The Statistical Physics of Stock Markets. Thought of the Day 143.0

This video is an Order Routing Animation

The externalist view argues that we can make sense of, and profit from stock markets’ behavior, or at least few crucial properties of it, by crunching numbers and looking for patterns and regularities in certain sets of data. The notion of data, hence, is a key element in such an understanding and the quantitative side of the problem is prominent even if it does not mean that a qualitative analysis is ignored. The point here that the outside view maintains that it provides a better understanding than the internalist view. To this end, it endorses a functional perspective on finance and stock markets in particular.

The basic idea of the externalist view is that there are general properties and behavior of stock markets that can be detected and studied through mathematical lens, and they do not depend so much on contextual or domain-specific factors. The point at stake here is that the financial systems can be studied and approached at different scales, and it is virtually impossible to produce all the equations describing at a micro level all the objects of the system and their relations. So, in response, this view focuses on those properties that allow us to get an understanding of the behavior of the systems at a global level without having to produce a detailed conceptual and mathematical account of the inner ‘machinery’ of the system. Hence the two roads: The first one is to embrace an emergentist view on stock market, that is a specific metaphysical, ontological, and methodological thesis, while the second one is to embrace a heuristic view, that is the idea that the choice to focus on those properties that are tractable by the mathematical models is a pure problem-solving option.

A typical view of the externalist approach is the one provided, for instance, by statistical physics. In describing collective behavior, this discipline neglects all the conceptual and mathematical intricacies deriving from a detailed account of the inner, individual, and at micro level functioning of a system. Concepts such as stochastic dynamics, self-similarity, correlations (both short- and long-range), and scaling are tools to get this aim. Econophysics is a stock example in this sense: it employs methods taken from mathematics and mathematical physics in order to detect and forecast the driving forces of stock markets and their critical events, such as bubbles, crashes and their tipping points. Under this respect, markets are not ‘dark boxes’: you can see their characteristics from the outside, or better you can see specific dynamics that shape the trends of stock markets deeply and for a long time. Moreover, these dynamics are complex in the technical sense. This means that this class of behavior is such to encompass timescales, ontology, types of agents, ecologies, regulations, laws, etc. and can be detected, even if not strictly predictable. We can focus on the stock markets as a whole, on few of their critical events, looking at the data of prices (or other indexes) and ignoring all the other details and factors since they will be absorbed in these global dynamics. So this view provides a look at stock markets such that not only they do not appear as a unintelligible casino where wild gamblers face each other, but that shows the reasons and the properties of a systems that serve mostly as a means of fluid transactions that enable and ease the functioning of free markets.

Moreover the study of complex systems theory and that of stock markets seem to offer mutual benefits. On one side, complex systems theory seems to offer a key to understand and break through some of the most salient stock markets’ properties. On the other side, stock markets seem to provide a ‘stress test’ of the complexity theory. Didier Sornette expresses the analogies between stock markets and phase transitions, statistical mechanics, nonlinear dynamics, and disordered systems mold the view from outside:

Take our personal life. We are not really interested in knowing in advance at what time we will go to a given store or drive to a highway. We are much more interested in forecasting the major bifurcations ahead of us, involving the few important things, like health, love, and work, that count for our happiness. Similarly, predicting the detailed evolution of complex systems has no real value, and the fact that we are taught that it is out of reach from a fundamental point of view does not exclude the more interesting possibility of predicting phases of evolutions of complex systems that really count, like the extreme events. It turns out that most complex systems in natural and social sciences do exhibit rare and sudden transitions that occur over time intervals that are short compared to the characteristic time scales of their posterior evolution. Such extreme events express more than anything else the underlying “forces” usually hidden by almost perfect balance and thus provide the potential for a better scientific understanding of complex systems.

