Financial Complexity & Nonlinear Dynamics

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This book presents a high-level overview to the application of complexity theory to finance, first presenting the more theoretical side to complexity finance before going on to illustrate more practical examples of nonlinear dynamical modeling applied to capital markets.

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Section 1 Overview

In a recent interview George Soros captured much of the predicament that finance as a science finds itself in today when he said "The efficient markets hypothesis has failed and it is recognized that it has failed and therefore economist need to find a new understanding of financial markets…. this is what science is, it is a trial and error. Unfortunately, we don't have a properly developed alternative and that is what we are looking for." He goes on to say that the approach to finance that we developed under the efficient markets paradigm is not applicable to the real world and that he, in fact, made his money betting against the efficient markets hypothesis.

Since the financial crisis, much of economic and financial theory has been called into question. We are increasingly recognizing the limitations of the many kinds of financial models that are dependent upon assumptions of linearity and equilibrium; that agents are rational and independent and that the future will resemble the past.

We come to increasingly recognize that linear development is but one kind of change, nonlinearity is another and of equal importance, if we are to build a more comprehensive understanding of financial systems. When systems involve synergies and feedback then they become nonlinear. You can get cascading effects that take the system out of equilibrium and into phase transitions and that these periods of what seems to be chaos, in fact, have their own kind of dynamics. By understanding the science of nonlinear dynamics we stand a much better chance of seeing and dealing with these periods of exponential and fundamental transformation.

This is indeed an exciting time for economics and finance as after almost two centuries of studying equilibrium solutions economists are beginning to study the emergence of non-equilibria and the general evolution of patterns in the economy. That is, we are starting to study the economy out of equilibrium and increasingly doing this through a computer-based algorithmic approach.

This new complexity approach is certainly a paradigm shift, one of its creators W. Brian Arthur describes the essence of this change in perception when he notes "really it is a shift from looking at the world in reductionist terms, from the top down and imagine everything holding everything else in equilibrium where not much is changing at all, to looking at the world as alive everything is affecting everything."

A key tool in this new approach is agent-based modeling (ABM) that gives us an inherently dynamic vision of markets, as patterns are continually being created and recreated through endless computations across complex networks of interaction; just as we see in the real world. When seen in this way financial markets show themselves not as mechanical, deterministic systems always moving towards stability and equilibrium but instead more like an ecosystem continuously evolving and creating new structures and patterns.

The complexity approach brings into focus connectivity and networks. After the 2008 financial crisis, many economists have come to the view that the very networked architecture of the financial system plays a central role in shaping the dynamics of the system - even more so now that it has become globally interconnected and interdependent. That to properly understand the vulnerabilities and opportunities we have to look at the networks of connections. Here again, complexity science provides us with a new set of models and computer tools for understanding financial network structure and dynamics. The science and mathematics of networks is now almost fifty years old and advancements are being made every year; these insights from network theory can be of critical value in providing the theoretical underpinnings to finance as a more mature science.

This changing paradigm of complexity is already proving critical to rethinking financial theory. The science of finance is very young and is changing very fast, but by integrating complexity theory we believe that it may well be key to actually studying finance as a science in a much more realistic way than we have done in the past. The importance of rethinking our approach to finance can't be understated, as professor Andrew Lo notes "you all know the saying 'seeing is believing' I would argue that other times things need to be believed to be seen, our narrative changes our behavior which changes reality, that's what I want to leave you with, the fact that we need new narratives in finance both from the perspective of financial advisers but also from a societal perspective, finance is a means to an end not an end unto itself."

1. What are Financial System?

Dealing with complexity requires shifting our focus so as to look at not just the parts to a system but also the overall macro system as a whole. Ideally, this means formulating some kind of overall systemic model of the financial organization we are dealing with. Even if this model may appear very basic it helps to structure our reasoning and place our more focused analytical understanding within a broader conceptual framework. Finance serves the function of accounting for and exchanging economic value. Financial systems allow funds to be stored and moved between economic actors; they enable individuals and organizations to share and exchange ownership with the associated risks and returns. A key distinction in financial systems can be made between systems designed to enable immediate economic exchange or systems designed to enable the longer-term exchange of ownership through investment; this can be thought of as a distinction between liquid capital and investment capital.

