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More gen- any attempt to model and regulate an economic system. erally, bottom-up risk factors associated with a given in- Thus, we consider the notion of causal completeness in strument can influence the prices, such as price impact or the context of being able to compare complex models by instantaneous liquidity effects in the presence of their ability to explain phenomena or quantities which herding26,79,102. Hence, price evolution may be driven can be observed or measured. by both bottom-up and top-down causation, where the It is suggested by Ellis36 that there are at least five latter are described by exogenous factors, leading to in- different classes of top-down causation30,35,36, listed here novations in couplings and feedbacks through the many with slight variation in nomenclature: layers of hierarchy within a financial system: between countries, companies, investment funds, hedging strate- TDC1 Algorithmic top-down causation gies, fund managers, dealers and traders, and bids and asks in the price discovery process. TDC2 Top-down causation via non-adaptive informa- At different levels, which are characterised by different tion control phenomena, one expects different theories to function. At TDC3 Top-down causation via adaptive selection some levels in the hierarchy there may be little stability in these effective theories, while stability may be enhanced TDC4 Top-down causation via feedback control of at other levels. The effective theories that function at the adaptive goals level of the price discovery process, bounded by the mar- ket micro-structure at various exchanges on very TDC5 Top-down causation with adaptive selection of time-scales, may be more stable than the dominant ef- adaptive goals fective theories which drive prices at the level of invest- ment strategies that operate on much longer time scales. In this paper, we present a simple hierarchical causal- These in turn may be coupled together in complex and ity model which incorporates a plausible approximation interesting ways that are fundamentally nonlinear111. of empirically measurable interactions, with a variety of It is also the case that different theories may be effec- top-down and bottom-up causality features, as a path tive at the same level. The number of different plausible to revising prevailing theoretical foundations in finan- theories is a decreasing function of available information, cial economics. To substantiate the model proposed, we since deeper or broader insight about system workings provide a candidate for each of the distinct causation can eliminate theories which become inconsistent under classes listed above, to give insight into the complex na- the incorporation of new information. ture of causation and chance in financial markets. Parts In the next section we give an overview of how markets of our hierarchical model remain consistent with existing may be mapped into a multilevel causality framework, arbitrage-free approaches to factor modelling111. with consideration of key concepts. In Section III we To describe our model, we commence with a review discuss specific agent-based models to be incorporated in of a seminal construct in the modelling of equity mar- a hierarchy of complexity and in Section IV we clarify ket causality. In , any (theoret- how risk, information and innovation lead to emergent ically) riskless portfolio with an expectation of positive prices in a concrete multilevel model. In Section V we future returns must cost something now90,91. Expected provide a general discussion of exemplars for actors which discounted returns can be estimated via a linear factor drive top-down causation and our conclusion is given in model, based on market-wide risk premia and historic, Section VI. (mostly) de-correlated -specific returns. In the fac- tor model of Ross90,91, risk premia which impact prices may be due to exogenous or endogenous contributions. II. TOWARDS A HIERARCHICAL VIEW OF Price changes which are different from noise are under- FINANCIAL MARKETS stood to be caused by top-down exogenous factors. Nev- ertheless, the resulting co-movements in prices may be A. Reductionist perspectives describable by endogenous factors within a given level in the hierarchy of complexity, due to efficient assimila- Notions of market equilibrium and price tion of exogenous information. In this model, systematic efficiency24,41,73,74 are fundamental to orthodox models deviations from predictions provided by the explanatory in finance. Equilibrium dynamics in our application re- factors cannot be sustained. Hence, over longer horizons late to how prices are influenced by risk and information the only return that a portfolio should will be a and how these are used by various agents in the system. risk-free rate plus the risk premia associated with a finite It useful to recall alternative views of how markets can set of common risk factors. be considered to function in this regard: It is generally accepted that real market dynamics are much less trivial. Deviations from prices predicated by EMH Efficient Market Hypothesis:37,93 Here prices are prevailing models can arise. Such events may trigger fully rational, they reflect fair-value and are the trade based on an expectation of profit due to detection best predictors of future prices because price of a mispricing, even though the anomaly may only be changes can only be due to unpredictable news. 3

This is a very broad aggregative perspective. This complexity, in tandem with the set of causation classes is considered to be a normative idealised state that that act on these levels. Prices in such a system are captures salient features of in equilib- strongly influenced by both top-down risk factors as well rium. Weakened forms of this perspective attempt as information variables that constitute bottom-up and to reconcile this with actual market behaviour. top-down state-variables. The prevalence of such drivers

