reciprocalmatrices. By examining the various properties of the mentioned comparison matrices, we demonstrate thatthese mathematical objects are among the most suitable structures, if not the only ones, to model the pro-cedures that underlie agents’ thought processes, especially those that relate to their judgment mechanisms.We then extend the comparison matrices to higher dimensions and establish an important result on relationof the polyadic decomposition of our constructed tensors and the consistency of the judgments cast by theagents throughout time. Once having constructed the mentioned tensors based upon the pairwise compari-son matrices and investigated their various algebraic properties, we utilize them to lay out the foundationsfor defining the financial chaos index.Being a robust estimator of the market’s realized volatility, subsequently raises an important question onthe relation of our proposed financial chaos index as compared to other indexes that measure rather themarket’s implied volatility. Our further contribution in this work is to examine the relationship between thefinancial chaos index and VIX, which is a predominant index for measuring the market’s implied volatility.Investigating the relationship between the financial chaos index and VIX, in turn enables us to characterizethe relation between the equity and option markets. We systematically study the kinematics and dynamicsof the financial chaos index and VIX using a variety of tools. Namely, we characterize the relationshipbetween these two indexes by carrying out their fractional cointegration analysis, which in turn leads toidentification of their long-run equilibrium relationship. Furthermore, by using the orthogonalized impulse- heory and Applications of Financial Chaos Index
The remainder of this paper and its structure are organized as follows: In Section 2, we define the axiom ofpairwise comparative judgment, under which various algebraic properties of the pairwise comparison ma-trices, and in particular reciprocal pairwise comparison matrices, are elucidated. In this section, we furtherdefine a third-order spatio-temporal tensor object using the so-called constitution criterion which embedsthe stock market information, which is then followed by the definition of the so-called consensus tensor.Further, in Section 3 various tools are employed to characterize the relationship between our proposed indexand the market’s implied volatility index (VIX).
Throughout the paper, scalar quantities, vectors and matrices are denoted using lowercase lightface (e.g., x ), lowercase boldface (e.g., x ) and uppercase boldface (e.g., X ) letters, respectively, where all vectors arepresumed to be column ones. Also, boldface calligraphic letters (e.g., X ) are used to denote the tensors. Wefurther use R ≥ and R > to denote the fields of positive and strictly positive real numbers, respectively.Tensors are algebraic objects which extend the notion of matrices to higher dimensions. The number ofdimensions of a tensor is referred to as the order or modes of the tensor, i.e., X ∈ R I × I ···× I N indicatesan N th-order tensor having K = (cid:81) Nn =1 I n elements in total. Analogous to matrix rows and columns, mode-n fibers of tensors are derived by fixing every index but one, while fixing all but two indexes of the tensorsyields hyperplanes known as slices . The Frobenius norm of X is further defined as follows: (cid:107) X (cid:107) F = (cid:112) (cid:104) X , X (cid:105) = I (cid:88) i =1 I (cid:88) i =1 · · · I N (cid:88) i n =1 (cid:113) x i i ...i n . (1.1)An important technique for tensor decomposition which finds a wide range of applications in science is the polyadic decomposition which approximates X by the sum of rank- tensors (also called atoms ), i.e., X ≈ (cid:98) X = R (cid:88) r =1 a r ◦ b r ◦ c r , (1.2)where the notation ” ◦ ” represents vector outer product , and R is a given positive integer known as the rankof (cid:98) X . It is worth mentioning that in the case where R is identified to be a minimal rank, the decomposition Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami given by equation (1.2) is referred to as the canonical polyadic decomposition (CPD). Furthermore, thetensor X is said to be a rank- tensor if it can be written as the outer product of three vectors, i.e., X = a ◦ b ◦ c , (1.3)where a ∈ R I , b ∈ R I and c ∈ R I . We consider a general exchange economy consisting of a universe S := { S , S , . . . , S N } of N assetstraded by M (cid:29) agents over a finite time horizon T := { t , t , . . . } ⊂ { , , . . . , T } , where T denotesthe maximum number of discrete time periods under consideration. Given the set of alternatives S , we arethen concerned with the situation where an omniscience agent, say agent m , whose judgment is assumed totypify the collective judgment of the market participants, performs a complete set of pairwise comparisonson alternatives contained in S and assigns a strictly positive quantity µ ( t ) m to her degree of preference onevery comparison at time t ∈ T . It will be shown that under mild conditions of the so-called pairwisecomparative judgment axiom and the strict positivity of µ ( t ) m for every t ∈ T , the preference gauge assignedby the agent m to every comparison is bound to take on form of a ratio between two strictly positivequantities. Various consequences of such an implication and its phenomenological aspects that underlie thefaculty of judgment in human beings will be also discussed throughout this section. Let us assume that for the agent m , there exist a directed and complete edge-weighted comparison multi-graph G ( t ) m = ( S , E ( t ) m ) with N nodes, whose edge set E ( t ) m contains the preference degrees assigned bythe agent to every pair of the assets that are compared at time t ∈ T . A walk or path on G ( t ) m is definedas a sequence S , S , . . . , S k of nodes where ( S i , S i +1 ) ∈ E ( t ) m for i = 1 , , . . . , k − , whose length isdetermined by the number of edges traversed to reach the terminal node starting from the initial node. Inthis setting, we presume that traversing a self-loop would increase the walk length by an increment. By acycle we understand a closed walk with identical initial and terminal nodes. The weight matrix associatedto G ( t ) m is defined as follows: Definition 1 (PCM) . A square matrix A ∈ A ( t )PCM ⊂ R N × N> is said to be a pairwise comparison matrix(PCM) if its elements elicit the agent’s preference degrees on pairs of assets that are compared at a giventime t ∈ T , where A ( t )PCM represents the family of all pairwise comparison matrices at time t . heory and Applications of Financial Chaos Index ( i, j ) -th element of the k -th power A k of the pairwise comparison matrix A can be interpreted as the preference degree of asset S j over S i based on a walk of length k performed on thecomparison multigraph G ( t ) m . Then completeness of G ( t ) m guarantees that the weight matrix A is irreducibleas every pair of nodes are connected by a path of arbitrary a certain length. Moreover, since every pair ofthe nodes in G ( t ) m is connected via a certain path of length k , the matrix is primitive, and such a primitivityproperty strengthens the irreducibility condition by further positing k -connectivity of G ( t ) m .Let us now introduce the relations ≺ k , ∼ k and (cid:22) k which denote k - length strict preference , indifference and preference-indifference relations on S , respectively. These relations, which extend those proposed in(Fishburn, 1970; Suppes, 1999) for the special case k = 1 , which are defined as follows: ∀ µ ( t ) m ∈ R > , ∃ ≺ k , ∃ ∼ k : ( S i ≺ k S j ) ∨ ( S j ≺ k S i ) ∨ ( S i ∼ k S j ) . (2.1)The formula (2.1) can be perceived as the fact that the agent m believes that for all her subjective preferencedegrees measured at a given time t ∈ T , there exist relations ≺ k and ∼ k such that the asset S j would beeither strictly better than the asset S i based on a walk of length k on G ( t ) m (e.g., S i ≺ k S j ) or strictly worse(e.g., S j ≺ k S i ), or S i and S j should be judged as being indifferent from one another (e.g., S i ∼ k S j ).We assume that ≺ k is a strict partial order and define the k -length indifference ∼ k as the absence of strictpreference such that x ∼ k y ⇔ ( x (cid:54)≺ k y ) ∧ ( y (cid:54)≺ k x ) . (2.2)We are then ready to formulate the so-called pairwise comparative judgment axiom as follows:Axiom (PCJ) . The agent m is said to satisfy a pairwise comparative judgment (PCJ) axiom, if for every t ∈ T , k ≥ , i, j, l ∈ { , , . . . , N } and µ ( t ) m > , there exist relations ≺ k , (cid:22) k and ∼ k such that(i). S i ≺ k S j ⇐⇒ µ ( t ) m ( S i ≺ k S j ) > µ ( t ) m ( S j ≺ k S i ); (ii). S i ∼ k S j ⇐⇒ µ ( t ) m ( S i ∼ k S j ) = µ ( t ) m ( S j ∼ k S i ); (iii). µ ( t ) m ( S i (cid:22) k S j ) = µ ( t ) m ( S i (cid:22) k S l ) µ ( t ) m ( S l (cid:22) k S j ) . The properties ( i ) and ( ii ) stated in the PCJ Axiom imply that for every x, y, z ∈ S , the k -length prefer-ence ≺ k and indifference ∼ k relations satisfy the reflexivity of indifferences, irreflexivity of preferences,symmetry of indifferences, asymmetry of preferences, transitivity of preferences and transitivity of indif-ferences. It is also noted that the k -length preference-indifference relation (cid:22) k is defined as the union of Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami k -length preference and indifference relations, i.e., x (cid:22) k y ⇔ ( x ∼ k y ) ∨ ( x ≺ k y ) . (2.3)As a result, transitivity of ∼ k amounts to the fact that (cid:22) k is transitive and complete, which in turn impliesthat (cid:22) k is a weak order. We point out that in the special case k = 1 , part ( iii ) of the PCJ Axiom is oftenreferred to as the consistency property of PCMs. It is used for quantifying the judgments consistently andensures interconnectivity among all activities which in turn permits their quantitative assessment. Let uspropose the following statistic for measuring the average degree of consistency on G ( t ) m : cDeg( E ( t ) m ) = 1 N N (cid:88) i,j,l =1 I [ (cid:15) ( t ) il (cid:15) ( t ) lj = (cid:15) ( t ) ij ] . (2.4)Here, one has cDeg = 1 for fully consistent G ( t ) m , while for cDeg( G ( t ) m ) values reasonably close to wewould rather refer to G ( t ) m as a near-consistent comparison multigraph. Furthermore, were it to be cDeg = 0 ,the comparison multigraph G ( t ) m would be inconsistent , rendering impossibility of existence of a quantitativescheme which would allow any associative comparison performed among N triads of assets.In light of the PCJ Axiom, we present the following series of technical lemmas and theorems on propertiesof PCMs and their special sub-classes. Lemma 1 (Rank Property) . Under the PCJ
Axiom , it holds for the agent m that for every t ∈ T , ∀ A ∈ A ( t )PCM : rank ( A ) = 1 . (2.5) Proof.