Phase transitions, critical points, extreme events seem to be so pervasive in stock markets that they are the crucial concepts to explain and, in case, foresee. And complexity theory provides us a fruitful reading key to understand their dynamics, namely their generation, growth and occurrence. Such a reading key proposes a clear-cut interpretation of them, which can be explained again by means of an analogy with physics, precisely with the unstable position of an object. Complexity theory suggests that critical or extreme events occurring at large scale are the outcome of interactions occurring at smaller scales. In the case of stock markets, this means that, unlike many approaches that attempt to account for crashes by searching for ‘mechanisms’ that work at very short time scales, complexity theory indicates that crashes have causes that date back months or year before it. This reading suggests that it is the increasing, inner interaction between the agents inside the markets that builds up the unstable dynamics (typically the financial bubbles) that eventually ends up with a critical event, the crash. But here the specific, final step that triggers the critical event: the collapse of the prices is not the key for its understanding: a crash occurs because the markets are in an unstable phase and any small interference or event may trigger it. The bottom line: the trigger can be virtually any event external to the markets. The real cause of the crash is its overall unstable position, the proximate ‘cause’ is secondary and accidental. Or, in other words, a crash could be fundamentally endogenous in nature, whilst an exogenous, external, shock is simply the occasional triggering factors of it. The instability is built up by a cooperative behavior among traders, who imitate each other (in this sense is an endogenous process) and contribute to form and reinforce trends that converge up to a critical point.

The main advantage of this approach is that the system (the market) would anticipate the crash by releasing precursory fingerprints observable in the stock market prices: the market prices contain information on impending crashes and this implies that:

if the traders were to learn how to decipher and use this information, they would act on it and on the knowledge that others act on it; nevertheless, the crashes would still probably happen. Our results suggest a weaker form of the “weak efficient market hypothesis”, according to which the market prices contain, in addition to the information generally available to all, subtle information formed by the global market that most or all individual traders have not yet learned to decipher and use. Instead of the usual interpretation of the efficient market hypothesis in which traders extract and consciously incorporate (by their action) all information contained in the market prices, we propose that the market as a whole can exhibit “emergent” behavior not shared by any of its constituents.

In a nutshell, the critical events emerge in a self-organized and cooperative fashion as the macro result of the internal and micro interactions of the traders, their imitation and mirroring.



Adjacency of the Possible: Teleology of Autocatalysis. Thought of the Day 140.0


Given a network of catalyzed chemical reactions, a (sub)set R of such reactions is called:

  1. Reflexively autocatalytic (RA) if every reaction in R is catalyzed by at least one molecule involved in any of the reactions in R;
  2. F-generated (F) if every reactant in R can be constructed from a small “food set” F by successive applications of reactions from R;
  3. Reflexively autocatalytic and F-generated (RAF) if it is both RA and F.

The food set F contains molecules that are assumed to be freely available in the environment. Thus, an RAF set formally captures the notion of “catalytic closure”, i.e., a self-sustaining set supported by a steady supply of (simple) molecules from some food set….

Stuart Kauffman begins with the Darwinian idea of the origin of life in a biological ‘primordial soup’ of organic chemicals and investigates the possibility of one chemical substance to catalyze the reaction of two others, forming new reagents in the soup. Such catalyses may, of course, form chains, so that one reagent catalyzes the formation of another catalyzing another, etc., and self-sustaining loops of reaction chains is an evident possibility in the appropriate chemical environment. A statistical analysis would reveal that such catalytic reactions may form interdependent networks when the rate of catalyzed reactions per molecule approaches one, creating a self-organizing chemical cycle which he calls an ‘autocatalytic set’. When the rate of catalyses per reagent is low, only small local reaction chains form, but as the rate approaches one, the reaction chains in the soup suddenly ‘freeze’ so that what was a group of chains or islands in the soup now connects into one large interdependent network, constituting an ‘autocatalytic set’. Such an interdependent reaction network constitutes the core of the body definition unfolding in Kauffman, and its cyclic character forms the basic precondition for self-sustainment. ‘Autonomous agent’ is an autocatalytic set able to reproduce and to undertake at least one thermodynamic work cycle.

This definition implies two things: reproduction possibility, and the appearance of completely new, interdependent goals in work cycles. The latter idea requires the ability of the autocatalytic set to save energy in order to spend it in its own self-organization, in its search for reagents necessary to uphold the network. These goals evidently introduce a – restricted, to be sure – teleology defined simply by the survival of the autocatalytic set itself: actions supporting this have a local teleological character. Thus, the autocatalytic set may, as it evolves, enlarge its cyclic network by recruiting new subcycles supporting and enhancing it in a developing structure of subcycles and sub-sub-cycles. 

Kauffman proposes that the concept of ‘autonomous agent’ implies a whole new cluster of interdependent concepts. Thus, the autonomy of the agent is defined by ‘catalytic closure’ (any reaction in the network demanding catalysis will get it) which is a genuine Gestalt property in the molecular system as a whole – and thus not in any way derivable from the chemistry of single chemical reactions alone.