17,95 are time-varying and depend on market conditions as NMH Noisy Market Hypothesis: Here there is a mean- well as actor interactions that exploit the market models ingful fair-price for companies, but speculative through various types of top-down causation. Key ques- noise trading pushes prices away from this fair- tions are: to what extent can an adequate characteriza- value. This is a defense of the efficient market hy- tion of financial markets be carried out in a bottom-up pothesis in that it provides a mechanism for prices manner and which features of financial markets can only to deviate from fully rational prices. It is assumed be possible through top-down causation within a hierar- that there is a mean-field theory that can capture chy of complexity? the market’s salient features after the noise has been averaged away. B. Levels of complexity AMH Adaptive Market Hypothesis:42,78 Here prices do not necessarily reflect fair-values . The market is a machine that sets prices. On the one hand, in- Modelling financial market dynamics via a large num- efficiencies can be substantial, and on the other , ber of interacting agents, who function at various levels patterns in prices may disappear as agents evolve within a financial market, is of interest because it is a nat- profitable strategies to exploit them. This repre- ural interpretation of real-world interactions. Relatively sents a form of scientific pluralism and is a signif- simple models can be built to capture essential features icant departure from the efficient market view in in the hierarchy of complexity. that there is much more than just noisy deviations In our approach, asset prices are key observable quan- from rational prices. There are instead many no- tities of the system and reflect how the system acts on tions of fair-value and there is no aggregate notion information, as conditioned by the prevailing classes of of a fully rational price. causation driven by actors who make decisions. We use the idea of actors, which serve as are exemplars CDEM Critical Dynamical Equilibrium Market:21 Here of causation types, to describe how effective theories at price diffusions are the result of a critical dynami- different levels can be used to account for complexity and cal equilibrium between persistent effects (liquidity adaption in a given financial market. takers which correlate orders) and anti-persistent The following list summarises a hierarchy of complex- effects (liquidity providers which generate mean- ity at which effective theories hold, coupled to top-down reverting forces). No-one can know the fair-value and bottom-up causation, in a manner consistent with reference price, even if it existed. This is the state- theoretical foundations: ment that there is no such thing as fair-value and there is no mean-field theory that can capture all Level 1 Individual agents generate buy and sell decisions salient market features. for an asset to generate aggregate price information via an Ising model of agents on a lattice coupled to Clearly these philosophical perspectives on what is at- an external field. tainable offer very different modelling paradigms, rang- ing from one where accurate prices exist as fundamen- Level 2 Emergence of aggregate asset prices follows from a tal information, which can be discovered, to the as- Potts model for asset prices. This provides a frame- sumption that prices are emergent phenomena deter- work for bottom-up price emergence from collective mined by agents acting on partial and potentially biased interactions in price dynamics which include top- 17,84 information . down influences, modelled by an external field. To consider examples of the type of reductionism, one need only look at Keynesian economics, as described Level 3 At this level, we propose an affine model which de- by Shiller3: instead of rational bottom-up agents, there scribes aggregate asset pricce fluctuations, driven are animal spirits that lead to aggregate behaviour via by net market interpretation of bottom-up informa- bottom-up causality which is greater than that allowed tion variables and shared market risk factors. As- within the efficient market perspective of bottom-up ra- suming that additional influences amount to - tionality. The approach is still reductionist and places term, uncorrelated noise, this level can be repre- emphasis on the unique attributes of the agents, in par- sented by a neo-classic arbitrage pricing theory ap- ticular, their behavioural characteristics make the whole proach. merely as an aggregate of its parts. In view of consensus that financial markets are more Level 4 Prices are impacted by shared, top-down financial than reductionist13,31,42, we propose a mechanistic com- and economic factors which arise beyond trading bination of causal linkages between levels in a hierarchy of activity in financial markets. 4