First, note that by consistency property ( iii ) of the PCJ Axiom, the matrix A can be written asfollows: A = a a · · · a N a a · · · a N ... ... . . . ... a N a N · · · a NN = a a · · · a N a a a a · · · a a N ... ... . . . ... a N a a N a · · · a N a N . Subsequently, observe that each row of this matrix is a constant multiplier of first row. Hence, rank( A ) =1 . (cid:4) Now, let us provide some intuition behind Lemma 1. For a strictly positive matrix (which is the case forany PCM), the rank of the matrix indicates the number of vertices on the convex hull of the data containedin the matrix. Thus, rank ( A ) = 1 delineates that a single class is sufficient to cluster the contained data,further implying existence of strong dependencies among various entries of the matrix. In other words, the heory and Applications of Financial Chaos Index rank ( A ) = 1 , implies that there exist only one piece of information embedded within the matrix A .In turn, Lemma 1 yields Lemma 2 (Reciprocity Property) . Under the PCJ
Axiom , for every A ∈ A ( t )PCM constructed by the agent m , we have that for each t ∈ T , ∀ i = 1 , , . . . , N : a ii = 1 , ∀ i, j = 1 , , . . . , N : a ij = 1 a ji · (2.6) Proof.
By Lemma 1, rank( A ) = 1 . Thus, the matrix A can be written as the outer product of two vectors,i.e., A = u ◦ v (cid:124) , where u , v ∈ R N> . Subsequently, using the consistency property ( iii ) of the PCJ Axiom,for every i, k = 1 , , . . . , N , we get a ii = u i v i = ( u i v k )( v k v i ) = ( u i v i )( u k v k ) = ⇒ a ii = a ii ( u k v k ) = ⇒ u k v k = 1 . It is then evident that u k = 1 v k , ∀ k ∈ { , , . . . , N } . (2.7)In view of equation (2.7), the elements of A are essentially expressed by element-wise division of vectors u and v , i.e., a ij = u i /v j . This then leads to a ij = 1 /a ji which completes the proof. (cid:4) The assertions of Lemma 2 suggest us to concentrate on a particular subclass of A ( t )PCM which satisfythe reciprocity property at every t ∈ T . It will be shown that equation (2.6) fulfilled for members ofsuch subclass of PCMs make them a preferred choice for modeling the procedures that underlie agents’judgment faculties. In other words, employing the above-mentioned subclass of PCMs enables us to connectmechanisms of the agents’ ways of thought to their quantification of the so-called secondary qualities . Inthis respect, it is worth to provide the following definition. Definition 2 (RPCM) . A square matrix A ∈ A ( t )RPCM ⊂ A ( t )PCM is said to be a reciprocal pairwise compar-ison matrix (RPCM) if(i) A has unit diagonal elements, i.e., a ii = 1 for all i = 1 , , . . . , N ,(ii) A has reciprocal off-diagonal elements, i.e., a ij = a ji for all i, j = 1 , , . . . , N ,where A ( t )RPCM represents the family of all reciprocal pairwise comparison matrices at time t which is aproper subclass of A ( t )PCM . Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
An important, plausibly unique, characterization property pertaining to RPCMs is stated in the followinglemma.
Lemma 3 (Scaled Self-similarity Property) . Under the PCJ
Axiom and for the agent m , the followinggeneralized idempotent relation ∀ k ≥ , ∀ A ∈ A ( t )RPCM : A k = N k − A (2.8) holds for every t ∈ T .Proof. It is straightforward. (cid:4)
We clarify possible implications of Lemma 3 as follows. For any RPCM associated to agent m ’s set ofbeliefs, Lemma 3 asserts that the diagonal elements of its k -th power are equal to those of the originalRPCM (whose diagonal elements are all ones) times N k − . Thus, we may write ∀ k ≥ , ∀ i ∈ { , , . . . , N } , µ ( t ) m ( S i ∼ k S i ) = N k − µ ( t ) m ( S i ∼ S i ) . (2.9)To gain more insight on implications of the scaled self-similarity property of RPCMs and comprehend therationale behind the results provided by equation (2.9), it deserves to distinguish between S i ∼ k S i and S i ∼ S i from the information-theoretic point of view. To this end, we first note that S i ∼ S i correspondsto traversing a self-loop on G ( t ) m , i.e., comparing an asset to itself. However, the information gain for such acomparison is clearly zero. Hence, we may refer to the event of performing the comparison S i ∼ S i as a redundant action . On the other hand (and in contrast to traversing self-loops), communicating the cycles of G ( t ) m , i.e., S i ∼ k S i where k ≥ , can potentially increase the amount of information gain for the agent. Thisstems from the fact that the subjective gauge of the agent about dominance relations existing among assetsbecomes more refined each time she assimilates the information pertaining to the cycles of G ( t ) m whoselengths are strictly greater than one.Now, since the relation S i ∼ k S i corresponds to a cycle of length k − for every i ∈ { , , . . . , N } ,there exist at most k − intermediary nodes in each cycle. According to the maximum entropy principle ,we can assume that the agent’s assimilation of information on each intermediary node of the cycle followsthe discrete uniform probability model with each individual probability equal to ( N ) k − . Hence, it is to beexpected that agent m would continue exploring the cycles of G ( t ) m rationally until she commits the so-calledredundant action, after which she stops the exploration. Define a new random variable Z which denotes the heory and Applications of Financial Chaos Index m to exploit the cycles of G ( t ) m until she commits a redundant action.Evidently, Z is a geometrically distributed random variable with the probability of success (traversing aself-loop) equal to ( N ) k − . As a result, the expected number of Bernoulli trials that takes for agent m tocommit a redundant action is E [ Z ] = 1( N ) k − = N k − . (2.10)In turn, this implies that the information gain gets increased N k − -fold by traversing the cycles of length k − on G ( t ) m , e.g., compare to equation (2.9).It is worth to mention that the kinds of interpretations like the one provided above which are consistent withcommon sense, potentially amplify the importance of RPCMs as being tools deemed appropriate for model-ing agents’ underlying mechanisms of judgment, further providing justifications for the use of comparativejudgment framework in a broader sense.Furthermore, the scaled self-similarity property of RPCMs leads to an interesting fractal phenomenon per-taining to RPCMs which occurs on hyperbolic plane, as stated in the following theorem. Theorem 1 (Fractal Property) . Under the PCJ
Axiom and for the agent m , the following hyperbolic self-similarity relation ∀ k ≥ , ∀ A ∈ A ( t )RPCM : sinh( A ) = sinh( N ) N A (2.11) holds for every t ∈ T .Proof. Using the matrix exponential for A and by resorting to implications of Lemma 3, we may write exp( A ) = ∞ (cid:88) k =0 A k k ! = I N + 1 N ( e N − A . (2.12)Furthermore, equation (2.12) implies that exp( − A ) = I N + 1 N ( e − N − A . (2.13)By equations (2.12) and (2.13), we then get exp( A ) − exp( − A ) = 1 N ( e N − e − N ) A , (2.14)which can be expressed in terms of the sine hyperbolic functions as presented below sinh( A ) = sinh( N ) N A . (2.15) (cid:4) Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
Corollary 1.
Under the PCJ
Axiom and for the agent m , the following relation ∀ k ≥ , ∀ A ∈ A ( t )RPCM : Tr[sinh( A )] = sinh( N ) (2.16) holds for every t ∈ T .Proof. It is straightforward. (cid:4)
Besides, the above lemmas yield that eigenvalues of an RPCM are either N or (with algebraic multiplicity N − ). Lemma 4 (Characteristic Polynomial) . For the agent m and every A ∈ A ( t )RPCM where t ∈ T , the charac-teristic polynomial for A takes on the following form: p A ( λ ) = λ N − N λ N − . (2.17) Proof.
It is well-known that the characteristic polynomial of a square matrix is given by p A ( λ ) = λ N − [Tr( A )] ( λ N − ) + · · · + ( − N det( A ) . (2.18)By Lemma 1, all the terms of p A ( λ ) except the first two leading terms equal zero. It remains to note that Tr( A ) = N for any RPCM, which yields p A ( λ ) = λ N − N ( λ N − ) . (2.19)In turn, this completes the proof. (cid:4) Using the implications of Equation (2.17), we provide a necessary condition on the consistency property ( iii ) of the PCJ Axiom in terms of the permanents of the RPCMs, as stated in the following theorem. Theorem 2 (Matrix Permanent Property) . Under the PCJ
Axiom and for the agent m , the following relation ∀ k ≥ , ∀ A ∈ A ( t )RPCM : perm( A ) = N ! (2.20) holds for every t ∈ T , where perm( A ) represents the permanent of the matrix A .Proof. In order to proved the theorem, it is sufficient to demonstrate that the matrices A and J N (all-ones matrix of order N ) are similar. This assertion simply follows from the fact that both of the matricesmentioned above share a common characteristic polynomial p ( λ ) = λ N − N ( λ N − ) . (2.21) heory and Applications of Financial Chaos Index perm( A ) = N ! if and only if A can be obtained from J N by a finite sequence of elementary operations, e.g., see (Akbari,Ariannejad, and Tajfirouz, 2016; Wang, 1974) for more technical details. (cid:4) Next, we use equation (2.17) for the characteristic polynomial of A to determine the eigenvalues of thematrix. Lemma 5 (Perron-Frobenius Eigenvalue) . For the agent m , it holds that ∀ A ∈ A ( t )RPCM , λ max = N (2.22) for every t ∈ T , where λ max represents the largest eigenvalue of A , also called the Perron-Frobeniuseigenvalue.Proof. Recall that for a matrix A with strictly positive elements, the Perron-Frobenius theorem states that A has a positive real eigenvalue λ max , which is strictly greater than the absolute values (moduli) of allthe other eigenvalues. On the other hand, by Lemma 4 the roots of the characteristic polynomial of A areeither or N . In addition, Lemma 1 implies that rank( A ) = 1 . Hence, A has a single nonzero eigenvalue λ max = N with all the remaining eigenvalues equal to zeros. (cid:4) An another interesting property of RPCMs is that they belong to a family of matrices for which there is arelation between their rank, order and trace (see equation (2.23)).
Lemma 6 (Trace-Order-Rank Property) . Under the PCJ
Axiom and for the agent m , the specific charac-teristics of a generic matrix A ∈ A ( t )PCM satisfies the following relationship: Tr( A ) = N · rank( A ) . (2.23) Proof.