Financial systems enable the exchange of products and services via liquid capital where currencies function as a shared medium of exchange enabling people to fluidly exchange underlying economic resources. Investment capital is concerned with the allocation of assets and liabilities over space and time, with associated risks and returns.

To facilitate this recording and exchanging of economic value a financial system converts economic claims to ownership and liabilities into an information based form of a financial asset or liability. As such we can say a financial system is an information form of the real economy; it is an information system for the recording and exchanging of value.

Finance quantifies underlying value within the economy and creates an information representation of that in the form of what we call a financial asset. Financial assets derive their value from a contractual claim on an underlying economic asset.

The point of this is that information can be more easily stored, processed and exchanged than real economic assets. This linking, recording and exchanging of economic value creates a network of interlinked assets and liabilities, a financial system.


A system is a set of elements and relations between these elements through which they form an interconnected whole. We can then represent an element(node) in the system as a financial asset or financial entity. A financial asset is a claim to some economic resource - when the value is negative it can be termed a liability. A financial entity is an individual or organization with an accounting record of assets and liabilities represented as a balance sheet. A node in the system can be represented by a single absolute value of the size of their assets. Connections represent the exchange or linkages between assets and liabilities between different organizations.

To serve its function a financial system has to be able to record and move assets from one entity to another; from those who have savings to those who need it for investment; from those who are buying to those who are selling a good or service through a currency; from one generation to another through inheritance; from individuals to public administration through taxes; from low interest nations to locations of high returns through stocks, bonds and loans; for spreading risk through insurance, for joint investment via special purpose vehicles. This relationship between those that have capital (the investors/buyers) and those that need it(debtors/sellers) forms the core of what financial systems are and do.

Nodes in the network make decisions about how to allocate their capital so as to obtain the economic resources they desire through exchange or investment. Nodes exchange resources or invest in other nodes to generate a return on their investment or obtain the things they desire. Financial assets are used as the medium of exchange. They serve as a standardized medium of known value for which goods and ownership can be exchanged as an alternative to bartering.

The type of transaction - and type of financial security used to enable it - can be seen to exist on a spectrum of liquidity, which defines how widely accepted and rapidly a financial asset can be converted. Economic exchange is done through liquid capital, such as fiat currencies. Investment is done through capital markets in the form of various capital market instruments, such as bonds, stock, commodities, derivatives. These are all claims to ownership or claims to a portion of a revenue stream. A derivative instrument is a contract that derives its value from one or more underlying entities. Financial intermediaries - such as banks, insurance companies, hedge funds and various forms of market makers etc - perform the function of aggregating resources, spreading investments and enabling exchanges.

Risk Returns

Financial entities make exchanges to obtain the things they desire. All exchanges involve a dynamic between risk and returns. Risk defines the potential of a financial loss and returns defines a gain e.g. when we exchange liquid capital for some good it may or may not deliver the functionality we hoped for, when we invest in a company it may or may not deliver returns, these are forms of risk.

Through financial instruments like loans, bonds, shares etc. financial entities connect their risks and returns with others within contractual agreements. Participants in the market aim to price assets based on their underlying value, their risk level and their expected rate of return.

Thus the core of a financial system is the relationship between creditor and debtor and the risk-return ratio of that connection, which defines the contractual agreements as to prices, dividends, liabilities etc.

Much of economics can be understood in terms of investment, risk and returns; buying a house or bicycle, starting a company, a government choosing to build a new bridge in the hope that it will stimulate the economy. Combining assets makes it possible to spread risk and returns and thus engage in larger investments without any one party needing to take on the full risk or provide the full capital investment cost.

As the cost of the transaction goes down resources can be more easily moved around in the system and assets and liabilities shared. A financial system is a type of information system, thus the form of the financial system is heavily contingent upon the underlying information technology used to enable it. The financial system's capacity to share assets and liabilities is relative to the level of the underlying information technology's efficiency at recording, organizing and exchanging financial information. The complexity of the financial system is fully contingent upon the information technology used to operate the network. Coupled with this social factors such as legal frameworks are key to the form of the financial system.

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