# Complexity theory and Finance (Zeidan and Richardson)

**Complexity Theory and the Financial Crisis: A Critical Review**

**Rodrigo M Zeidan and Kurt Richardson**

**Abstract**

We present a critical review of the impact of complexity theory on the analysis of financial markets, concentrating on the differences between Econophysics and

Econobiology. Both approaches show that there are limitations in our current understanding of the banking system, which can lead to regulatory problems, and in turn may have contributed to the inability of most academics and economic agents (including the FED) in predicting the crisis.

**Introduction**

The effects of the Financial Crisis are still being felt, and speculations on the predictability of the crisis and what measures should be taken to prevent another one are hot topics in financial circles. The purpose of this article is to analyze the financial crisis through the lens of complexity theory and to ponder which explanations can be given and which path should be open if one considers the world through the complexity theory lens.

The main advantage of using a complexity-based approach is that it meets head-on a common discontent of many researchers after the crisis: that current financial models

IBMEC/RJ and ISCE, respectively. Contact information: rodrigo_zeidan@hotmail.com and kurt@iscepublishing.edu. were incapable of signaling the crisis. But there are no easy answers, because one of the main characteristics of complexity theory is incompressibility (for more, see Richardson (2008) and Ciliers (2005)), which puts a limit on the information that can be extracted from complex systems, and which also necessarily leads to theoretical

pluralism. If we take the reasonable assumption that financial systems are complex in nature, there is no single theory that could explain every feature of each system, which makes a simple, decisive explanation for the crisis impossible. However, that does not mean that we cannot extract predictions, lessons, policies, or insights from thinking complexly about financial systems. In fact, it is just the opposite; thinking deeply on the impacts of complexity into financial system can lead to different kind of analyses that should bring testable implications for financial research. Here we focus on one possible explanation for the crisis, that the true nature of systemic risk is ignored by the Central Bank, due to an incomplete understanding of the true nature of financial markets and thus its available actions will be inherently curbed until it properly models the banking system.

**Complexity in Financial Markets: A Primer**

Richardson (2008) divides the approaches of how complexity thinking can impact management studies into three broad schools: the neo-reductionist school, the

metaphorical school, and the critical pluralist school. Although the impact of complexity research on management studies is still in its infancy, the complexity school in financial market is much more developed. There is no question that complexity matters in financial markets, and the literature is developing in such a way that even investor strategies are now being designed based on complexity concepts (for instance, Brunnermeier and Oehmke (2009) point out three different ways in which boundedly rational investors can deal with complexity). In fact, there are two main general approaches to complexity in economics and finance,

that of Econophysics and Econobiology. Both are bottom-up, population-based approaches (Rickles, 2010), but differ sharply in their methods to deal with complexity.

Econophysics is a term coined by Stanley et al (1996) and is preoccupied with the application of the physics of complex system into financial and economics markets [see, for instance, Mantegna and Stanley (1999) and Rickle (2010)], while Econobiology takes its lessons from biological complexity models (e.g. Arthur, 1995)). Because both approaches have its origins clearly defined, the implications for complexity studies in finance are finely traceable.

First a caveat: while in the realm of Finance complexity models are much more common than in management studies, the confusion that Richardson (2008) observed between complexity and complicatedeness still persists, as can be seen in the examples of Caballero and Simsek (2009) and Bushman et al (2004), both of which use complexity in the common usage of the word. There is nothing wrong with either works, but here we analyze complexity in its technical form.

Using Richardson’s classification, we can easily put the majority of Econophysics in the neo-reductionist school and much of Econobiology in the metaphorical school. This classification is so straightforward that it is uninteresting at first sight. However, when applied to a broader perspective of financial systems, this classification allows for some interesting insights, as will be shown later.

Rickle (2010) summarizes the Econophysics literature and the relationship with complexity, arguing that: ”according to the econophysicist’s conception of financial markets the prices (of assets) are viewed as fluctuating macroscopic variables that are determined by the interactions of vast numbers of agents”, and presents some important findings regarding asset returns that can be explained by econophysics, but not necessarily by regular financial models, such as fat tails, volatility clustering, and volatility persistence as examples. The methods of Econophysics are based on the tools of dynamical physics, like statistical mechanics and chaos theory, for instance. Since those tools are numerous, there are a multitude of different Econophysics models that try to deal with particular questions in financial markets, but all share the same epistemology – i.e. are based on data analysis first and hypothesis formulation second.

The idea of a paradigm shift brought by Econophysics to financial and economic modelling is strong: “Econophysics is viewed (by most of its practitioners) as a

revolutionary reaction to standard economic theory that threatens to enforce a paradigm shift in thinking about economic systems and phenomena” (Rickles, 2010, p.13).