Level Model Processes Causation Type Exemplar Actor 6 Meaning and Ethics TDC1 (section V A) “Predictors” 5 Market-structure. TDC2 (section V B) “Profiteers” 4 Economic Factors. Macro-economic and Finan- TDC3 (section V C) “” (Section IV) cial market generated risk TDC4 (section V D) “Traders” factors. TDC5 (section V E) “Regulators, Rulers” 3 Asset price aggregation Risk factors, information vari- (Section IV C) ables, and noise combine to TABLE II: Actors serve as exemplars for causations. They form aggregate assets prices. are able to act through-out the hierarchy of complexity given 2 Asset price emergence The collective behaviour of in Table I. (Section IIIB) assets generate noise in the presence of different kinds of information. The use of an actor model as a possible computation 1 Agent interactions Agents buying and selling an metaphor to describe exemplars of possible core classes (Section IIIA) individual asset in the pres- of top-down causation is relevant as a complement to an ence of different kinds of in- agent based modelling approach, which is effective for formation to generate prices. capturing bottom-up interactions117. Financial markets TABLE I: Hierarchy of complexity given by levels and associ- are evolving open-systems, a feature that poses interest- ated processes. Levels can be coupled both up and down the ing problems for many computational models, for exam- hierarchy through various feedbacks. Each level has a prevail- ple, a bottom-up agent based modeling framework and ing effective theory that can be implemented as a model. actor models offer a relatively simple capacity to cope naturally with indeterminism and concurrency. Three core features characterise an actor61,62: (i) the Level 5 Market structure incorporates the regulatory ability to process information, (ii) the ability to store in- framework prevailing within a market, across mar- formation, and (iii) the ability to communicate. When kets, as well as more generic market structure, seg- combined with it functional axioms: (A1) actors can cre- mentation and market micro-structure. This level ate more actors, (A2) actors can send messages and (A3) influences and determines the nature of arbitrage actors decide what to do with the next message. Al- pricing and the attainable ranges for prices. though messages are delivered at most once, without du- plication, the actor model is a many-to-many model in Level 5 Beliefs, meaning and ethics create and influence be- which the use of a metaphor of identity of an actor and an haviour in the greater economy. This level of causa- actor’s address can enable a single actor to be associated tion level abstracts how an economy may be organ- with one or more addresses and enable one address to be ised psychologically through agent beliefs, anticipa- associated with one or more actors by their identity. This tions, expectations, interpretations and ethics13. creates the potential for a very fluid dynamic between ac- The broad view considered is summarised in Table I, tors that has the possibility of high-concurrency. where we represent structures and processes acting across The next feature of interest is the actor model’s ability 61 the levels of the hierarchy of complexity to capture the to generate and handle indeterminism . Indeterminism key averaging scales of the system. is a foundational feature of any system that is truly con- current, because there is nothing to ensure or tag the order in which messages need to be managed. Hence, C. Computational implementations the system has to order things by itself and in the ac- tor model, indeterminism is handled naturally using the Agent-based modelling is a well-established discipline concept of arbiters, which take simultaneous inputs and within heterodox economics14,33,47,65,77,106. We identify generate sequential outputs. actors 61,62, which serve as exemplars for the types of cau- A consequence of the actor model for computation sation under consideration. An actor is a computational is that a global consensus is not possible. In the lan- model, which is used here to characterise a specific type of guage of spin-models, this can be understood in terms top-down causation, and actors incorporate universal fea- of the phenomenon that there is no mean-field theory. tures of the level at which they function. Hence, one may However, this feature encodes a deep form of pluralism differentiate between the concepts of agents and actors as within a given system, where local arbitration is possi- one differentiates between bottom-up, interacting model ble but one local view can really be quite different from components and top-down causation. Actors may explic- another. Such local positions can be inconsistent but itly accommodate concurrency and indeterminism and still be functionally important to the viability of the sys- the fact that there is a reduced set of causation classes tem itself. Thus, the actor model can be used to encode leads to a reduced set of actors, differentiated by their top-down causation, as distinct from the bottom-up cau- actions. sation which is natural to the agent based model perspec- Table II describes possible actors functioning across tive. the hierarchy of complexity. Conceptual issues relating to cohesion and coupling97 5 are at the heart of approaches to fragility and robust- III. AGENT-BASED PERSPECTIVES ness in complex software systems. Here, we recall that a goal of software development is to build a system with A. An Ising market model for a single stock with low-coupling and high-cohesion in order to create a 97 system which is both robust and fit-for-purpose . Cohe- Key strategies applied in agent based models to gen- sion relates to the degree to which elements in a system erate prices for a single asset are26,44,79: belong together in terms of their function or responsi- bilities. This is differs from procedural coupling within • do what you neighbours do, which captures herd a system, which is concerned with the degree to which behaviour of noise traders, for example as in the processes and components of the system depend or rely Lux-Marchesi model79, and on each other. • do what the minority does, which captures the be- Since similar features are relevant for the modern fi- haviour of traders with opinion about the values of nancial system, the actor model approach incorporates trade-able asset. economically meaningful elements: from a regulatory perspective, one would be aim to avoid tight-coupling These conflicting interactions have been combined into (strong internal dependencies) in a financial market, par- a simple spin model by Bornholdt20 where the standard ticularly when the system may have low-cohesion and el- nearest neighbour interactions capture the neighbour in- ements of the system do not belong or connect. teractions, while the coupling to the global observables is captured via the global magnetization of the system. In Bornholdt model the local field hi(t), which is used to D. Emergence describe net demand or net price increment, takes on the form: 7 Aristotle claimed that N N 1 h (t)= J s − αC (t) s (t). (1) i ij j i N j The whole is something over and above its j=1 j=1 X X parts, and not just the sum of them all... Here the model has i =1,...,N spins with orientation si(t)= ±1. The first term is the local Ising Hamiltonian Emergence refers to the concept that novel and with nearest-neighbour interactions Jij = J and Jij =0 coherent, distinguishable structures or patterns can for all other interactions. This induces order based on a arise during the process of self-organization in complex 6,7,28,50,87 given scale of interaction. The second term is the global systems . Our approach is narrowly confined to coupling where a given strategy employed by the i-th how random interactions lead to some form of weak emer- 6,28,50 agent is captured by the spins Ci(t). gence due to feedbacks . The local field determines the dynamics of the spins Even though noise provides additional degree’s of free- according to dom that are important for adaption17, it has been shown that by adding uncorrelated noise to oscillators one can 1 si(t + 1) = +1 with p+ = , (2) induced explosive synchronization, i.e., very sharp phase 1+ e−2βhi(t) 76,98 transitions . Barab´asi and Albert show that for suf- si(t +1) = −1 with p− =1 − p. (3) ficiently heterogeneous network topologies, simple corre- lations can induce explosive phase transitions15. The simple case described by Bornholdt sets Ci(t) = +1. This creates a force that aligns the minority of spins in Thus, the role of noise within the hierarchy of complex- the system. These types of traders can be considered as ity needs to be considered with care, since the related ca- fundamentalists as each agents tries to align relative to pacity for creating opportunities for innovation can pro- fundamental value and as such are contrarian in nature; vide pathways to tight-coupling and lead to fragility or this would suppress large fluctuations in the system. even catastrophic failure. The case with Ci(t)= −1 would cause alignment with In order to understand the nature of the couplings the majority of agents in the system and as such can be across the hierarchies, phenomenological features within considered a proxy for chartists where agents follow the a given level must be identified to allow quantification broad consensus views and as such herd with the global of coupling strengths. A mechanistic approach towards behaviour; this cause global alignment of the system and monitoring may be futile within a highly adaptive com- as such allows large fluctuations in the system. plex system, such as a financial market, since it is only For a fixed ratio of chartist and fundamentalist in the in hindsight that tightly-coupled paths of causation may system the chartist would dominate. Therefore, an in- become apparent. It may be more feasible to identify teraction term allowing for transitions between chartists observable universal features of various top-down causa- and fundamentalist was introduced20: tion types at different levels in the hierarchy of complex- ity, while discerning how bottom-up features may trigger N events based on tightly-coupled top-down and bottom-up Ci(t +1)= −Ci(t) if αsi(t)Ci(t) sj (t) < 0. (4) j=1 causation features. X 6

Here the idea is that an agent in the majority will choose • a stock’s price can include unique information, i.e. the strategy Ci(t) = +1 while the agent in the minority the model should include non-systematic behaviour will choose the strategy Ci(t) = −1 so that each agent which is unique to a particular stock. chooses a risky strategy for the prospect of higher re- One can apply the super-paramagnetic ordering of a turns, but at a cost. q−state Potts model directly for cluster identification18. Agents will try to anticipate losses given a participa- In a Potts model market, each stock may only belong to tion threshold and attempt to switch to that asset which one of q−states18,49,75 and each state can be represented will become the next majority or minority asset depend- by a cluster of similar stocks. Cluster membership is ing on the . This model takes on a simpli- indicative of some commonality among the cluster mem- fied local field when strategy adjustments are considered bers and each stock has a component driven by state instantaneous: dynamics and a component influenced by stock specific

N N noise. In addition, there may be global couplings that 1 influence all the stocks, represented by the external field h (t)= J s − αs s , (5) i ij i i N j to model a market mode. j=1 j=1 X X In this approach, the cost function can be interpreted