Kauffman’s definitions on the basis of speculative chemistry thus entail not only the Kantian cyclic structure, but also the primitive perception and action phases of Uexküll’s functional circle. Thus, Kauffman’s definition of the organism in terms of an ‘autonomous agent’ basically builds on an Uexküllian intuition, namely the idea that the most basic property in a body is metabolism: the constrained, organizing processing of high-energy chemical material and the correlated perception and action performed to localize and utilize it – all of this constituting a metabolic cycle coordinating the organism’s in- and outside, defining teleological action. Perception and action phases are so to speak the extension of the cyclical structure of the closed catalytical set to encompass parts of its surroundings, so that the circle of metabolism may only be completed by means of successful perception and action parts.

The evolution of autonomous agents is taken as the empirical basis for the hypothesis of a general thermodynamic regularity based on non-ergodicity: the Big Bang universe (and, consequently, the biosphere) is not at equilibrium and will not reach equilibrium during the life-time of the universe. This gives rise to Kauffman’s idea of the ‘adjacent possible’. At a given point in evolution, one can define the set of chemical substances which do not exist in the universe – but which is at a distance of one chemical reaction only from a substance already existing in the universe. Biological evolution has, evidently, led to an enormous growth of types of organic macromolecules, and new such substances come into being every day. Maybe there is a sort of chemical potential leading from the actually realized substances and into the adjacent possible which is in some sense driving the evolution? In any case, Kauffman claims the hypothesis that the biosphere as such is supercritical in the sense that there is, in general, more than one action catalyzed by each reagent. Cells, in order not to be destroyed by this chemical storm, must be internally subcritical (even if close to the critical boundary). But if the biosphere as such is, in fact, supercritical, then this distinction seemingly a priori necessitates the existence of a boundary of the agent, protecting it against the environment.

Revisiting Catastrophes. Thought of the Day 134.0

The most explicit influence from mathematics in semiotics is probably René Thom’s controversial theory of catastrophes (here and here), with philosophical and semiotic support from Jean Petitot. Catastrophe theory is but one of several formalisms in the broad field of qualitative dynamics (comprising also chaos theory, complexity theory, self-organized criticality, etc.). In all these cases, the theories in question are in a certain sense phenomenological because the focus is different types of qualitative behavior of dynamic systems grasped on a purely formal level bracketing their causal determination on the deeper level. A widespread tool in these disciplines is phase space – a space defined by the variables governing the development of the system so that this development may be mapped as a trajectory through phase space, each point on the trajectory mapping one global state of the system. This space may be inhabited by different types of attractors (attracting trajectories), repellors (repelling them), attractor basins around attractors, and borders between such basins characterized by different types of topological saddles which may have a complicated topology.

Catastrophe theory has its basis in differential topology, that is, the branch of topology keeping various differential properties in a function invariant under transformation. It is, more specifically, the so-called Whitney topology whose invariants are points where the nth derivative of a function takes the value 0, graphically corresponding to minima, maxima, turning tangents, and, in higher dimensions, different complicated saddles. Catastrophe theory takes its point of departure in singularity theory whose object is the shift between types of such functions. It thus erects a distinction between an inner space – where the function varies – and an outer space of control variables charting the variation of that function including where it changes type – where, e.g. it goes from having one minimum to having two minima, via a singular case with turning tangent. The continuous variation of control parameters thus corresponds to a continuous variation within one subtype of the function, until it reaches a singular point where it discontinuously, ‘catastrophically’, changes subtype. The philosophy-of-science interpretation of this formalism now conceives the stable subtype of function as representing the stable state of a system, and the passage of the critical point as the sudden shift to a new stable state. The configuration of control parameters thus provides a sort of map of the shift between continuous development and discontinuous ‘jump’. Thom’s semiotic interpretation of this formalism entails that typical catastrophic trajectories of this kind may be interpreted as stable process types phenomenologically salient for perception and giving rise to basic verbal categories.