By Lemma 3, we have A k = ( N k − ) A , ∀ k ≥ . (2.24)For k = 2 , equation (2.24) yields that A = N A . (2.25)The proof then follows from the fact that equation (2.25) implies the validity of equation (2.23). See(Harville, 2018) for more details on this implication. (cid:4) Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
Further on, since for any RPCM we have
Tr( A ) = λ max , in such case Lemma 6 stipulates that λ max = N · rank( A ) . (2.26)Equation (2.26) can be interpreted as follows: It is known that if we take any vector corresponding to thelargest eigenvalue λ max , then the action of the matrix on that specific vector attains its maximum value.Next, recall that rank( A ) quantifies the amount of information that is preserved by such action. Hence,equation (2.26) means that the maximum action of the matrix equals its complexity (provided that theorder N of the matrix signifies its complexity) multiplied by the amount of information preserved by suchaction. In turn, this provides an additional useful property for RPCMs and substantiates their use to modelthe thought processes of particular agents. Namely, equation (2.26) guarantees that the possibility for theloss of information is eliminated inasmuch as action of RPCMs on vectors chosen along the eigenvectorcorresponding to λ max are taken into account. This is due to the fact since rank( A ) = 1 for RPCMs, thenreducing the dimensionality of the data set would require one to project each data point along some unitvector. Moreover, it is of great interest to choose the unit vector in a way that projection of the data pointsalong it would retain as much of the variation of the data points as possible. To achieve this goal, it iswell-known that the eigenvector corresponding to λ max can be chosen for the purpose of dimensionalityreduction since such an eigenvector is the direction along which the data set would have the maximumamount of variance.Next, another important aspect of RPCMs that deserves attention pertains to sensitivity of their eigenvaluesto perturbations of their elements. A possible approach for conducting such sensitivity analysis can becarried out by studying the condition number of the matrix since the change in the output of a function fora small changes in its input arguments can be quantified by the condition number of the function. For asimple eigenvalue λ j , it was shown in (Wilkinson, 1988) that cond( λ j ; A ) , expressed in terms of the leftand right eigenvectors of A , measures the sensitivity of λ j to small perturbations in elements of the matrix.Yet, another useful representation for cond( λ j ; A ) is given in (Smith, 1967) whose author puts forward thefollowing relationship cond( λ j ; A ) = (cid:107) adj( λ j I − A ) (cid:107) p (cid:48) A ( λ j ) , (2.27)where (cid:107) A (cid:107) denotes the largest singular value of A , and p (cid:48) A ( λ j ) is the derivative of the characteristicpolynomial of A w.r.t. λ which is evaluated at λ = λ j . For the matrix A , being an RPCM with a simple heory and Applications of Financial Chaos Index λ max = N , then equation (2.27) simplifies as follows: cond( λ = N ; A ) = (cid:107) adj( N I − A ) (cid:107) N N − · (2.28)Therefore, it becomes clear that for a matrix A ∈ A ( t )RPCM with a sufficiently large order N , whose majorityof the elements are from the same order of magnitude and none of its elements is near-zero or near-infinity,the eigenvalue of the matrix would be relatively insensitive to changes occurred in all of its elements, unlessthe structure of the matrix undergoes a considerable perturbation. We refer to this property of the largesteigenvalue of RPCMs as robustness . Lemma 7 (Robustness of Perron-Frobenius Eigenvalue) . The largest eigenvalue λ max of a generic matrix A ∈ A ( t )RPCM is robust w.r.t. perturbations occurred in the matrix, provided that the elements of the matrixare homogeneous, i.e., they are uniformly bounded from above and below.Proof. See (Saaty, 1993). (cid:4)
An application of the sensitivity analysis for RPCMs is illustrated by the following example:
Example 1.
Consider the following matrix A ∈ A ( t )RPCM A = α β α γ β γ , where α, β, γ > . Then, it can be shown that the adjugate of I − A is as follows: adj(3 I − A ) = α + βγ β + αγ α + γβ βα + 2 γ β + αγ αβ + γ . This enables us to compute cond( λ = 3; A ) . We forego to provide an explicit relation for the condi-tion number due to its excessive length. Now, assume that the original matrix A is constructed usingthe triplet ( α, β, γ ) = (1 . , . , . and the perturbed matrices are created using the triplets (1 . , . , . , (0 . , . , . , (1 . , . , . and (5 . , . , . . Then it canbe shown that the condition number for the original matrix is . , and for the perturbed matrices thecondition numbers are equal to . , . , . and . , respectively. It is then observed thatperturbations occurred in the neighborhood of the original matrix do not influence the condition numbersubstantially, whereas an structural change in the matrix, as in the latter case, leads to a huge change inthe value of the condition number. Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
Besides, the derivation of (cid:107)·(cid:107) is often too involved and complex. Hence, one can potentially obtain anoverestimate of cond( λ = 3; A ) by resorting to the well-known inequality (cid:107)·(cid:107) ≤ (cid:107)·(cid:107) F . Yet, anotherconservative upper bound on cond( λ = 3; A ) is obtained by observing that adj(3 I − A ) = + 2 α β α γ β γ + βγ αγ γβ βα αγ αβ . Then, cond( λ = 3; A ) would be bounded from above by summing the Frobenius norms of each decomposedmatrix, i.e., an upper bound on cond( λ = 3; A ) is (cid:18) N N − (cid:19) max (cid:40) max( N, α, β, γ, αγ, βγ ) , α, β, γ, αγ, βγ ) (cid:41) · In turn, this implies that insofar as the elements of the matrix A are uniformly bounded, perturbations inthe neighborhood of A do not affect its maximum eigenvalue substantially. The concluding properties of RPCMs we discuss here pertains to the eigenvalues of their perturbed matrices.Namely, the following lemma holds.
Lemma 8.
The Perron-Frobenius eigenvalue λ max = N for the matrix A if and only if the matrix A ∈A ( t )RPCM . Otherwise, λ max > N for any matrix being a perturbation of A .Proof. See (Saaty, 1993). (cid:4)
Furthermore, the following lemma delineates the relation between Perron-Frobenius eigenvalues of RPCMsand their consistency properties (property ( iii ) of the PCJ Axiom).
Lemma 9.
The matrix A ∈ A ( t )RPCM for some t ∈ T , is consistent if and only if the Perron-Frobeniuseigenvalue of the matrix is λ max = N .Proof. See (Saaty, 1993). (cid:4)
As per Lemmas 8 and 9, any perturbation of the matrix A ∈ A ( t )RPCM can be seen as a departure fromthe fulfillment of the condition ( iii ) of the PCJ Axiom. This is due to the fact that λ max = N for any A ∈ A ( t )RPCM at every t ∈ T . Therefore, at any given point of time t , one may write the definition ofconsistency as follows: a ( t ) il a ( t ) lj = a ( t ) ij , i, j, l = 1 , , . . . , N. (2.29) heory and Applications of Financial Chaos Index G ( t ) m for the agent m yields cDeg( E ( t ) m ) = 1 usingequation (2.4) for all t ∈ T . In turn, this permits the agent m to quantify her judgments to an extent that shecan perform associative comparisons among all N triplets of assets. In this section, we employ tensors as a natural structure to embody the information pertaining to the mutualinteractions among price changes in the assets. Let us start by defining the so-called pairwise comparisontensors and their reciprocal counterparts.
Definition 3 (PCT) . A third-order spatio-temporal tensor A ∈ A |T | PCT ⊂ R N × N ×|T | > is said to be a pairwisecomparison tensor (PCT) if it is formed by concatenating |T | number of PCMs as its frontal slices, where A |T | PCT denotes the family of PCTs having temporal dimensions |T | . Definition 4 (RPCT) . A generic tensor A ∈ A |T | RPCT is said to be a reciprocal pairwise comparison tensor(RPCT) if each of its frontal slices is an RPCM, where A |T | RPCT denotes the family of RPCTs having temporaldimensions |T | , which is a proper subclass of A |T | PCT . In what follows, we extend the concept of matrix inconsistency introduced in (Saaty, 1977; Saaty, 1993)to the domain of tensors and define a function to measure the inconsistencies of PCTs. Keeping this inmind, recall Lemma 8 which states that the largest eigenvalue λ max of any matrix obtained by perturbing itsassociated A ∈ A ( t )RPCM would have its largest eigenvalue being greater than N . More so, it was discussedthat the consistency property ( iii ) of the PCJ Axiom is satisfied only if a given matrix belongs to A ( t )RPCM (e.g., see Lemma 9). Hence, one may envisage the statistic λ max − N as being a measure of departure of thejudgments cast by the agent m from the consistency condition of the PCJ Axiom. An average perturbationstatistic can then be defined as ( λ max − N ) / ( N − . Definition 5 (Inconsistency Function) . For a tensor A ∈ A |T | PCT , the inconsistency function ψ : T → R |T |≥ is defined as follows: ψ ( t ) := λ ( t )max − NN − · (2.30) Here, λ ( t )max represents the largest eigenvalue of the PCM corresponding to the t -th frontal slice of A . In turn, Definition 5 suggests one to examine temporal inconsistencies which agent m may confront in theprotean of making her judgments. It can be stated that the judgments cast by the agent m are temporallyconsistent if her judgments at every point of time t ∈ T satisfy the condition ( iii ) of the PCJ Axiom. That6 Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami is, the inconsistency function for a pairwise comparison tensor A ∈ A |T | PCT is zero if every frontal slice of A is an RPCM. In other words, in an ideal case for which the reciprocal pairwise comparison tensors areused for agent m ’s absorption of the market information, one can ensure that the mechanisms underlyingthe agent’s subjective opinion remain persistent over time, i.e., µ ( t ) m = µ m for all t ∈ T (one may find thissituation similar to the scenario involving stationary frequencies of an ergodic Markov chain).Such regularity of the inconsistency measures obtained for a PCT can also be defined by using the con-cept of approximate entropy (ApEn). In order to define the ApEn for a sequence of inconsistency mea-sures { ψ (1) , ψ (2) , . . . } of length |T | , firstly two blocks of longitude l ≤ |T | are defined by ˘ ψ ( i ) = { ψ ( i ) , . . . , ψ ( i + l − } and ˘ ψ ( j ) = { ψ ( j ) , . . . , ψ ( j + l − } , and then the distance between themis calculated using d ( ˘ ψ ( i ) , ˘ ψ ( j )) = max k =1 ,...,l ( | ψ ( i + k − − ψ ( j + k − | ) . Thereafter, the followingstatistic C li ( r ) = 1 |T | − l + 1 |T |− l +1 (cid:88) j =1 I [ d ( ˘ ψ ( i ) , ˘ ψ ( j )) ≤ r ] (2.31)is defined, where the summation term counts the number of consecutive blocks of longitude l which aresimilar to the given block ˘ ψ ( i ) within a given resolution r . Subsequently, the logarithmic frequency withwhich longitude- l blocks that are close together stay together for the next increment defines the ApEn forthe given sequence of inconsistency measures, i.e., ˚ ψ = 1 |T | − l + 1 |T |− l +1 (cid:88) i =1 C li ( r ) . (2.32)ApEn defined by equation (2.32) enjoys several mathematical and statistical properties among which thefollowing are noteworthy: 1- ApEn is a statistic which is robust, meaning that it is insensitive to artifactsor outliers; 2- ApEn is not altered by translations or scaling applied uniformly to all elements containedwithin the considered sequence of inconsistency measures; 3- nonlinearity causes greater values of ApEn,e.g, see (S. M. Pincus and Goldberger, 1994); 4- ApEn is model-free, which makes it a promising statisticfor the analysis of data series whose generating sources are unknown; and 5- Computation of ApEn requiresequally-spaced measurements over time, e.g, see (S. M. Pincus, 1991).The above-mentioned properties of ApEn, among others, make it a suitable candidate to measure the regu-larity of the inconsistency measures. This has been formally defined as follows: Definition 6 (Regularity Measure) . For a tensor A ∈ A |T | PCT , the regularity measure of inconsistencies ˚ ψ : R |T |≥ → R ≥ is defined by the approximate entropy of the inconsistency function ψ ( t ) given by equa-tion (2.30) . heory and Applications of Financial Chaos Index ˚ ψ , one can then quantify the regularity of the inconsistencies, e.g., tack-ling the question of whether a pairwise comparison tensor A ∈ A |T | PCT is regularly inconsistent (consistent),or it is irregularly inconsistent. Basically, the regularity measure ˚ ψ of inconsistencies estimates the amountof randomness found in ψ ( t ) without requiring any prior knowledge of the source that generates the time-dependent inconsistency values. In other words, lower values of ˚ ψ attest that ψ ( t ) is persistent, repetitiveand predictive in the sense that patterns repeat themselves throughout the series. On the other hand, highervalues of ˚ ψ imply independence between particular temporal values of ψ ( t ) and hence are indicators to-wards lower number of repeated patterns and smaller values of randomness, e.g., see (Delgado-Bonal andMarshak, 2019; S. Pincus, 2008) for more details on approximate entropy. Remark 1.