as a Hamiltonian whose low energy states correspond to with a global coupling constant α > 0. The first term aligns spins, while the second terms causes the spins to cluster configurations that are most compatible with the change sign if the global magnetization become large. data sample. Structures are identified with configura- tions, denoted S = {s }N , where the s denotes the To understand the interpretation of the model in the i i=1 i equivalence class or cluster to which the i-th object be- context of a financial market one can interpret the spins longs. The Hamiltonian takes on the form: states as demand for an asset. The magnetization term varies with net demand and represents a function of ag- 1 H = − Jij δ(si,sj ) − kisi. (7) gregate price change26,44: β s ,s ∈S i iXj X N Here, the spins si range over q-states and can be inter- 1 18,75 X(t)= sj . (6) preted as spins in the Potts model , with k serving N i j=1 to tune external influences. The first term represents X common internal influences and the second term rep- Price fluctuations are the observable property of the resents external influences. One can interpret the cou- system and a key success of this model is that it exhibits pling parameters Jij as being functions of correlation intermittency and clustering, as well as large coefficients49,75 in order to fix a distance function that jumps in returns, without necessitating fine tuning. One is decreasing with distance between objects. If all the could introduce an additional parameter to index each spins are related in this way, then each pair of spins is stock and in a more generic framework one should also connected by some non-vanishing coupling Jij = Jij (cij ). consider the excess demand in more generality, as a func- The case where there is only one cluster can be thought tion of price and in order to solve a market of as a ground state. On the other hand, if the system equation where the aggregate magnetization is becomes more excited, then it could break up into ad- zero across the entire system. ditional clusters. Each cluster would have specific Potts For the descriptive purposes used here, we assume lin- magnetizations, even though the nett magnetization may earity in demand function with a constant market depth be zero. Generally, correlations may be both a function to allow us to view the magnetization as a measure of the of time and a temperature factor in order to encode both induced price changes118 . the evolution of clusters as well as the hierarchy of clus- We now move to a Potts model description of the price ters modelled as a function of temperature. dynamics of collections of stocks to introduce a second We seek the the lowest energy state that fits the data. level in our hierarchy of complexity. In order to parameterize the model efficiently one can choose to make the Noh ansatz83 and use this to de- velop a maximum-likelihood approach49 rather than ex- B. A Potts model market for interacting stocks plicitly solving the Potts Hamiltonian numerically18,75. In this investigation, we ignore the second term when fit- Stocks whose prices evolve under correlated dynam- ting data because we include shared factors directly in ics may be group into co-evolving collections or clusters. later sections when we discuss information and risk and A key idea underlying such a group based stock pricing the influence of these on price changes. model is that we seek to reflect two distinct features: To interpret the system observables as being the clus- ter configurations which evolve through time, the price • a group of stocks have something in common, i.e. increments associated with the i-th stock may be given the shared behaviour within a groups of stocks can by: be described by modelling a the systematic compo- 2 nent. Xi(t)= gsi ηsi + 1 − gsi ǫi, (8) q 7

where the cluster related influences are driven by ηsi where ri,t+1 is the excess return of the i-th stock and and the stock unique effects by ǫi. Both parameters are rp,t+1 are the excess returns on the factor-mimicking assumed to be mutually independent Gaussian random portfolios that proxy potential shared risks, here for the variables, with unit variance and zero mean116. The rel- p-th risk factor. This shared risk could be of a statisti- ative contribution of the group component to returns is cal nature, for example the eigenmodes of the correlation 109 controlled by the intra-cluster coupling parameter gsi . matrices of asset price returns , or they could be of a This model encodes the idea that stocks that have fundamental nature. The ǫi,t are zero-mean noise contri- something in common are in the same cluster but also butions which are uncorrelated with the risk factors, but that stock membership in clusters is mutually exclusive may be correlated across assets. and that intra-cluster correlations are positive. Now suppose the following serves as a model for the conditional expected returns: IV. HIERARCHIES, RISK AND EMERGENCE: P A MULTILEVEL MODEL Et[ri,t+1]= αi,t + βi,p,tEt[rp,t+1]. (11) p=1 We exhibit how risk and information may be incorpo- X rated in a level within a hierarchical system such that If this is substituted into Eqn. (10) and ri,t+1 is replaced the basic features of arbitrage pricing theory are recov- withr ˜i,t+1, to indicate that the quantity is now the mea- ered as a level within the hierarchy of causation. The sured return (wherer ˆ is used to denote estimated re- level is described by a family of linear risk and informa- turns), then we obtain a conventional pricing model used tion models, where a rationale for the family of models in financial economics40: is reviewed in detail in Wilcox and Gebbie111. We note that more advanced stochastic modelling may be used to P analyse derivative securities in incomplete markets under r˜i,t+1 = αi,t + βi,p,tr˜p,t+1 + ǫi,t+1. (12) p=1 assumptions of semi-martingale dynamics for underlying X 34,48,86 processes . Early convention wisdom was argued that if risk factors First, we describe three simplified assumptions which in the model explain returns, then the term, αi,t, characterise a simple risk-based pricing model: should be vanishingly small. The typical approach is to • Agents are only interested in means and covari- use time-series regression to estimate a static for a ˆ ˆ 2 ances for a view of the underlying distribution of as- sample period, i.e. Et[βi,p,t] ∼ βi,p = Σi,p/σˆp based on set prices and their relationship to underlying risks. the sample estimated covariance matrix. A limitation of this model is that there may be vari- • Expected returns in excess of a risk-free asset only ables that influence asset prices in practice which have depend on their relationship to shared risk factors little to do with shared risks. We refer to these variables to reflect the idea that non-systematic behaviour as shared information variables. which is specific to a particular stock is short-lived.

• There is a unique risk-free asset for tractable risk- B. Shared information neutral modelling perspectives. Haugen and Baker (1996)56 provide a useful approach Next we review a standard model for shared risks driv- in financial economics for incorporating shared informa- ing asset prices, using the idea of no-arbitrage pricing. tion variables. Their model has a similar form to that We then perturb this view by directly including informa- for shared risks as in Eqn. (11), but expected returns tion variables. Et[ri,t+1] are now estimated in terms of lagged attributes. The latter are typically company or asset specific at- A. Shared risk tributes which are bottom-up information variables, de- noted by θ˜i,m,t, for the m-th source of information rel- The Fama and French38,40 model for accounting for evant to the i-th stock at time t. The cross-sectional temporal risks associated with unanticipated returns, model specification takes on a linear form as follows: M ri,t+1 − Et[ri,t+1], (9) r˜i,t+1 = αt+1 + δm,t+1θ˜m,i,t + ǫi,t+1 (13) m=1 has served as a standard approach to using arbitrage pric- X ing theory. More generally, the times-series of excess re- The conditional expected returns for the i-th stock can turns may be modelled as follows: be estimated by:

P M ri,t+1 − Et[ri,t+1]= βi,p,t [rp,t+1 − Et[rp,t+1]] + ǫi,t+1(10), Et[ri,t+1]=Et[αt+1]+ Et[δm,t+1]θ˜i,m,t ∼ αi,t, (14) p=1 m=1 X X 8 where the expected pay-off, Et[δm,t+1], to the lagged in- only have short-term departures from being negligibly formation variable can be estimated by a variety of means small, and β’s are allowed to be time-dependent. to compute the expected conditional returns. Using the above insight, we propose the following fac- This model is not consistent with the Fama-French40 or torisation of the right-hand-side of Eqn. (14) for a unify- arbitrage pricing theory models. However, it includes fea- ing framework to incorporate top-down and bottom-up tures which support avenues for top-down and bottom-up economic information. The model is built on the two feedbacks. In addition to the model’s practical useful- sources of causality: ness, for small deviations from the arbitrage pricing the- ory framework, one can combine information variables • generic information variables, denoted Zk,t for the with risk variables and retain a risk interpretation for k-th source of top-down information, where Zk,t asset pricing in the long-time limit. referred specifically to macro-economic information in46, and

C. Information, risk and causation • company specific variables, θi,m,t as in Haugen and Baker56, which serve as bottom-up informa- tion variables , now in a single unified framework. Ferson and Harvey46 combine time-dependent risk fac- tors in Eqn (11), the Et[rp,t+1], with lagged asset specific Following the approaches of Haugen and Baker56 and information variables by conditioning the β coefficients Ferson and Harvey46 by effectively conditioning the with time-dependence as follows: βi,p,t = βi,p,t(θi,m,t) := coefficients of both the top-down market information 0 M 1 and bottom-up stock-specific information terms using b i,p + m=1 b m,pθi,m,t . This is analogous to the cross-sectional regression on previous-time top-down and manner in which the time-t expectation of δ is com- h P i m,t+1 bottom-up information. This yields this following repre- puted from θ for Eqn. (14). Hence, the following i,m,t sentation for coefficients: model specification for the conditional expected returns is obtained: K 0 αi,t = α i + αi,kZ˜k,t, k P X=1 E [r ]= α + β (θ )E [r ]. (15) P K M t i,t+1 i,t i,p,t i,m,t t p,t+1 0 2 ˜ 1 ˜ p=1 βi,p,t = b i,p + b i,k,pZk,t + b m,pθi,m,t . X p=1 " k m=1 # X X=1 X Thus, by substitution, the time t + 1 measurements can be obtained by expanding Eqn. (10) to obtain the Putting it all together we obtain: following explanatory model for returns: K P,K 0 1 ˜ 2 ˜ P,M r˜i,t+1 = α i + α i,kZk,t + b i,k,pZk,tr˜p,t+1 1 ˜  k=1 p=1  r˜i,t+1 = αi,t + b m,pθi,m,tr˜p,t+1  X kX=1   p=1  P,M  mX=1     1  P + b m,pθ˜i,m,tr˜p,t+1   + b0 r˜ + ǫ , (16) p=1 i,p p,t+1 i,t+1 mX=1 p=1 X P 0 where writing in the original Haugen and Baker variables + b i,pr˜p,t+1 + ǫi,t+1. (18) p=1 would give the following representation: X

M This gives a theory that is driven by the returns of

r˜i,t+1 = αi,t + δm,t+1θ˜i,m,t shared risk as proxied by the factor-mimicking portfolios ( m=1 ) rp,t but conditioned by top-down information variables X P Zk,t (for example slowly changing economic features) and 0 + b i,pr˜p,t+1 + ǫi,t+1. (17) bottom-up information variables θi,m,t (for example asset p=1 specific features) in the presence of potentially correlated X noise ǫi,t. Since the estimated coefficients in Eqns (14) and This particular framework remains consistent with ar- (15) are obtained by cross-sectional regression, the bitrage pricing theory111 and illustrates that, even within model is nonlinear and has been supported by empiri- arbitrage pricing theory, departures from equilibrium are cal evidence111. The combined model has two important permitted through either top-down or bottom-up causa- model features. Firstly, shared risk factors include time- tion between different levels in the hierarchy of complex- dependence on the information variables and secondly, ity. the model conforms with arbitrage pricing theory as long The finally modification we make to our model is to as the non-vanishing bias terms, the so-called α terms, modify the noise term to accommodate emergence and 9

Variable Type Description Symbol Level Causation Type Typical Response to Noise

Risk factors The factor mimicking port- rp,t+1 5-4 TDC1 (section V A) Noise is a hinderance and under- folio for the p-th risk fac- mines the adopted model. tor at time t + 1 from state TDC2 (section V B) Noise effects are reduced because variables Zk,t and θm,t. a fixed goal is aimed at averaging Top-down The k-th shared state vari- Zk,t 5-3 away noise. information able at time t. TDC3 (section V C) Noise facilitates or is exploited in Bottom-up The m-th state variable for θi,m,t 5-2 the long-term through adaption. information the i-th asset at time t. TDC4 (section V D) Noise both facilitates and is cre-

Cluster specific The noise term for si-th ηsi,t+1 5-2 ated for adaption under selection. noise clusters at time t. TDC5 (section V E) Noise both facilitates and is cre- Asset specific The noise term for i-th as- ǫi,t 5-1 ated and system goals and selec- noise set at time t. tion criterion are adapted to ex- ploit accordingly. TABLE III: The key variables in the model as driven by noise formation using spin models and the top-down and bottom-up TABLE IV: The table summarises some typical responses of information variables that describe both top-down risk factors the causation classes to noise and the feedbacks associated as well as there time-dependence. Actors are both able to with noise. influence the state-variables as well as select for them as part of the various causation classes. We have demonstrated a model which couples together different parts of a hierarchy of complexity and which complexity via innovations in coupling in the agent based permits random innovation leading to innovation, into model. The key variables for the combined model are a single pricing equation. Before concluding, we dis- summarised in Table III. cuss specific examples of top-down causation and possible roles. D. Emergence