One of the simpler catastrophes is the so-called cusp (a). It constitutes a meta-diagram, namely a diagram of the possible type-shifts of a simpler diagram (b), that of the equation ax4 + bx2 + cx = 0. The upper part of (a) shows the so-called fold, charting the manifold of solutions to the equation in the three dimensions a, b and c. By the projection of the fold on the a, b-plane, the pointed figure of the cusp (lower a) is obtained. The cusp now charts the type-shift of the function: Inside the cusp, the function has two minima, outside it only one minimum. Different paths through the cusp thus corresponds to different variations of the equation by the variation of the external variables a and b. One such typical path is the path indicated by the left-right arrow on all four diagrams which crosses the cusp from inside out, giving rise to a diagram of the further level (c) – depending on the interpretation of the minima as simultaneous states. Here, thus, we find diagram transformations on three different, nested levels.

The concept of transformation plays several roles in this formalism. The most spectacular one refers, of course, to the change in external control variables, determining a trajectory through phase space where the function controlled changes type. This transformation thus searches the possibility for a change of the subtypes of the function in question, that is, it plays the role of eidetic variation mapping how the function is ‘unfolded’ (the basic theorem of catastrophe theory refers to such unfolding of simple functions). Another transformation finds stable classes of such local trajectory pieces including such shifts – making possible the recognition of such types of shifts in different empirical phenomena. On the most empirical level, finally, one running of such a trajectory piece provides, in itself, a transformation of one state into another, whereby the two states are rationally interconnected. Generally, it is possible to make a given transformation the object of a higher order transformation which by abstraction may investigate aspects of the lower one’s type and conditions. Thus, the central unfolding of a function germ in Catastrophe Theory constitutes a transformation having the character of an eidetic variation making clear which possibilities lie in the function germ in question. As an abstract formalism, the higher of these transformations may determine the lower one as invariant in a series of empirical cases.

Complexity theory is a broader and more inclusive term covering the general study of the macro-behavior of composite systems, also using phase space representation. The theoretical biologist Stuart Kauffman (intro) argues that in a phase space of all possible genotypes, biological evolution must unfold in a rather small and specifically qualified sub-space characterized by many, closely located and stable states (corresponding to the possibility of a species to ‘jump’ to another and better genotype in the face of environmental change) – as opposed to phase space areas with few, very stable states (which will only be optimal in certain, very stable environments and thus fragile when exposed to change), and also opposed, on the other hand, to sub-spaces with a high plurality of only metastable states (here, the species will tend to merge into neighboring species and hence never stabilize). On the base of this argument, only a small subset of the set of virtual genotypes possesses ‘evolvability’ as this special combination between plasticity and stability. The overall argument thus goes that order in biology is not a pure product of evolution; the possibility of order must be present in certain types of organized matter before selection begins – conversely, selection requires already organized material on which to work. The identification of a species with a co-localized group of stable states in genome space thus provides a (local) invariance for the transformation taking a trajectory through space, and larger groups of neighboring stabilities – lineages – again provide invariants defined by various more or less general transformations. Species, in this view, are in a certain limited sense ‘natural kinds’ and thus naturally signifying entities. Kauffman’s speculations over genotypical phase space have a crucial bearing on a transformation concept central to biology, namely mutation. On this basis far from all virtual mutations are really possible – even apart from their degree of environmental relevance. A mutation into a stable but remotely placed species in phase space will be impossible (evolution cannot cross the distance in phase space), just like a mutation in an area with many, unstable proto-species will not allow for any stabilization of species at all and will thus fall prey to arbitrary small environment variations. Kauffman takes a spontaneous and non-formalized transformation concept (mutation) and attempts a formalization by investigating its condition of possibility as movement between stable genomes in genotype phase space. A series of constraints turn out to determine type formation on a higher level (the three different types of local geography in phase space). If the trajectory of mutations must obey the possibility of walking between stable species, then the space of possibility of trajectories is highly limited. Self-organized criticality as developed by Per Bak (How Nature Works the science of self-organized criticality) belongs to the same type of theories. Criticality is here defined as that state of a complicated system where sudden developments in all sizes spontaneously occur.


CUSUM Deceleration. Drunken Risibility.


CUSUM, or cumulative sum is used for detecting and monitoring change detection. Let us introduce a measurable space (Ω, F), where Ω = R, F = ∪nFn and F= σ{Yi, i ∈ {0, 1, …, n}}. The law of the sequence  Yi, i = 1, …., is defined by the family of probability measures {Pv}, v ∈ N*. In other words, the probability measure Pv for a given v > 0, playing the role of the change point, is the measure generated on Ω by the sequence Yi, i = 1, … , when the distribution of the Yi’s changes at time v. The probability measures P0 and P are the measures generated on Ω by the random variables Yi when they have an identical distribution. In other words, the system defined by the sequence Yi undergoes a “regime change” from the distribution P0 to the distribution P at the change point time v.