A pairwise comparison tensor A ∈ A |T | PCT is said to be regularly consistent if (˚ ψ = 0) ∧ ( ∀ t : ψ ( t ) = 0) , (2.33) and regularly inconsistent if (˚ ψ = 0) ∧ ( ∃ t : ψ ( t ) > . (2.34) Otherwise, A is said to be irregularly inconsistent if (˚ ψ > ∧ ( ∃ t : ψ ( t ) > , (2.35) where the magnitude of ˚ ψ delineates the amount of irregularity. The RPCTs constructed for agent m according to Definition 3, reflect the ideal situations in which the agentsatisfies the conditions of the PCJ Axiom throughout time. In reality, however, the values of the comparisontensor could deviate substantially from the ideal case. In order to illustrate this phenomenon, consider tensor A ∈ A |T | RPCT whose perturbed tensor is denoted by ˜ A ∈ A PCT and referred to as the consensus pairwisecomparison tensor . The following theorem shows that ˜ A has to be a rank- estimate of A to guarantee thatthe inconsistency function (2.30) can be recovered. Theorem 3 (Consensus PCT) . Let A ∈ A |T | RPCT be a tensor whose associated consensus tensor is denotedby ˜ A ∈ A PCT . Also, let ˜ A be a rank- tensor, i.e., ˜ A = ˜ z ◦ (˜ x ◦ ˜ y (cid:124) ) , which solves the following Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami constrained polyadic decomposition model: min ˜ x , ˜ y , ˜ z (cid:107) A − ˜ z ◦ (˜ x ◦ ˜ y (cid:124) ) (cid:107) F (2.36a) s . t . ˜ z t (˜ x ◦ ˜ y (cid:124) ) > N , t = 1 , , . . . , |T | , (2.36b) ˜ x , ˜ y ∈ R N> , (2.36c) ˜ z ∈ R |T |≥ . (2.36d) Here, N denotes an N × N matrix whose entries are all zeros, and inequality (2.36b) is evaluatedcomponent-wise. Then, the elements of the vector ˜ z depend on the largest eigenvalues of the frontal slicesof ˜ A (up to some positive scaling).Proof. Consider the matrix A ∈ A ( t )RPCM at a given time t ∈ T , and let u and v represent the left and righteigenvectors of its associated Perron-Frobenius eigenvalue λ max , respectively, i.e., u (cid:124) A = λ max u (cid:124) and Av = λ max v , where strict positivity of A renders u , v ∈ R N> . Then, by Perron-Frobenius theorem we have lim k →∞ (cid:18) A λ max (cid:19) k = (cid:18) v (cid:124) u (cid:19) uv (cid:124) · (2.37)However, Lemma 3 implies that A k = N k − A . Hence, A ∼ (cid:18) λ k max N k − (cid:19) (cid:18) v (cid:124) u (cid:19) uv (cid:124) = λ max (cid:18) v (cid:124) u (cid:19) uv (cid:124) , (2.38)which implies that A is a rank- matrix multiplied by a constant which depends on its largest eigenvalue λ max .On the other hand, the rank- polyadic decomposition of the tensor A ∈ A |T | RPCT can be written as the outerproduct of a rank- matrix by a vector which resides on the temporal dimension of the tensor, i.e., A ≈ ˜ A = ˜ z ◦ (˜ x ◦ ˜ y (cid:124) ) , (2.39)where ˜ x , ˜ y ∈ R N> and ˜ z ∈ R |T |≥ . Therefore, elements of the vector ˜ z can be regarded as the terms whichdepend on the largest eigenvalues of the corresponding matrices which pertain to frontal slices of the con-sensus tensor ˜ A .Consequently, in order to recover the inconsistency function ψ ( t ) at each time instant, one can first obtain ˜ A by solving model (2.36) and then proceed by computing the largest eigenvalue of the approximated tensorat each of its frontal slices, and thereafter evaluate the inconsistency function given by equation (2.30). (cid:4) heory and Applications of Financial Chaos Index ψ ( t ) and polyadicdecomposition of RPCTs. Remark 2.
In order to recover the inconsistency measures, one may first approximate the tensor A ∈A |T | RPCT by solving model (2.36) and then proceed by computing the largest eigenvalue of the approximatedtensor ˜ A at each of its frontal slices. Subsequently, the inconsistency measure at each time instant is derivedby evaluating the function ψ ( t ) given by (2.30) . It is further noted that the error of approximation in rank- polyadic decomposition of an RPCT is directlylinked to its average inconsistency measure, as elucidated in the following example. Example 2.
Let r ( t ) ∈ R N> denote the vector of lag- l daily rates of return for N assets, i.e., the lag- l ratesof return for assets are computed using the following relation: r ( t ) = C ( t ) C ( t − l ) , (2.40) where C ( t ) denotes the adjusted closing price for the assets at time t ∈ T .In our simulations, we considered N = 811 assets, selected to be those that have ever appeared in the listof S & P at a time period from January to December for which their price information did notcontain any missing values . Also, in our datasets, the lag- rates of return were comprised of |T | = 7558 and |T | = 360 daily and monthly time steps, respectively.The RPCTs were constructed for each given lag value (shown schematically in Figure 1), which was thenfollowed by computation of their associated inconsistency function ψ ( t ) , average inconsistency measure ¯ ψ ( t ) as well as their regularity measure ˚ ψ ( t ) . The values for the lag parameter l we considered in oursimulations ranged from through .Figures 2 to 4 depict, respectively, the rank- poliadic decomposition errors, average inconsistency mea-sures and regularity measures for different values of the lag parameter l . It is clear from these figures thatthe error of polyadic decomposition grows nonlinearly by increasing values of the lag parameter, whereasthe growth of the average inconsistency measures exhibits a linear trend. These observations were alsoverified empirically, and led to the formulation of the following relations: (cid:15) ( l ) ≈ (cid:15) (1) √ l, (2.41) Data were retrieved from the Center for Research in Security Prices (CRSP) provided by Wharton Research at the University ofPennsylvania (WRDS). All models were solved on Amazon Web Services (AWS) using a server with Xen hypervisor, cores and TB of RAMmemory. Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
Figure 1: Schematic of an RPCT. and ¯ ψ ( l ) ≈ ¯ ψ (1) l, (2.42) where (cid:15) ( l ) and ¯ ψ ( l ) denote the relative error of approximation and average inconsistency measure at lag l , respectively. Furthermore, Figure 4 implies that ˚ ψ ( t ) decreases monotonically as the value of the lagparameter increases. In the above discussion, we analyzed the mechanisms of judgment casting for a specific agent in the market.However, our ultimate goal is to expand our analysis to the entire participants of the market. For this pur-pose, we are to make a reasonably realistic assumption based on which the closing prices of assets are theonly public source of information available or being of interest to the agents. Then, our analytical frame-work developed for an individual agent can be extended to encompass the totality of the agents by notingthat, for an individual agent, what distinguishes different RPCTs from each other pertains to the lag crite-rion used in the computation of the rates of return. Thus, the problem of multiple subjective criteria existingacross the market (as there are M (cid:29) agents who construct their RPCTs using their particular lag values),can potentially be reduced to that of finding an appropriate value for l which represents the lag value usedby majority of the agents. Let us call the criterion constructed above, which is based on such a lag valuethe constitution criterion (denoted by C ); we adapt the term constitution from (Gibbard, 2014) where theauthor addresses Arrow dilemma (Arrow, 2012) and refers to amalgamation of individual preferences as aconstitution.We reckon that the best constitution criterion is the one corresponding to the lag value using which theconstructed consensus PCT yields smallest value of the average inconsistency ¯ ψ ( t ) and the largest value ofthe regularity of inconsistencies ˚ ψ ( t ) . This is because larger values of ˚ ψ ( t ) indicate higher independence heory and Applications of Financial Chaos Index Lag R e l a t i v e E rr o r ( % ) Figure 2: Plot of the relative errors for rank- polyadic decomposition at different lag values. Lag . . . . . . . A v e r ag e I n c o n s i s t e n c y M e a s u r e × − Figure 3: Plot of the average inconsistency measures at different lag values.