Some Econophysics models are by now mainstream in the financial literature, especially those dealing with volatility clustering and persistence, and with fat tails in asset returns – in this area statistical mechanics modelling thrives, due to the inherent characteristics of probabilistic function in asset pricing and returns.

However, important critiques of econophysics have surfaced, even when the discipline was in its infancy. One particular relevant critique comes from Mirowski (1989), where the author argues that much of Econophysics is problematic due to epistemological considerations (specifically, that there is no theory behind the transposition of physics models to economics), even though it can be argued that financial systems are much more akin to physical systems than most economics models. Econophysics then helps in a better understanding of the statistical properties of financial systems. Econobiology, on the other hand, has a different approach to the complexity of financial systems.

Arthur (1995) offers a paradigmatic view of Econobiology processes. In Econobiology models, the complex systems develop bottom-up through agent-based interactions, as in Econophysics models, but the difference is that while in the former the locus of analysis is agent interactions and the processes of emergence, in the latter the goal is to determine the statistical properties of the results of the agents’ interactions. The difference is not only on the mechanics of both approaches, but also in their epistemology – Arthur (1995, p.20) expresses the view that one should look into economics and financial markets in “psychological terms: as a collection of beliefs, anticipations, expectations, and interpretations; with decision-making and strategizing and action-taking predicated upon these beliefs and expectations”.

The bottom-up approach of Econobiology models is preoccupied with emergent properties from agent interactions. It is very useful to analyze innovation activity, for instance, because this kind of activity fits nicely as an emergent property of agents searching for innovation (since Nelson and Winter’s (1982) seminal work, evolutionary economics modelling of innovative activity has become mainstream in the industrial organization literature). For financial models, Arthur et al (1995) show how a simple model of stock valuation can be radically transformed by changing a simple assumption

on the behavior of rational investors, now assuming that investors are heterogeneous in

their expectations. The result is that (p.24) “Under heterogeneity, deductive logic leads

not just to indeterminate expectations but also to unstable ones. (…) Given differences

in agent expectations, deductive expectation formation becomes indeterminate, and so

even perfectly rational investors cannot form expectations in a determinate way”.

The main avenue of research of Econobiology models in finance is then the inductive

approach through the use of simulation. The idea is to populate the simulator with

heterogeneous agents and identify the properties of the emergent features of the

interactions. Arthur et al (1996) studied asset pricing under inductive reasoning using a

simulation with 100 artificial investors each with 60 expectational models. The main

results are that, if agents believe in the standard model of finance, their beliefs and the

standard model is evolutionary stable and thus upheld; and that the market possesses a

non-trivial

*psychology*, with evolving strategies and different periods of volatility in the

price series (sometimes consistent with GARCH processes). The advantage of this kind

of models is that regular stylized facts of financial models (such as stocks crosscorrelations)

can be generated instead of being simply observed. More recent models

have stretched the use of simulation (Ponzi and Aizawa, 2000) and evolutionary models

to refine the analyses of emergent properties in financial markets, with the use of neural

networks (Azzini and Tettamanzi, 2006), biological algorithms (Brabazon and O’Neil,

2006), and grammatical evolution (Adamu and Phelps, 2009). The main problems with

evolutionary finance models are that there is still a reliance on simulations, with many

results still exploratory; and that the models offer a different way to look at financial

markets but are too scant in terms of definite predictions.

Here we concentrate on network topology and other models to analyze some of the

possible dynamics of financial systems. Which is the best approach to analyze this

important feature of the crisis? In the next section we review some works in each

approach to complexity in finance, and try to extract some insights to help explain the

building of the crisis.

- Complexity Models, Risk Creation and the Financial Crisis.

There are many proposed explanations for the crisis: lack of regulation; cheap money;

irrational exuberance; “creative” exotic financial instruments; and excessive risk-taking.

Here we want to concentrate on the fact that the problem of banking regulation is more

fundamental than simply choosing the level of regulation – we may not understand

important features of the banking system dynamics.

Take risk, for instance. Regular financial micro models divide risk in idiosyncratic and

systemic components. For regulation purposes the focus is usually on systemic risk, the

component that can affect the whole financial system. The proponents of financial

deregulation usually claim that markets are efficient – i.e. that it operates on the basis of

a “Law of Conservation of Risk”, in which banks are efficient in allocating risk throughout

the system, and risk is neither created nor destroyed, merely shuffled around efficiently.