The CUSUM (cumulative sum control chart) statistic is defined as the maximum of the log-likelihood ratio of the measure Pv to the measure P on the σ-algebra Fn. That is,

Cn := max0≤v≤n log dPv/dP|Fn —– (1)

is the CUSUM statistic on the σ-algebra Fn. The CUSUM statistic process is then the collection of the CUSUM statistics {Cn} of (1) for n = 1, ….

The CUSUM stopping rule is then

T(h) := inf {n ≥ 0: max0≤v≤n log dPv/dP|Fn ≥ h} —– (2)

for some threshold h > 0. In the CUSUM stopping rule (2), the CUSUM statistic process of (1) is initialized at

C0 = 0 —– (3)

The CUSUM statistic process was first introduced by E. S. Page in the form that it takes when the sequence of random variables Yis independent and Gaussian; that is, Yi ∼ N(μ, 1), i = 1, 2,…, with μ = μ0 for i < 𝜈 and μ = μ1 for i ≥ 𝜈. Since its introduction by Page, the CUSUM statistic process of (1) and its associated CUSUM stopping time of (2) have been used in a plethora of applications where it is of interest to perform detection of abrupt changes in the statistical behavior of observations in real time. Examples of such applications are signal processing, monitoring the outbreak of an epidemic, financial surveillance, and computer vision. The popularity of the CUSUM stopping time (2) is mainly due to its low complexity and optimality properties in both discrete and continuous time models.

Let Yi ∼ N(μ0, σ2) that change to Yi ∼ N(μ1, σ2) at the change point time v. We now proceed to derive the form of the CUSUM statistic process (1) and its associated CUSUM stopping time (2). Let us denote by φ(x) = 1/√2π e-x2/2 the Gaussian kernel. For the sequence of random variables Yi given earlier,

Cn := max0≤v≤n log dPv/dP|Fn

= max0≤v≤n log (∏i=1v-1φ(Yi0)/σ ∏i=vnφ(Yi1)/σ)/∏i=1nφ(Yi0)/σ

= 1/σ2max0≤v≤n 1 – μ0)∑i=vn[Yi – (μ1 + μ0)/2] —– (4)

In view of (3), let us initialize the sequence (4) at Y0 = (μ1 + μ0)/2 and distinguish two cases.

a) μ> μ0: divide out (μ1 – μ0), multiply by the constant σ2 in (4) and use (2) to obtain CUSUM stopping T+:

T+(h+) = inf {n ≥ 0: max0≤v≤n i=vn[Yi – (μ1 + μ0)/2] ≥ h+} —– (5)

for an appropriately scaled threshold h> 0.

b) μ< μ0: divide out (μ1 – μ0), multiply by the constant σ2 in (4) and use (2) to obtain CUSUM stopping T:

T(h) = inf {n ≥ 0: max0≤v≤n i=vn[(μ1 + μ0)/2 – Yi] ≥ h} —– (6)

for an appropriately scaled threshold h > 0.

The sequences form a CUSUM according to the deviation of the monitored sequential observations from the average of their pre- and postchange means. Although the stopping times (5) and (6) can be derived by formal CUSUM regime change considerations, they may also be used as general nonparametric stopping rules directly applied to sequential observations.


Knowledge Limited for Dummies….Didactics.


Bertrand Russell with Alfred North Whitehead, in the Principia Mathematica aimed to demonstrate that “all pure mathematics follows from purely logical premises and uses only concepts defined in logical terms.” Its goal was to provide a formalized logic for all mathematics, to develop the full structure of mathematics where every premise could be proved from a clear set of initial axioms.

Russell observed of the dense and demanding work, “I used to know of only six people who had read the later parts of the book. Three of those were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated.” The complex mathematical symbols of the manuscript required it to be written by hand, and its sheer size – when it was finally ready for the publisher, Russell had to hire a panel truck to send it off – made it impossible to copy. Russell recounted that “every time that I went out for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up.”