Lag . . . . . R e g u l a r i t y M e a s u r e Figure 4: Plot of the regularity measures at different lag values. Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami among inconsistencies of PCMs at different points of the time horizon. Thus, opting for the largest value of ˚ ψ ( t ) ensures that G ( t ) C under consideration will more likely to remain independent at different time instants,making it possible to draw more accurate conclusions on relations existing among the assets at intersequenttime instants. By employing the concepts and tools presented in Sections 1.1 and 1.2, we introduce a stock market indexwhich captures chaotic behavior existing in asset prices. Let us recall that the consensus PCT derived byusing some appropriately chosen constitution criterion could potentially embody the stock market informa-tion about mutual asset price changes in an effective way. In turn, this enables us to formally define thefinancial chaos index as follows:
Definition 7 (FCIX) . Consider a consensus PCT given by ˜ A ∈ A |T | PCT whose associated RPCT (denotedby A ∈ A |T | RPCT ) is constructed using lag- l rates of return for some value of l as its constitution criterion.The financial chaos index FCIX :
T → R |T |≥ is then defined via the inconsistency function given byequation (2.30) as follows: FCIX( t ) := ψ ( t ) = λ ( t )max − NN − , (2.43) where λ ( t )max represents the largest eigenvalue of the matrix corresponding to the t -th frontal slice of ˜ A . Wefurther denote by FCIX t (or by ψ t ) the time series which takes on the observed FCIX values. For instance, consider the data stated in Example 2 and recall that the lag- rate of return was deemedsuitable for the purpose of constitution criterion. Using the lag- rates of return for computing FCIX t , wesketch the plot of annotated monthly FCIX t in Figure 5 for a time period from January 1990 to December2019, respectively. Note that we obtain the monthly FCIX t by taking average of the daily FCIX t valuesduring each month. As it is seen from these figures, the FCIX t spikes whenever the stock market undergoeschaotic fluctuations in response to various anomalous political, geopolitical, economical, financial, fiscal,health and psychological events. For instance, such spikes were observed during the first and second Gulfwars, the explosion of the dot-com bubble, September 11, the stock market crash, SARS coronoviruspandemic in , failure of Lehman Brothers, the debt-ceiling dispute in , Chinese stock marketcrash in , OPEC oil cut, the world-wide stock market down fall, and the recent US-China tradetensions, to mention but a few. Also, for practical reasons addressed in the sequel, we utilize the notion ofthe quarterly FCIX t , which is defined as the average of daily FCIX values during each quarter. heory and Applications of Financial Chaos Index Time (monthly) . . . . . . F i n a n c i a l C h ao s I nd e x × − G o v e r n m e n t s hu t d o w n G u l f W a r II nd i a n ec o n o m i cc r i s i s B l a c k W e dn e s d a y M e x i c a np e s o c r i s i s M a d - c o w d i s e a s e A s i a n (cid:28) n a n c i a l c r i s i s R u ss i ag r e a t d e p r e ss i o n S a m b a e (cid:27) ec t E a r l y s r ece ss i o n D o t - c o m bubb l e G r e a tr ece ss i o n C r i s i s i n T u r k e y S e p t e m b e r tt a c k s S t o c k m a r k e t c r a s h G u l f W a r II S A R S c r o n a v i r u s p a nd e m i c I r a n i a nnu c l e a r p r og r a m w o rr i e s U . S . h o u s i n g bubb l e B N PP a r i b a s c r i s i s E c o n o m i c S t i m u l u s A c t Lehman and TARPMarch , crisis G r ee k c r i s i s / H N v i r u s p a nd e m i c D e b t - ce ili n g c r i s i s U k r a i n i a n c r i s i s J C P O A C h i n e s e s t o c k m a r k e t c r a s h B r e x i t O PE C o il c u t w o r l d - w i d e s t o c k m a r k e t d o w n U S - C h i n a tr a d e t e n s i o n s Figure 5: Plot of the annotated monthly
FCIX t portrayed during January 1990-December 2019. The term volatility is a notoriously slippery concept and bears an intricate nature and meaning which makesits quantification a daunting task. One widely-conventional definition of the volatility ascribes it to theamount of dispersion of returns for a given set of assets. However, we argue here that the dispersion statisticand its variants are not the only available choices for quantification of volatility, and our developed financialchaos index is in fact an alternative measure for the market volatility, which is superior to some dispersion-based measures from a certain point of view.Let us consider the tensor A ∈ A |T | RPCT constructed using some constitution criterion C as a model forembedding the stock market information whose associated comparison multigraph at each time instant t ∈ T is given by G ( t ) C = ( S , E ( t ) C ) . Further, recall that the value of FCIX t would become zero for some t = t only if the t -th frontal slice of A would satisfy the PCJ Axiom. In other words, the fulfillment ofthe consistency property of the PCJ Axiom would be equivalent to existence of a set of edge weights E ( t ) C based on which all activities on G ( t ) C are interconnected.Besides, it was shown in Section 2.2 that small perturbations in the neighborhood of A would have negli-gible impact on the value of FCIX t + dt , implying presence of a similar set of interconnecting patterns on G ( t + dt ) C , a feature also known as near-consistency. On the contrary, a relatively large value of FCIX t + dt as compared to FCIX t would indicate a major structural change in A for the time period [ t , t + dt ] .4 Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
Hence, drastically different interconnecting patterns would rather appear at time t + dt as compared tothose delineated by E ( t ) C .Moreover, the interconnecting patterns pertaining to E ( t + dt ) C compared to the ones found on E ( t ) C wouldlack the near-consistency property to a great extent if FCIX t + dt is relatively large. In view of this, makinginference on dominance of relations existing among the assets at t = t + dt would become a near impossibletask due to the lack of proper edge weights E ( t + dt ) C permitting existence of a quantitative comparisonscheme among all the assets involved in G ( t + dt ) C .Insofar as the above-mentioned rationale is concerned, one possible definition for the market volatility thatwe propose is formulated in terms of the financial chaos index, which is presented below. Definition 8 (Market Volatility) . Let the mutual information on the asset prices in the market be modeledusing the tensorial formulation based on a given constitution criterion C , and let E ( t ) C be a random edge setrelated to the comparison multigraph G ( t ) C at time t = t . Then, the market is said to be at a high-volatile(or high-chaos) regime at time t = t if the random RPCM given by A ( t ) C associated to E ( t ) C satisfies atleast one of the following four properties:1. (sensitivity) ∃ δ > , ∀ (cid:15) > N →∞ P (cid:104) (cid:13)(cid:13)(cid:13) adj( N I − A ( t ) C ) (cid:13)(cid:13)(cid:13) N N − < δ (cid:105) < (cid:15),
2. (consistency) ∃ δ > , ∀ (cid:15) > N →∞ P (cid:34) N (cid:80) i,j,l =1 I [ a ( t ) il a ( t ) lj = a ( t ) ij ] N < δ (cid:35) < (cid:15),
3. (discrepancy) ∃ δ > , ∀ (cid:15) > N →∞ P (cid:104) (cid:13)(cid:13) A ( t ) − A ( t + dt ) (cid:13)(cid:13) F max { (cid:13)(cid:13) A ( t ) (cid:13)(cid:13) F , (cid:13)(cid:13) A ( t + dt ) (cid:13)(cid:13) F } < δ (cid:105) < (cid:15),
4. (homogeneity) ∃ δ > , ∃ K > , ∀ (cid:15) > N →∞ P (cid:34) N (cid:80) i,j =1 (cid:0) I [ a ( t ) ij > K ] + I [ a ( t ) ij < K ] (cid:1) N < δ (cid:35) < (cid:15). According to the definition of the market volatility provided above,
FCIX t = t can be regarded as a suitablestatistic to quantify the amount of market volatility at time t ∈ T . In the following, we present severalproperties of our developed market volatility statistic: heory and Applications of Financial Chaos Index FCIX t = t is a robust estimator of the market volatility at time t ∈ T as it relieson the largest eigenvalue of the RPCM at t = t , and hence, it possesses all the properties of theinconsistency measure ψ ( t = t ) defined by equation (2.30).2. In contrast to various dispersion-based approaches, the financial chaos index does not require samplepoints (of size greater than one) realized from a particular asset’s returns to quantify the marketvolatility.3. In contrast to various dispersion-based approaches, which utilize some value- or capitalization-weighting scheme to quantify the market volatility, the financial chaos index captures a differentfeature which pertains to quantifying the market volatility by measuring the inconsistency of mutualchanges in assets’ returns. A brief inspection of Figure 5 shows that the monthly
FCIX t data stream is made of consecutive regimesthat are separated by abrupt changes, owing to the fact that the underlying model producing the data switchesmultiple times among various regimes. In such a situation, the realization of the monthly FCIX t could befurther characterized by resorting to retrospective change-point detection methods. In this framework, thetime series which are also referred to as signals, are segmented into several homogeneous sub-signals.The literature on the segmentation of time series by various change point detection methods is vast. A recentand thorough review of these methods is reported in (Truong, Oudre, and Vayatis, 2018). In our case, weperform the change-point detection on a high-dimensional mapping of ¯ ψ t which is implicitly defined by a kernel function . This method is non-parametric and model-free, and it can be used to segment the time serieswithout having any prior knowledge on the form of the underlying probability distribution that generates thedata, see (Arlot, Celisse, and Harchaoui, 2019; Desobry, Davy, and Doncarli, 2005; Harchaoui and Capp´e,2007; Harchaoui, Moulines, and Bach, 2009) for more details. In short, ¯ ψ t is first mapped onto a reproducingHilbert space H (i.e., a Hilbert space in which the point evaluation of functions is a certain continuous linearfunctional) for which the associated kernel function is denoted by k ( · , · ) : R T≥ × R T≥ → R . Further, therelated mapping function f : R T≥ → H is defined by f ( ¯ ψ t ) = k ( ¯ ψ t , · ) ∈ H , leading to the followingdefinitions for the inner-product and norm (cid:104) f ( ¯ ψ s ) | f ( ¯ ψ t ) (cid:105) H = k ( ¯ ψ s , ¯ ψ t ) , (2.44)6 Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami and (cid:13)(cid:13) f ( ¯ ψ t ) (cid:13)(cid:13) H = k ( ¯ ψ t , ¯ ψ t ) , (2.45)respectively, for any samples ¯ ψ s , ¯ ψ t ∈ R T≥ .Next, assume that the kernel k ( · , · ) is translation invariant, i.e., k ( ¯ ψ s , ¯ ψ t ) = φ ( ¯ ψ s − ¯ ψ t ) , ∀ s, t, (2.46)where φ is a bounded continuous positive definite function on R . Then, the mapping f ( · ) is shown to trans-form piecewise i.i.d. signals into piecewise constant signals within the feature space H . That is, the signalbecomes piecewise constant