Risk, instead of being a static feature of financial systems, is part of a dynamic process

where it is destroyed and created in the course of trading activity. Shifting risk may allow

for more efficiency in terms of costs to market agents, but what may be lacking in

regular economic models is the notion that systemic risk is more than the sum of its

parts. Recent models are specifically looking into general equilibrium banking models,

because as Acharya (2009, p.2) observes: “The standard theoretical approach to the

design of bank regulation considers a “representative” bank and its response to

particular regulatory mechanisms such as taxes, closure policy, capital requirements,

etc.” Such partial equilibrium approach has a serious shortcoming from the standpoint of

understanding sources of, and addressing, inefficient systemic risk. In particular, it

ignores that in general equilibrium, each bank’s investment choice has an externality on

the payoffs of other banks and thus on their investment choices”.

Acharya’s model shows that “optimal regulation should be “collective” in nature and

should involve the joint failure risk of banks as well as their individual failure risk”. For

that, he simply defines systemic risk as “the joint failure risk arising from the correlation

of returns on asset-side of bank balance-sheets”. Both definition and conclusion are the

jumpstart for our complexity-based analysis of financial markets. The idea that systemic

risk is based on a joint-failure is simple enough, as is the idea that regulation should be

collective. Another interesting network banking model is that of Caballero and Simsek

(2009), where the authors propose a model of a financial crisis based on the costs of

gathering information on the soundness of their trading activity. Financial crises happen

when, after an emergence of acute financial distress in a part of the system, the cost of

information gathering becomes too unmanageable for banks, and they have no option

but to withdraw from loan commitments and illiquid positions. All these models would be

considerably enhanced by looking into the world through a complexity lens.

2.1. Econophysics Modelling of the Banking System

There is no single econophysics model of the banking system, but recent work can

enlighten some of the important features of financial markets.

The main avenue of research considers the banking system as a network of payment

flows, and studies its topology, with the underlying risk determining the stability of the

network. One simple illustration of this is presented in graph 1, with an example of the

the topology of the Danish network payment system, from Rørdam and Bech (2009).

Graph 1 – The Network Topology of the Danish Payment System (Rørdam and Bech,

2009)

(Rørdam and Bech, 2009)

In the figure the white nodes are the biggest banks. This topology highlights one of the

most discussed aspects of the banking crisis,- that some banks are too big to fail. This

is measured through the systemic impact of each node on the network stability – a

network is more stable if nodes can disappear without loss of payment flows.

Inaoka et al (2004) develop a network banking model based which measures the

network stability and is based on power laws and cumulative distributions. The result is

that the banking network shows a power-law degree distribution. Also, the main

characteristic of the network, in this case, is that it is much more efficient than stable –

i.e. there is a trade-off between banks searching for efficiency and the whole stability of

the network. Becher et al (2008) don’t observe this trade-off in their analysis of the

topology of the UK payment system, concluding that “liquidity is able to flow efficiently

around the network and that the network is quite resilient to shocks (p.1)” – even taking

a node off the network doesn’t undermine the ability of banks to have payments flow

freely. Pröper et al (2008), in another topological analysis, this time on the Dutch

payment system, find some indication of a lack of efficiency – the network is small,

compact and sparse, using a mere 12% of possible connections in the long run, and

also of stability, stating that the network is subject to possible serious instability.

Another issue of network topology is the network connectivity, which is related to the

problem of collective bankruptcy. Aleksiejuk et al (2001) present a model of a banking

network, with the goal of relating collective bankruptcies to self-organized criticality. In

their model the main parameter is the mean number of interbank connections, q. Two

competing processes increase (A) or decrease (B) q (p.5):

- Due to the random character of the stochastic variable i(t) and the random selection

of a financial partners from the nearest neighbors.

- Due to collective bankruptcies, because new banks that replace bankrupts are not

involved in the interbank market at the beginning of their activity.

The authors conclude that (p.6) “if the value of q is below the percolation threshold then

the mean size of contagion cluster is small and the process A overcomes the process B.

It follows that in the course of time the number of interbank connections q is growing up

driving the system to the critical state. If the value of q exceeds a critical value (the

percolation threshold) then a large contagion cluster spreading all over the system can

appear.”

There are other works concerning banking system network topology, but the ones

presented here show the relevance of this kind of analysis to the stability analysis of

banking systems. There are, of course, still many unresolved issues regarding which

features are relevant to regulatory policy, and, more importantly, what should be done

once different network characteristics are identified.

Network topology is hardly the only way to analyze financial markets with Econophysics

models. For instance, the analysis of entropy in financial markets yields important

information on risk creation (Michael and Johnson, 2004), while Bartolozzi and Thomas

(2004) analyze a stochastic cellular automata model and conclude that crashes or

bubbles are triggered by a phase transition in the state of the bigger clusters present in

the network system.