E. Block variables and hierarchies of complexity In our model, we address scenarios where stocks group together randomly on short-time scales due to liquid- ity, or lack thereof, asynchronicity of trading or market It is well understood that money serves the purpose of micro-structure effects, even if econometric similarities smoothing over mismatches in supply and demand at the and shared risk between assets still drive stock cluster- level of direct bartering to facilitate transactions. More ing over longer horizons. Such incidence could introduce generally, the ability to transact instantaneously can be measured by estimating liquidity. Black17, Ellis36 and non-trivial lead-lag and feedback effects. To model this 58 idea, we apply the group model of Noh83 (following the Kirman and Helbing observe that noise is a fundamen- approach of Giada and Marsili49). This enables us to tal feature of complex adaptive systems because it adds slack by allowing additional degrees of freedom to the consider a cluster noise term ηsi for unanticipated price changes (due to the random fission and fusion of mu- set of options available in a given systems. We consider tually exclusive groups of assets) into order to describe how noise modelling in complex systems facilitates fea- unanticipated returns in Eqn. (10) as follows: tures such as the ability or incentive to adapt through selection. Typical responses of the top-down causation ri,t+1 − Et[ri,t+1]= αi,t classes to noise generated in complex adaptive systems P are given in Table IV. + βi,p,t [rp,t+1 − Et[rp,t+1]] One approach to modeling a hierarchy of complexity p=1 within a spin system is by using block variables, following X the renormalisation group approach29,51,53,71,102,113. The +g η + 1 − g 2ǫ . (19) si si,t+1 si i,t rationale is that larger block domains can be representa- 119102,113 Here, as before, si denote Pottp spins of the cluster that tive of higher complexity structures that arise the i-th stock currently resides in and ns denote finite, in critical systems with structures on different scales. but variable number of such clusters in the market at Such systems may typically exhibit one of three broad any given time. The binding strength of the si-th clus- regimes: (i) the regime where noise effects are mini- ter components is given by gsi , the uncorrelated cluster mal and ordering across the system is maximal, (ii) the specific noise is described by ηsi,t and the stock specific regime when noise effects dominate the system and or- uncorrelated noise which is independent of random clus- dering effects are minimal, and (iii) the critical regime ter emergence is denote by ǫi,t for the i-th stock. where changes in the scale of the system do not change The final step in our model specification is to pass to the behaviour or properties of the system as the system conditional expected returns in Eqn. (19) above, anal- balances between order and possibility of saturation by ogous to the manner that expected future returns were noise. described in Eqn. (10). The first limitation of a block variable view is that 10 stocks are not seen as emergent properties of the sys- ing away of noise effects. Neither view may be entirely tem. Secondly, we have already posited that a complex wrong. If there is indeed a hierarchy of complexity within financial market cannot be determined fully by using only the system, there may be times at which both can be bottom-up averaging implicit in such a block variable ap- right or wrong. Hence, a hierarchy of complexity can be proach. Thus, representing a stock’s price as magnetiza- consistent with the realities of scientific pluralism inher- tion of a particular block may not be sufficiently realistic. ent in many complex adaptive systems. Nevertheless, as an approximation towards modeling a Thus, even the choice of parameters in pricing models hierarchy of complexity, we argue that the analogies are and how models are used becomes related to the type of useful. causation models adopted by agents in the system. In We have provided particular, ad hoc representation the next section we discuss as a selection of examples for and approximation of the processes driving price incre- classes of actors in models of financial markets. ments via emergence, whereby the stock itself, in the sense of the existence of a company, is not an emergent property of the system, but the price of a stock is. V. TOP-DOWN CAUSATION ACTORS More generally, one may model companies in a complex exchange economy as represented by stock prices, which We discuss the causality classes and the actors used to arise as emergent properties of the system, with top-down represent them, as described in Table II, with interest in and bottom-up causation features in agreement with the how they respond to noise, as described in Table IV, and more general situation. This would entail a progression in respond to and influence the information variables Zk,t models from many agents trading a single stock, to agents and θi,m,t, that in turn influence the risk factors rp,t, as trading groups of stocks, and so on, up to the point where described in Table III. The five types of actors reviewed one has many agents trading the market, and groups of are discussed with more general examples in Ellis36, §4.2. markets within a global system. As in models for physical Our discussion is by no means comprehensive and further systems, the insight of averaging to identify dominant examples may be expanded44. dynamics in each level can help build an effective model that agrees with the observed phenomenology at each scale. A. Algorithmic top-down causation In the scheme described in this paper, the top-down risk and information variables, (rp,t and Zk,t), respec- This view of causation in financial markets is essen- tively, can be view as external fields in the effective tially a deterministic, dynamical systems view. This can models, while the bottom-up information variables θi,m,t serve as an interpretation of financial markets whereby serve to model structural features of the system itself. markets are understood through assets prices, represent- This offers a simplified, but quantitative perspective of ing information in the market, which can be described by the interaction complexities which can and do occur in a dynamical system model. This is the ideal encapsulated financial markets. in the Laplacian vision63: Viewing the system through a lens of such a hierarchy of complexity can lead to insights at various levels. At We ought to regard the present state of the a given level, one may consider attempting to regulate universe as the effect of its antecedent state interactions, for example to be constrained by produc- and as the cause of the state that is to fol- 9 tivity or transformative goals . However, this may be low. An intelligence knowing all the forces ineffective if either there is significant adaptation at time- acting in nature at a given instant, as well scales which are shorter than those of regulatory response as the momentary positions of all things in or emergence is susceptible to noise effects or random the universe, would be able to comprehend in bottom-up interactions, possibly triggered by regulation one single formula the motions of the largest itself. bodies as well as the lightest atoms in the Top-down control can also induce noise effects in lower world, provided that its intellect were suffi- levels, which may amplify or induce bottom-up effects ciently powerful to subject all data to anal- that may swamp any attempts of control the system. ysis; to it nothing would be uncertain, the In such cases, impacts on one set of system variables future as well as the past would be present to Zk,t and θi,m,t may induce new states, for example of its eyes. The perfection that the human mind shared risks, and modify evolution dynamics, as reflected has been able to give to astronomy affords but in changes in other connected variables. a feeble outline of such an intelligence. [1814] When viewing the system from lower levels in a hier- archy of complexity, one may well become increasingly Here boundary and initial conditions of variables convinced of a critical dynamical equilibrium view of the uniquely determine the outcome for the effective dy- markets. However, an efficient market view may seem namics at the level in hierarchy where it is being ap- valid when viewing the same system through the lens plied. This implies that higher levels in the hierarchy of pricing models with time-scales which allow averag- can drive broad macro-economic behavior, for example: 11 at the highest level there could exist some set of differ- dominant class of actors operating in this paradigm as ential equations that describe the behavior of adjustable the Profiteers. While the use of goals may damp out the quantities, such as interest rates, and how they impact effects of randomness as the system is driven towards pre- measurable quantities such as gross domestic product, selected goals17120, top-down causation of the profiteers aggregate consumption. can also lead to necessary conditions for financial crises. The literature on the Lucas critique94 addresses limi- The history of finance is littered with examples of corpo- tations of this approach. Nevertheless, from a completely rate malfeasance that fit into this category of top-down ad hoc perspective, a dynamical systems model may of- causation, from WorldCom and Enron, and through to fer a best approximation to relationships at a particular the various banking scandals resulting in billions of dol- level in a complex hierarchy. lars worth of litigation2 Example: Predictors: This system actor views Particularly striking is the ease with-which collusion in causation in terms of uniquely determined outcomes, financial systems may occur when the profit motive dom- based on known boundary and initial conditions. Predic- inates decision making. This can occur when bottom-up tors may be successful when mechanistic dependencies in information variables, θi,m,t, concerning individual cor- economic realities become pervasive or dominant. porate credit ratings, debt cover or book-values, etc, An example of a predictive-based argument since the which initially drive the system, become manipulated Global Financial Crises (2007+) is the bipolar Risk- with disinformation, but are nevertheless presented in a On/Risk-Off description for preferences27,81, whereby in- sanitized manner. Through agent responses, this can in vestors shift to higher risk portfolios when global assess- turn influence the behaviour of shared-risks in the market ment of riskiness is established to be low and shift to which arise out of aggregating the information in θi,m,t low risk portfolios when global riskiness is considered and Zk,t. to be high. Mathematically, a simple approximation of the dynamics can be described by a Lotka-Volterra (or