once it is mapped in the high-dimensional feature space H if it is comprisedof independent random variables with piecewise constant distributions under the assumption that k ( · , · ) istranslation invariant, see (Sriperumbudur et al., 2008) for more details.Then, our change-point detection problem is aimed to detect mean-shifts in the embedded signal whose costfunction is considered to be the average scatter measure (Harchaoui and Capp´e, 2007) given by cost( ¯ ψ t ) := b (cid:88) t = a +1 (cid:13)(cid:13) f ( ¯ ψ t ) − ¯ µ (cid:13)(cid:13) H , (2.47)where ¯ µ ∈ H denotes the empirical mean of the embedded sub-signal { f ( ¯ ψ t ) } bt = a +1 . Note that explicitcomputation of the mapped data is not required since it can be shown with some effort that the kernel costfunction given by equation (2.47) can be expressed as follows: cost( ¯ ψ t ) = b (cid:88) t = a +1 k ( ¯ ψ t , ¯ ψ t ) − b − a b (cid:88) s,t = a k ( ¯ ψ s , ¯ ψ t ) . (2.48)To compute the kernel cost function given by equation (2.48), we employ the Gaussian kernel (also knownas the radial basis function ). This yields that cost( ¯ ψ t ) = ( b − a ) − b − a b (cid:88) s,t = a exp( − γ (cid:13)(cid:13) ¯ ψ s − ¯ ψ t (cid:13)(cid:13) ) , (2.49)where γ denotes a bandwidth parameter.For a fixed number of change points, say K (cid:63) , the change-point detection problem translates to the followingdiscrete optimization problem: min K (cid:63) (cid:88) k =0 cost( { ¯ ψ t } t k +1 t = t k ) (2.50a) s . t . | t | = K (cid:63) , (2.50b) t < t < · · · < t K (cid:63) < t K (cid:63) +1 = |T | , (2.50c) heory and Applications of Financial Chaos Index Time (monthly) . . . . . . F i n a n c i a l C h ao s I nd e x × − Figure 6: Schematic of the obtained segments based on monthly
FCIX t during January 1990-December 2019. where t = [ t , t , · · · , t K (cid:63) , t K (cid:63) +1 ] (cid:124) denotes a vector containing the change points augmented by twodummy indexes t := 0 and t K (cid:63) +1 := |T | . By noting the additive nature of the objective function (2.50a),this optimization problem can be solved recursively by the dynamic programming that relies on the obser-vation that (2.50) is equivalent to the following optimization problem: min cost( { ¯ ψ t } t (cid:63) t =0 ) + K (cid:63) − (cid:88) k =0 cost( { ¯ ψ t } t k +1 t = t k ) (2.51a) s . t . t (cid:63) ≤ |T | − K (cid:63) , (2.51b) t (cid:63) = t < t < · · · < t K (cid:63) − < t K (cid:63) = |T | . (2.51c)This model delineates that once the optimal partition having K (cid:63) − elements of all sub-signals is known,then the first change point of the optimal segmentation is expected to be relatively easily evaluated.Figure 6 depicts the results of solving model (2.51) for the realization ¯ ψ t of a monthly FCIX t duringthe time period from January 1990 to December 2019 using the Gaussian kernel cost function given byequation (2.49). This figure indicates that the monthly FCIX t switches cyclically among multiple regimes.8 Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
In this section, we exploit the relationship between financial chaos index and
CBOE volatility index (VIX)which is a prominent measure for the market’s expected volatility implied by S&P options. Our goalwould be to test whether
FCIX t and VIX t , also referred to as the fear index , both reflect similar patternsof unusual behavior pertaining to the stock market. Then, we investigate their joint long-run behavior aswell as their short-run kinematics, and also their possible causal relations. Thereafter, we borrow a varietyof tools developed in information theory to model the dynamics of the realized and implied volatility byformulating a time-dependent dynamical model for the coupled FCIX t - VIX t system.In Section 2.3 we showed that the financial chaos index is a representative statistic for estimating the mar-ket volatility. However, it remains an important task to investigate the relationship between the realizedvolatility measured by FCIX t and the implied volatility which indicates the forward-looking expectationof the volatility of the market which is measured by VIX t . For this purpose, the use of cointegration tech-niques deems necessary for co-behavior analysis of two time series. By cointegrating these two time series,the possible short-run and long-run behavior existing between the series can potentially reveal themselves.More specifically, by analyzing the cointegrated relationship between FCIX t and VIX t , a long-run equi-librium trajectory can be defined, such that any departure from that path induces equilibrium correction that move the coupled system back towards their stable trajectory. Figure 7 depicts the plots of standardizeddaily FCIX t and VIX t during a time period from January 1990 to December 2019, based on which clearpatterns of co-movement are observed between the series. As a preliminary step towards cointegration analysis of daily realizations of
FCIX t and VIX t , each timeseries is first examined individually by performing the Augmented Dickey-Fuller (ADF) and
Kwiatkowski,Phillips, Schmidt and Shin (KPSS) tests for unit roots and stationarity, respectively. Based upon our exper-iments, both time series reject the null hypotheses of stationarity and presence of a unit root. As a generalrule of thumb, if a time series rejects both the unit root and statianarity tests, it is commonly the indicationof a situation in which the considered time series is fractionally integrated. That is, for a specific time series,say X t , and for some value of the fractional integration order d , X t is said to be fractionally integrated oforder d , denoted by X t ∈ I ( d ) , in case ∆ d X t ∈ I (0) , i.e., ∆ d X t is fractionally integrated of order zero.For our purpose, the parameter d is considered to take its values from the interval ( − . , ∞ ) . Further, the heory and Applications of Financial Chaos Index Time (business days) − S t a nd a r d i ze d I nd e x Financial Chaos Index (FCIX)CBOE Volatility Index (VIX)
Figure 7: Plots of the standardized daily
FCIX t vs VIX t during January 1990-December 2019. employed fractional difference operator ∆ d is defined by ∆ d X t = ∞ (cid:88) n =0 π n ( − d ) X t − n , (3.1)where the coefficients π n ( u ) are obtained as follows: π n ( u ) = u ( u + 1)( u + 2) . . . ( u + n − n ! (3.2)which follows from the binomial expansion (1 − z ) − u = (cid:80) ∞ n =0 π n ( u ) z n , e.g., see (Jensen and Nielsen,2014; Johansen and Nielsen, 2014) for more details on this expansion and efficient estimation of fractionaldifferences.Next, we plot the corresponding sample autocorrelation function and estimated power spectral density foreach time series. Figures 8 and 9 suggest that the autocorrelations for both time series decay hyperbolicallywhich is the signature of fractional (long memory) time series, contrary to the geometric decay whichpertains to short memory processes (the case of d = 0 ). Furthermore, Figures 10 and 11 reveal that the massof spectrum is concentrated near the zero frequency for both time series, substantiating the considerationthat the zero-frequency mass of each process is proportional to λ − d ( λ denoting the frequency) for therespective values of the parameter d , a feature which is characteristics of the fractional time series.Subsequently, we compute the fractional integration order of each univariate series by resorting to the0 Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami applications of extended local Whittle estimator (Abadir, Distaso, and Giraitis, 2007) which is consistentfor d ∈ ( − . , ∞ ) . By choosing the trimming parameter as m = |T | . as recommended by the authors,estimates of the fractional integration orders are derived as being ˆ d = 0 . and ˆ d = 0 . for FCIX t and VIX t , respectively. These results indicate that both time series are nonstationary having stationaryincrements, since their corresponding estimated fractional integration orders ˆ d ∈ [0 . , . . To test for cointegration rank between
FCIX t and VIX t , we employ Johansen’s procedure proposed in(Johansen, 2008) which was further developed and expanded in a series of scholastic works within thepast decade, e.g., see (Dolatabadi, Nielsen, and Xu, 2016; Johansen and Nielsen, 2012; Johansen andNielsen, 2018; Johansen and Nielsen, 2019) for more details. The initial steps of this procedure are tospecify and estimate a fractionally cointegrated vector autoregressive (FCVAR) model of order p for X t = (FCIX t , VIX t ) (cid:124) and then determine the number of cointegrating vectors by performing a likeli-hood ratio test for the rank of the matrix Π = αβ (cid:124) , which is also called the long-run impact matrix . Thisis followed by estimating the resulting vector error correction model (VECM) by the method of maximumlikelihood. More specifically, the level FCVAR( p ) model in error correction form is given by ∆ d X t = α ∆ d − b L b ( β (cid:124) X t + ρ (cid:124) ) + p (cid:88) i =1 Γ i ∆ d L ib X t + ζ + (cid:15) t , (3.3)where L b = 1 − ∆ d is a fractional lag operator and b denotes the degree of fractional cointegration. Theinnovations { (cid:15) t } follow N (0 , Σ ) , which are further assumed to be uncorrelated over time. Also, α and β are × r matrices with ≤ r ≤ . These matrices play important roles in separating the short-run and long-run relationships between FCIX t and VIX t . For instance, α determines the corresponding error-correctionspeeds which distributes the influence of FCIX t and VIX t onto the temporal behavior of ∆ d X t , whereascolumns of β provide the basis vectors for the space of fractionally cointegrating relations. That is, β (cid:124) X t can be considered as the error-correction term, reflecting long-run equilibrium relations between FCIX t and VIX t . The matrices Γ i are further referred to as short-run impact matrices which capture the short-rundynamics of the variables. Besides, the parameter ρ represents a constant term in the model designating themean level of the long-run equilibria such that E [ β (cid:124) X t ] + ρ (cid:124) = 0 . An additional constant term giving riseto a deterministic trend in the levels of FCIX t and VIX t is further captured by the parameter ζ . Note thatin the case of d = 1 the trend would be linear, and if in addition b = d = 1 , then ρ will be absorbed in ζ .Subsequently, the fractional cointegration rank r can be determined by examining the following nested heory and Applications of Financial Chaos Index Lags (business days) − . . . . . . . A u t o c o rr e l a t i o n Figure 8: Autocorrelation function for
FCIX t . Lags (business days) − . . . . . . . A u t o c o rr e l a t i o n Figure 9: Autocorrelation function for
VIX t . . . . . . . Frequency − − − − P o w e r Sp ec t r a l D e n s i t y ( d B / H z ) Figure 10: Power spectral density for
FCIX t . . . . . . . Frequency − P o w e r Sp ec t r a l D e n s i t y ( d B / H z ) Figure 11: Power spectral density for
VIX t . Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
Table 1: Results of likelihood ratio test for cointegration rank.