2.2 Econobiology Modelling of the Banking System

Econobiology research in our present context begins with agent-based modelling of the

banking system. The main advantage is that the models treat banks as agents in a

biological population and analyze their process of evolution – in this sense, risk is one of

the determinants of a bank “evolution”. Because in a population agents are always

entering and leaving, the models have a clear parallel with financial markets. The main

problem is that it is difficult to get out of the metaphorical level towards a working agentbased

banking model, even though modelling risk is increasingly promising as an

avenue of research. However, some frontier research is now breaching the

metaphorical level into workable models. Here we present three classes of models

which are surprisingly effective in determining banking failures through the lens of

evolutionary biology.

The first kind of models use neural networks [beginning with Tam (1991) and Tam and

Kyiang (1992), and surveyed by Atiya (2001)]. “An artificial neural network (NN) is a

computational structure modelled loosely on biological processes. NNs explore many

competing hypotheses simultaneously using a massively parallel network composed of

non-linear relatively computational elements interconnected by links with variable

weights. It is this interconnected set of weights that contains the knowledge generated

by the NN” (Adya and Collopy, 1996, p. 481). The linkage to financial markets is direct,

because it is easy to think of banks being interconnected and generating information in

the process of transactions between them. Again, as in Econophysics modelling, we

arrive at networks as basis of the analysis, but the method of analysis is quite different.

The evolution of the neural network yield information on banking failures through

decaying processes due to loss of information.

Support vector machines (SVM) can also be used to model risk in financial markets

[e.g. Tay and Cao (2001) and Chen and Shih (2006)]. SVM is a type of learning

machine based on statistical learning theory. Instead of considering the network

structure as in neural networks models, it uses optimization procedures based on

learning processes, another important feature of biological processes.

The last class of banking models based on evolutionary biology briefly described here is

that of genetic algorithms (GA). A genetic algorithm is an optimization procedure based

the idea that evolution happens through changes in genes, hence concepts like

inheritance, mutation, and selection. If we think of banks as agents searching for

survival, a selection processes based on evolutionary biology can describe what

happens on financial markets. Min et al (2006) presents a model that uses GA to

improve on their model of SVM to analyze the probability of banking failures. In their

model, the authors were able to measure relative performance and conclude that it

outperforms regular financial models in detecting the probability of banking failures. In

this sense, their model would be complementary on Value-at-Risk internal models.

Table 1 summarizes the main implications and disadvantages of the Econophysics and

Econobiology approaches to financial markets. The table below is far from exhaustive,

its goal being merely to illustrate some of the research agenda of complexity theory in

financial markets.

Table 1 – Econophysics; Econobiology and Financial Markets.

Underlying

foundations

Methods Financial Markets Disadvantages

Econophysics

Statistical

mechanics, Selforganized

criticality;

Chaos Theory

Dynamical

Modelling;

Network Topology;

Power-Law

Distributions

Network Topology,

Volatility Clustering and

Persistence; Fat Tails;

Extreme Events

Narrow

explanations;

Theoretically

empty.

Econobiology

Evolutionary

biology; Bounded

rationality; Learning

theory

Agent-based

models;Genetic

Algorithms;

SVM;Simulation

Herding, Self-fulfilling

behavior, Adaptative

Market Hypothesis;

Uncertainty emergence;

Innovation.

Lack of

predictions;

Hard to model.

- Final Comments

The purpose of the paper was to present important contributions of complexity theory to

the analysis of financial markets, focusing on the distinction between Econophysics and

Econobiology and their possible contribution to the better understanding of financial

markets dynamics. It was our intention to bring new issues into the fold, instead of

answering decisively questions like why the major crisis of 2007-8 happened.

We followed the reasoning of authors like Mainzer (2009, p.1): “the world-wide crisis of

financial markets and economies is a challenge of complexity research. Misleading

concepts of linear thinking and mild randomness (e.g. Gaussian distributions of

Brownian motion) must be overcome by new approaches of nonlinear mathematics

(e.g., non-Gaussian distribution), modelling the wild randomness of turbulence at the

stock markets. Systemic crises need systemic answers”. Bouchaud (2008) goes

beyond it, calling for a scientific revolution for economics based on complexity theory.

Here we don’t take a strong position against regular financial and economic models,

and we don’t advocate that complexity theory will answer everything, because there is

no simple explanation or easy answers to be given.

However, the models and results here can bring to light some important questions

regarding the shaping of financial markets – if the banking system presents

characteristics like self-criticality, power law distributions, risk emergence, bounded

rationality, and other non-deterministic features, the problem of the regulator is much

more difficult than regular economic models allow for. It is just not a question of how

stringent to regulate the system or not, but a more fundamental question of what is

being regulated.

Further research on the complexity approach should greatly enhance our understanding

of financial markets, and a specific line of research could concentrate on developing an

encompassing theory to tie all the different approaches together to arrive, at some point,

at a complexity-based standard model of finance.

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