predator-prey) model. The excess-liquidity due to quan- C. Top-down causation via adaptive selection titative easing and the prevalence and ease of trading in exchange traded funds and , combined with Fixed, high-level goals guide outcomes of stock se- low interest rates and the increase use of automation, pro- lections in this class of causation. This paradigm is vided a basis for the risk-on/risk-off analogy for analysing geared towards optimal deployment of assets via max- large capital flows in the global arena. imization of an appropriate utility function (which acts In our Ising-Potts hierarchy, top down causation is fil- as a fixed meta goal), to allow risk-return decisions to tered down to the rest of the market through all the be made in the face of uncertainty. Such decisions are shared risk factors, rp,t and the top-down information more general than profit maximization. More specifi- variables, Zk,t, which dominate bottom-up information cally, in Markowitz optimization for example, which is variables, θi,m,t. At higher levels, bottom-up variables equivalent to maximising quadratic utility under assump- are effectively noise terms. Nevertheless, the behaviour tions of Gaussian increments, investors choose to maxi- of the traders in a lower levels can still become driven mize wealth using the means and covariances. by correlations across assets, based on perceived global Example: Investors: The influential actors in this riskiness. Thus, risk-on/risk-off transitions can have am- class may be referred to as Investors. To adopt a top- plified effects. down causation model with adaptive selections, investors require markets to be sufficiently efficient, with depar- tures from equilibrium that are short-lived, uncorrelated B. Top-down causation via non-adaptive noise. Such investors have an idealised sense of being able information control to diversify risk as a meta goal and, hence, the idea of being able to optimally deploy capital. Unpredictability In this class of causation, in-built and unchanging goals enters the dynamics due to randomness. In general, high determine the outcome. In general, a goal refers to any level selection criteria have the advantage of exploiting targeted outcome. In capitalist economics, the primary randomness in an ensemble of options121. goal of maximization of individual and corporate profit is From a theoretical point of view, this causation class one of the most pervasive, unchanging goals in financial is a dominant source of top-down causation in financial systems, second perhaps only to the objective of attaining markets. Deviations away from asset prices determined a living wage. As a result, the profit motive is encoded by the shared-risk factors are assumed to be sufficiently in a variety of structural features in the modern finan- small to become negligible and correlations in the noise, cial systems, including remuneration structures, the tax due to activities lower in the hierarchy, are also regarded regime, the legal framework for the flow of capital and as negligible. This means that the direct impact on in- the flow of labour, legal structures deployed to manage formation variables, Zk,t or θi,m,t are either random or corporations. have short-lived effects and that only aggregates effects Example: Profiteers: Since most other capitalist are of concern for risk variables rp,t. This may drive as- goals are subsumed by quests for profit, we refer to the set prices to cluster in groups via the shared risk-factors 12 when capital is deployed based on those shared-risks. quality1,52,84,85,89,104,105,111. Thus, Regulators can im- pact the activity and success of all the other actors, ei- ther directly or indirectly through knock-on effects. Ex- D. Top-down causation via feedback control of amples include the following: Investor behaviour could adaptive goals change the goal selection of Traders; change in the latter could in turn impact variables coupled to Traders activ- This class of causality is characterised by adaptively ity in such a way that Profiteers are able to benefit from selected goals, which respond to context and are used change in liquidity or use leverage as a mean to achieve to guide the outcomes36. Consider, for example, market profit targets and overcome noise. participants who incorporate information about demand Idealistically, Regulators may aim for increasing pro- and random fluctuations, to offset risks for more opti- ductivity, managing inflation, reducing unemployment mal trading. This can be carried out by implementing and eliminating malfeasance. However, the circumven- different trading strategies, contingent on prevailing con- tion of rules, usually in the name of innovation or by ditions. claims of greater insight on optimality, is as much part Example: Traders: Participants in the market of a complex system in which participants can respond which may serve as an exemplar for this causation class to rules. Tax arbitrages are examples of actions which include market-makers, who facilitate transactions at all manipulate reporting to reduce levies paid to a profit- times, and traders, who interact with the market on be- facilitating system. In regulatory arbitrage, rules may half of clients for fees. be followed technically, but nevertheless use relevant new The dynamic nature of goals, conditioned on past in- information which has not been accounted for in system formation, may cause groups of agents to herd based on rules. Such activities are consistent with goals of profi- common objectives22. Here the behaviours of agents are teering but are not necessarily in agreement with longer based on the information variables, Zk,t and θi,m,t, and term optimality of reliable and fair markets. shared-risks rp,t, with goals updated based on past states Rulers, i.e. agencies which control populations more of the system as well. We note that this type of causa- generally, also impact markets and economies. Exam- tion may include both bottom-up and top-down effects. ples of top-down causation here include segregation of Goals may be selected to guide outcomes based on prop- workers and differential assignment of economic rights to erties of the system which reside higher in the hierarchy, market participants, as in the evolution of local miners’ for example, a may observe the spread between rights in the late 1800’s in South Africa and the national a pair of tradeable securities and make trading decisions Native Land act of 1913 in South Africa, international in terms of price levels, while at the same time using her agreements such as the Bretton Woods system, the Mar- balance sheet to influence and move the prices themselves shall plan of 1948, the lifting of the gold standard in 1973 and the regulation of capital allocations and capital flows to trigger or action ad hoc outcomes. Noise variables ηsi may become significant if correlations in the noise become between individual and aggregated participants. dominant. Ideas on target-based goal selection are already in cir- culation in the literature on applications of viability the- ory and stochastic control in economics9,86. Such ap- E. Top-down causation with adaptive selection of proaches provide alternatives to the Laplacian ideal of adaptive goals attaining perfect prediction by offering analysable future expectations to regulators and rulers. Examples: Regulators and Rulers: Regulators attempt to act on a financial market based on the intel- ligent and reasonable formulation of rules. For example, VI. CONCLUSION changing the market micro-structure at the lowest level in the hierarchy, can change the way that asset prices In order to address the need for more realistic economic assimilate changes in information variables Zk,t or θi,m,t. models which include feedbacks, adaptive goal-seeking, Similarly, changes in accounting rules could change the emergence and the occurrence of crises, we have consid- meaning and behaviour of bottom-up information vari- ered the applicability of a particular hierarchical mod- 36 ables θi,m,t and changes in economic policy and policy elling approach by considering specific model interac- implementation can change the meaning of top-down in- tions. formation variables Zk,t and influence shared risk factors In Section IV, we provided a consistent set of coupled rp,t. models, incorporating causation at differently levels in In hierarchical analysis, theories and plans may be the hierarchy, driven by top-down sources, realised as embodied in a symbolic system to build effective and actors, and bottom-up information variables. The model robust models to be used for detecting deeper de- incorporates the possibility of emergence, via random in- pendencies and emergent phenomena8,54,59,82. Mecha- teractions, but is still consistent with no-arbitrage equity nisms for the transmission of information and asym- models at relevant scales, and can be calibrated for fore- metric information information have impacts on market casting, based on historic data. We find the actor-based 13 perspective of causation sources to be useful for identi- financial economics. Different actors may impact differ- fying feedbacks and non-trivial features, which influence ent components of the system and more than one theory whether the system is in fact fit-for-purpose, in the sense may be effective for a given level in the hierarchy under of having low-coupling and high-cohesion. Here, insights scrutiny. Thus, the approach is comprehensive and plu- can be derived from both the qualitative features of ac- ralistic, provided consistency constraints are maintained tors and actor interactions as well as the quantitative at the interfaces between models, for investigating multi- outputs of simulations. layer models in finance. Since economies can only be run once, unlike small experiments in controlled laboratory conditions, it be- comes completely relevant to scrutinise as much infor- mation as possible when stipulating and enforcing (or ac- Acknowledgments tively not enforcing) the economic impact of laws. While, no human-constructed model is ever likely to include all significant information and dynamics, increased comput- The authors thank George Ellis, Franco Bussetti, An- ing power, advances in complex systems modelling and toinette and Robert Wilcox, Ron Cross, Dieter Hendricks better quantification of system information may yield and Virginie Konlack for discussions on complexity and insight-producing simulations for better economic deci- financial markets. This work is based on the research sup- sion making. ported in part by the National Research Foundation of The specific instantiation of a hierarchical model dis- South Africa (Grant numbers 87830, 74223 and 70643). cussed in this paper is by no means a unique solution to The conclusions herein are due to the authors and the the challenge of finding relevant hierarchical models in NRF accepts no liability in this regard