Rank Log-likelihood LR statistic p -value0 37657.897 16.536 0.0021 37666.060 0.211 0.6462 37666.165 - -hypotheses H ( r ) : r = r , versus their alternatives H ( r ) : r = 2 . We may then formulate the error-correction fractional cointegrating null hypotheses and their alternativesas follows: (cid:40) H EC0 , : r = 0 , H EC0 , : r = 1 , (cid:40) H EC1 , : r = 2 , H EC1 , : r = 2 . These hypothesis could be tested in the following order H , and H , such that H , can only be rejectedif H , is rejected, i.e., the testing procedure gets terminated by returning the result r = 0 if one fails toreject H , , otherwise H , is tested to determine whether r = 1 or not. Note that a likelihood ratio (LR)test statistic called trace statistic is employed to examine these hypotheses whose details are provided in(Johansen and Nielsen, 2018).Table 1 reports the results for testing H , and H , obtained using the computer program provided in(Nielsen and Popiel, 2016). As per this table, H , gets rejected at significance level, which impliesthat r > . On the other hand, there is insufficient evidence to reject H , . Consequently, we may concludethat X t is cointegrated with one cointegrating vector (i.e., rank( Π ) = 1 ), which in turn implies that thereexist a common fractionally integrated trend driving both FCIX t and VIX t . It is worth mentioning that thiscommon trend is more likely to be the so-called implicit common efficient price which permanently inflictsmovements in both time series by following the new market information. Further, note that the optimal lagvalue in our experiments was identified to be p = 7 by fitting FCVAR models to the data and evaluating theperformance of various model selection criteria such as AIC, BIC and HQC.Furthermore, on the basis of the experiments which we have conducted, parameters of the model given inequation (3.3) were estimated as follows (standard errors in parentheses): ˆ d = 0 .
542 (0 . , ˆ b = 0 .
212 (0 . , ˆ ρ = 0 . , heory and Applications of Financial Chaos Index Time (business days) − . − . . . . . . Stable equilibrium relation -sigma range Figure 12: Plot of the long-run equilibrium between standardized daily
FCIX t and VIX t during January 1990-December 2019. ˆ α = (cid:2) .
482 0 . (cid:3) (cid:124) , ˆ β = (cid:2) . − . (cid:3) (cid:124) , ˆ ζ = (cid:2) − . − . (cid:3) (cid:124) , and Π (cid:86) = (cid:20) . − . . − . (cid:21) . In turn, the obtained estimates presented above enable us to characterize the long-run relationship betweenstandardized values of
FCIX t and VIX t . We illustrate this relationship schematically in Figure 12, wherethe horizontal axis represents a stable relation between variables. This figure indicates to a presence ofa long-run common movement between FCIX t and VIX t along the considered time horizon. Note that FCIX t and VIX t are tied together through cointegration, and they both respond to disturbances whichmake adjustments towards the long-run equilibrium relationship. In this respect, Figure 12 reveals thatthere exist several departures of large magnitude from the equilibrium behavior of the two time series. Suchfluctuations are observed to occur mainly during the time periods when the market has undergone severecrises, some of which for instance were listed on previous pages to have happened within the time frame − .In contrast to the analysis of the long-run equilibrium, identification of the short-run relations appear to be4 Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami a more difficult task since there usually are quite a few matrices of coefficients whose interpretations areoften not trivial. To surmount this obstacle, we incorporate the notions of the impulse-response function (IRF) and the cross-correlation function (XCF) along with the notion of approximate entropy within ourframework to study the existing short-run relations. As depicted in Figure 13, the time delay at which twosignals are aligned best corresponds to p = 0 , where the XCF is roughly . However, it is observed thatthe amount of XCF remains relatively high even when the time lag parameter is considered to be as largeas p = ± business days, implying that the impacts of any change in either of the time series could persistin the other series for several time periods ahead.In order to investigate the causal relations which might (or might not) exist within the coupled econom-ical system FCIX t − VIX t and track an impact that either time series makes on the other, we confineourselves to the framework of empirical causal analysis which emphasizes the orthogonalized impulse-response functions to conduct the causal analysis in between these two time series. Basically, an impulse-response function describes the evolution of the reaction from one time series along a specified time horizonafter a shock is inflicted on the other time series at a given moment. Then, in order to estimate the effectof a one-unit shock in (cid:15) t to variables of the system, we decompose the variance-covariance matrix into alower triangular matrix using the Cholesky decomposition . Let P denote the Cholesky decomposition of Σ such that PP (cid:124) = Σ and P is a lower triangular matrix, and assume that u t is such that u t = P − (cid:15) t . Ourexperiments, led to the following estimate for P : (cid:98) P = (cid:20) . . . . (cid:21) · (3.4)We note that the orthogonalized impulse response function would reflect a one standard-deviation impulseto u t since E [ u t u (cid:124) t ] = P − E [ (cid:15) t (cid:15) (cid:124) t ]( P (cid:124) ) − = I . (3.5)Figures 14 and 15 depict the effect of the impact of a one-unit shock in FCIX t on VIX t for businessdays ahead and vice versa, respectively. In order to characterize the regularity and information content ofthe orthogonalized IRFs, the approximate entropy of the signals are further computed, yielding . for FCIX t (cid:32) VIX t and . for VIX t (cid:32) FCIX t . These quantities imply that FCIX t is more sensitive tochanges in VIX t rather than the other way around, as the orthogonalized IRF for VIX t (cid:32) FCIX t attains ahigher value of approximate entropy, and hence, it has a higher degree of irregularity and unpredictability. Itis also worth to mention the synchronicity which is evident like that when both orthogonalized IRF graphs heory and Applications of Financial Chaos Index after the shocks were being imposed. Besides, a magnitude of response forthe case of VIX t (cid:32) FCIX t is considerably larger than its counterpart FCIX t (cid:32) VIX t , which stipulatesthat a one-unit shock in VIX t causes a more intense response pertaining to FCIX t as compared to the casewhere the shock is imposed on FCIX t .One rationale that partially accounts for the above-mentioned assessments is rooted in the fact that the fi-nancial chaos index is a robust estimator of the mutual price changes of the market assets. That is, any sub-stantial marginal increase in the value of FCIX t would demand majority of the underlying assets (S&P )to depart significantly from their steady-state price levels. As a consequence, any abrupt change in the valueof FCIX t due to a shock could potentially be an important indicator of some structural changes in the un-derlying assets. In contrast to abrupt changes in FCIX t , imposing a one-unit shock in FCIX t is less likelyto be a cause of significant structural changes in the asset prices. Thus, the response of such a shock cannothave a large magnitude since VIX t would respond sharply to a shock in FCIX t only if the underlying eq-uities (S&P ) utilized in its computation undergo major changes. The foregoing statement accounts forthe relatively lower magnitude of VIX t response to a shock in FCIX t , as depicted in Figure 14. This figurefurther discloses that an increase in the FCIX t values causes positive increments in VIX t response with thepositive effect peak to occur within the first few days, which is then followed by an oscillation until it meetsthe highest-magnitude response. Thereafter, the response of VIX t to a shock in FCIX t remains positive forthe remainder of the time horizon.Conversely, the response of FCIX t to a shock in VIX t has a slightly different behavior than that explainedabove. As depicted in Figure 15, VIX t has a positive contemporaneous effect on FCIX t and this positiveresponse persists during the whole time horizon. However, it is relevant to stress the presence of a steepdecline in response of FCIX t which occurs immediately after the mentioned contemporaneous effect attime instant t (the ”double dip” observed within [ t , t +5] ). In order to interpret this empirical observation,we find it useful to recall that the lower values of the financial chaos index are linked to a market in whichthe collective judgment of its participants remain consistent over time (here, we understand consistencyin the same sense as that described under the PCJ Axiom). Thus, a positive increase in FCIX t caused bythe contemporaneous effect could potentially be understood as a deviation from the collective consistencycondition . On the other hand, the response from FCIX t decreases immediately after a day is elapsed sincethe collective judgment of the participants in the market becomes more aware of the inconsistencies thathave occurred due to an increased amount of the expected volatility. Thus, the realized market volatility6 Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami reacts to this phenomenon by correcting its value. However, the response from
FCIX t spikes once againafter a few days, albeit this time on the reason that the implied volatility becomes realized in the market,which further makes the task of casting a series of consistent judgments less probable for the agents, whichin turn gives rise to increased values of the FCIX t response.To conclude our discussion, it can be said that VIX t has a stronger causal effect on FCIX t as comparedto the opposite. Hence, the implied volatility of the market plays a key role in characterization of themarket’s realized volatility, further implying that a set of feedback and feedforward causal relations governthe interactions between FCIX t and VIX t . In the previous subsection, we investigated the short- as well as long-run co-behavior of
FCIX t and VIX t and showed the existence of a potential bi-directional causal relation between these two time series. Inwhat follows, we consider an additional aspect of the dynamical relation existing between the series whichpertains to the scheme by which the information content flows from one time series to another and their pos-sible interactions. For this purpose, we first assume that the coupled system FCIX t - VIX t is a given isolatedsystem. We will then introduce and utilize the notions of transfer entropy and self-entropy to quantify thedynamics of the information flow within the coupled system by taking into account the temporal evolutionof the existing information dynamics.Let V and Z denote two dynamical processes which further represent a time-evolutionary coupled dynam-ical system, and let us assume that Z is assigned to be the target process. Then, one can employ somestandard information-theoretic measures to study the information relevant to Z (Cover and Thomas, 2012;Faes and Porta, 2014). A pivotal measure of such kind is the well-known Shannon entropy which quantifiesthe amount of information carried by Z in terms of its average uncertainty, i.e., H ( Z ) = − (cid:88) p ( z ) log p ( z ) , (3.6)where p ( z ) denotes the underlying probability mass function of the discrete random variable Z . Also,another important quantity to consider is the conditional entropy . It is defined by H ( Z | V ) = − (cid:88) p ( z, v ) log p ( z | v ) , (3.7)and represents the average uncertainty in Z when a realization of the random variable V is known.The dynamical properties of our coupled system can then be studied by introducing the notion of the tran- heory and Applications of Financial Chaos Index − − − −
20 0 20 40 60 80
Lags (business days) . . . . . . . S a m p l e C r o ss - C o rr e l a t i o n Figure 13: Plot of the sample cross-correlation between daily
FCIX t and VIX t during January 1990-December 2019. t t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + . . . . . . . . V I X R e s p o n s e × − Figure 14: Plot of the orthogonalized IRF for
FCIX t (cid:32) VIX t during January 1990-December 2019, ApEn = 0 . . t t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + t + . . . . . F C I X R e s p o n s e × − Figure 15: Plot of the orthogonalized IRF for
VIX t (cid:32) FCIX t during January 1990-December 2019, ApEn = 0 . . Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami sition probability which is the probability of transition of the discrete-time system from its past states to thecurrent one. Denote by Z t the present state of the process (time series) Z , while Z − t = [ Z t − , Z t − , · · · ] (cid:124) represents the state variable of the process Z , which contains the complete information related to the pastof the process Z . The self-entropy for Z is then defined as follows: S Z = (cid:88) p ( z t , z − t ) log p ( z t | z − t ) p ( z t ) , (3.8)where z t and z − t denote the corresponding realizations of Z t and Z − t , respectively. Basically, the self-entropy assesses the influence of the past states of the target process Z onto its present state. In otherwords, the self-entropy given by equation (3.8) measures the extent to which the uncertainty about Z t isreduced by the knowledge of Z − t . The self-entropy for the process Z ranges from zero to H ( Z t ) , wherethe zero corresponds to the situation where the information from the past states ( Z − t ) does not contribute toreduction of uncertainty in Z t , and the latter quantity is associated to the case where the whole uncertaintyis dissipated about Z t after acquiring the knowledge of Z − t .Besides, in order to assess the influence of imparting the past states of the process V t on Z t , i.e. to evaluatethe information contained in V − t , one can make use of the well-known transfer entropy measure (Schreiber,2000), which is given by the following formula: T V → Z = (cid:88) p ( z t , v − t , z − t ) log p ( z t | v − t , z − t ) p ( z t | z − t ) · (3.9)Similar to self-entropy, the smallest value T V → Z = 0 is attained when V − t does not induce an uncertaintyreduction in Z t beyond what is already contributed by Z − t . In contrast, the largest value of transfer entropy T V → Z = H ( Z t | Z − t ) is taken on when the totality of uncertainty about Z t that was not already dissipatedby the knowledge of Z − t is reduced by the information provided through imparting V − t . In other words,the direct information measure, signified by the transfer entropy, quantifies the loss of information whenassuming V t does not causally influence Z t when it actually does, e.g., see to (Amblard and Michel, 2013)for more details.Note that the self-entropy and transfer entropy measures introduced above, can also be expressed as follows: S Z = H ( Z t ) − H ( Z t | Z − t ) , (3.10)and T V → Z = H ( Z t | Z − t ) − H ( Z t | V − t , Z − t ) . (3.11) heory and Applications of Financial Chaos Index FCIX VIX . . .
678 1 . Figure 16: Computed diagram of information flow between
FCIX t and VIX t during January 1990-December 2019. The values of self-entropy and transfer entropy computed using daily realizations of FCIX t and VIX t during the time period from January 1990 to December 2019 are depicted in Figure 16, whereby it canbe seen that T VIX → FCIX is considerably larger than T FCIX → VIX . This suggests that the time-dependentinformation transfer is much stronger in one direction than the other one. This observation supports ourprevious discussions on orthogonalized impulse-response functions where it was shown that the magnitudeof impact on
FCIX t responding to a shock in VIX t was observed to be much greater than that of VIX t reacting to shock in FCIX t . Therein, it was also discussed that a shock to VIX t had a contemporaneouseffect on FCIX t , but not vice versa.A subsequent combination of the results obtained using the above information-theoretic measures, namely,the transfer entropy and self-entropy, with those derived by employing the orthogonalized impulse-responsefunctions leads us to draw a conclusion on the direction of causal relations between these two processes.Namely, we conclude that there is a substantial evidence favoring existence of a bi-directional causal re-lation between FCIX t and VIX t . Furthermore, this causal relation implies that VIX t influences FCIX t instantaneously with a significant dispense of information, whereas the opposite casual relation does notshare a similar instantaneous nature. Yet, it still contributes to a substantial transfer of information from FCIX t to VIX t , where the dispensed information are most likely to be transferred in a time-lagged manner.Next, we extend our analysis to the situation where one concerns with the evolution of information dynamicsbetween FCIX t and VIX t over time. To this end, we formulate the following dynamical system model todescribe the various mechanisms of exchange of information between the considered processes. Namely, Transfer entropy quantities were computed using the RTransferEntropy package in R programming language (Behrendt, Dimpfl,Peter, and Zimmermann, 2019), while the self-entropy quantities were computed using the ITS MATLAB toolbox for practical com-putation of information dynamics available online at which is developed based on themethodologies presented in (Faes, Porta, and Nollo, 2015; Faes, Porta, Nollo, and Javorka, 2017; Xiong, Faes, and Ivanov, 2017). Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami
F ViVrF αβθγ δ
Figure 17: Symbolic diagram of information flow between
FCIX t and VIX t . we propose the following time-dependent dynamical system model: ∂F∂t = ( γ + θ ) F − αF,∂V∂t = βV − δV, (3.12)where F := FCIX t and V := VIX t . Also, parameters α , β , γ , δ and θ represent T FCIX → VIX , T VIX → FCIX , S FCIX , S VIX , ApEn(iVrF) , respectively, where iVrF denotes the orthogonalized response function of
FCIX t to an impulse in VIX t . These parameters are also illustrated in Figure 17. It deserves to mentionthat, the term θ is involved in model (3.12) as a result of the contemporaneous effect that a one-unit changein VIX t exerts on FCIX t . Note that the approximate entropy can potentially be viewed as a measure ofinformation content in the orthogonalized IRF for VIX t (cid:32) FCIX t during the specified time horizon.In order to characterize the model (3.12) in more in-depth, we perform its stability analysis. It can be shownwith some effort that the four critical points of this dynamical system are as follows: ( F , V ) = ( αγ + θ , δβ ) , ( F , V ) = (0 , , ( F , V ) = ( αγ + θ , , and ( F , V ) = (0 , δβ ) · Note that these critical points are all feasible as
FCIX t and VIX t can take on zero values from the theoreticalpoint of view. However, such cases are not practical and we preclude them from our analysis. Also, withoutthe loss of generality, we impose a set of non-zero constraints on β and γ + θ . These constraints will preventunboundedness of the elements of ( F , V ) which can lead to the cases of unbounded realized as well asimplied types of market volatility.Next, the Jacobian matrix is obtained as J = (cid:18) γ + θ ) F − α θθ βV − δ (cid:19) . (3.13) heory and Applications of Financial Chaos Index ( F , V ) , which yield ( λ , λ (cid:48) ) = ( αδ + θ , δβ ) . Thereafter, we apply the center manifold theorem to study stability of the critical points by deriving theirdimensions of stable (
Dim( E S ) ), unstable ( Dim( E U ) ) and center ( Dim( E C ) ) manifolds. Using the previ-ously computed values of α = 0 . , β = 0 . , γ = 0 . , δ = 1 . and θ = 0 . , the first criticalpoint is evaluated to be ( F , V ) = (0 . , . . This critical point also has the following manifolds’dimensions: Dim( E S ) = 0 , Dim( E U ) = 2 , and Dim( E C ) = 0 , which indicates that in fact it is a saddle point.It is also interesting to note that the elements of ( F , V ) = (0 . , . reside on roughly all-time highsof their respective time series FCIX t ( max = 0 . and mean = 0 . ) and VIX t ( max = 80 . and mean = 19 . ). In other words, there could potentially exist hidden acting forces which prevent the marketfrom retaining its volatility state at the neighborhoods of extrema. This assertion can be verified by notingthat the average values for FCIX t and VIX t during the considered time period January -December are both substantially less than their extrema, an observation which could well be due to the fact that ( F , V ) is a source. In this paper, we first axiomitized the pairwise comparative judgments and demonstrated that the so-calledreciprocal pairwise comparison matrices were the only members of the family of pairwise comparison ma-trices that would satisfy the formulated axiom. Subsequently, various algebraic properties of the reciprocalpairwise comparison matrices such as their rank, trace, powers, eigenvalues, etc, were discussed, and it wasshown that the reciprocal pairwise comparison matrices are suitable structures to model the procedures thatunderlie agents’ thought processes and their mechanisms of casting judgments. Thereafter, the sensitivityof the eigenvalues of the reciprocal pairwise comparison matrices to perturbations in their entries were in-vestigated, and it was established that the largest eigenvalues of the considered matrices would be robustw.r.t. such perturbations.Next, we defined a special class of spatio-temporal tensors by extending the reciprocal pairwise comparisonmatrices to a tensor domain, which in turn enabled us to effectively embed the collective judgment of the2
Masoud Ataei, Shengyuan Chen, Zijiang Yang, M.Reza Peyghami agents throughout time. An inconsistency function was then developed based on the largest eigenvalues ofthe frontal slices of the constructed tensors, in order to examine the consistency of the judgments cast by theagents throughout time. A relationship between the mentioned inconsistency function and the rank- esti-mates of the constructed tensors was further established, upon which the financial chaos index was defined.Several properties of the financial chaos index were also studied, including its robustness for estimating themarket volatility, which enabled us to reliably perform the tasks of market segmentation by resorting to theapplications of the proposed index.Furthermore, we clarified the connection between the equity and option markets by studying the relationshipbetween the financial chaos index (as a measure of market’s realized volatility) and the VIX (as a measure ofmarket’s implied volatility). More so, the causal relations existing among the financial chaos index and VIXwere exploited using the orthogonalized impulse response functions and information-theoretic measures.Our computational results which pertain to the time period January 1990- December 2019 imply that thereexist a bidirectional causal relation between the processes underlying the realized and implied volatility ofthe stock market within the given time period, where it was shown that the later had a stronger causal effecton the former as compared to the opposite. Acknowledgments
This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC).The authors would like to gratefully acknowledge contributions of Professor Vladimir Vinogradov for hisfruitful discussions regarding the statistical procedures presented in this paper. The authors are further grate-ful to Professor Torben G. Anderson for providing constructive comments and feedback on the employedstatistical procedures as well as economical interpretation of the reported results.
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