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http://dx.doi.org/10.2139/ssrn.2283486 117 Top-down causation may be introduced agent based mod- 113 Wilson, K. G., (1979), Problems in physics with many els via environmental variables, for example, the use of an scales of length, Scientific American, 241, August, 158- external field when approximating agents with spin mod- 179 els, or through changing network structure or topology for 114 Wu, F., Y., (1982), The Potts model, Rev. Mod. Phys. the domain supporting the spin-models. 54, 235 - 268 118 In a linear model44,45 for price change we can consider 115 Zhang, L., Mykland, P.A., A¨ıt-Sahalia, A Tale of Two dp(t) 1 a price p(t) such that dt = λ M(t)p(t) for magneti- Time Scales: Determining Integrated Volatility With zation M(t) define as the aggregate sum of spin states Noisy High-Frequency Data, Journal of the American Sta- in the system as a representation of excess demand. Un- tistical Association, 100: 1394-1411 (2005) der discretization this implies that p(t +1) = p(t)ecM(t) 116 This form of the price model ensures that the self correla- to allow one to compute the price change : X(t) = tion of a stock is one and independent of the cluster cou- ln(p(t)) ln(p(t 1)) = cM(t 1). pling. This can be seen by computing the self correlation 119 Renormalisation− − group analysis− is a mathematical tool in- 2 E[Xi ] and using that clusters and stock unique process vented for the analysis of critical phenomena in systems are unit variance zero mean processes: which are characterised by structures on many different scales. These systems exhibit power law relationships be- g η g2 ǫ 2 g2 g2 . tween the system observables and control parameters E[( si si + q1 si i) ]= si + (1 si ) = 1 (20) − − 120 TG thanks GFR Ellis for interesting discussions on goal This is not a unique choice, another possible choice often oriented control in the presence of noise. used is: 121 TG thanks GFR Ellis for a discussion on how control sys- tems with high level selection criteria can exploit noise √gsi 1 2 1+ gsi E[( ηsi + ǫi) ]= = 1. (21) √1+ gsi √1+ gsi 1+ gsi