Momentous though it was, the greatest achievement of Principia Mathematica was realized two decades after its completion when it provided the fodder for the metamathematical enterprises of an Austrian, Kurt Gödel. Although Gödel did face the risk of being liquidated by Hitler (therefore fleeing to the Institute of Advanced Studies at Princeton), he was neither a Pole nor a Texan. In 1931, he wrote a treatise entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which demonstrated that the goal Russell and Whitehead had so single-mindedly pursued was unattainable.

The flavor of Gödel’s basic argument can be captured in the contradictions contained in a schoolboy’s brainteaser. A sheet of paper has the words “The statement on the other side of this paper is true” written on one side and “The statement on the other side of this paper is false” on the reverse. The conflict isn’t resolvable. Or, even more trivially, a statement like; “This statement is unprovable.” You cannot prove the statement is true, because doing so would contradict it. If you prove the statement is false, then that means its converse is true – it is provable – which again is a contradiction.

The key point of contradiction for these two examples is that they are self-referential. This same sort of self-referentiality is the keystone of Gödel’s proof, where he uses statements that imbed other statements within them. This problem did not totally escape Russell and Whitehead. By the end of 1901, Russell had completed the first round of writing Principia Mathematica and thought he was in the homestretch, but was increasingly beset by these sorts of apparently simple-minded contradictions falling in the path of his goal. He wrote that “it seemed unworthy of a grown man to spend his time on such trivialities, but . . . trivial or not, the matter was a challenge.” Attempts to address the challenge extended the development of Principia Mathematica by nearly a decade.

Yet Russell and Whitehead had, after all that effort, missed the central point. Like granite outcroppings piercing through a bed of moss, these apparently trivial contradictions were rooted in the core of mathematics and logic, and were only the most readily manifest examples of a limit to our ability to structure formal mathematical systems. Just four years before Gödel had defined the limits of our ability to conquer the intellectual world of mathematics and logic with the publication of his Undecidability Theorem, the German physicist Werner Heisenberg’s celebrated Uncertainty Principle had delineated the limits of inquiry into the physical world, thereby undoing the efforts of another celebrated intellect, the great mathematician Pierre-Simon Laplace. In the early 1800s Laplace had worked extensively to demonstrate the purely mechanical and predictable nature of planetary motion. He later extended this theory to the interaction of molecules. In the Laplacean view, molecules are just as subject to the laws of physical mechanics as the planets are. In theory, if we knew the position and velocity of each molecule, we could trace its path as it interacted with other molecules, and trace the course of the physical universe at the most fundamental level. Laplace envisioned a world of ever more precise prediction, where the laws of physical mechanics would be able to forecast nature in increasing detail and ever further into the future, a world where “the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule.”

What Gödel did to the work of Russell and Whitehead, Heisenberg did to Laplace’s concept of causality. The Uncertainty Principle, though broadly applied and draped in metaphysical context, is a well-defined and elegantly simple statement of physical reality – namely, the combined accuracy of a measurement of an electron’s location and its momentum cannot vary far from a fixed value. The reason for this, viewed from the standpoint of classical physics, is that accurately measuring the position of an electron requires illuminating the electron with light of a very short wavelength. But the shorter the wavelength the greater the amount of energy that hits the electron, and the greater the energy hitting the electron the greater the impact on its velocity.

What is true in the subatomic sphere ends up being true – though with rapidly diminishing significance – for the macroscopic. Nothing can be measured with complete precision as to both location and velocity because the act of measuring alters the physical properties. The idea that if we know the present we can calculate the future was proven invalid – not because of a shortcoming in our knowledge of mechanics, but because the premise that we can perfectly know the present was proven wrong. These limits to measurement imply limits to prediction. After all, if we cannot know even the present with complete certainty, we cannot unfailingly predict the future. It was with this in mind that Heisenberg, ecstatic about his yet-to-be-published paper, exclaimed, “I think I have refuted the law of causality.”

The epistemological extrapolation of Heisenberg’s work was that the root of the problem was man – or, more precisely, man’s examination of nature, which inevitably impacts the natural phenomena under examination so that the phenomena cannot be objectively understood. Heisenberg’s principle was not something that was inherent in nature; it came from man’s examination of nature, from man becoming part of the experiment. (So in a way the Uncertainty Principle, like Gödel’s Undecidability Proposition, rested on self-referentiality.) While it did not directly refute Einstein’s assertion against the statistical nature of the predictions of quantum mechanics that “God does not play dice with the universe,” it did show that if there were a law of causality in nature, no one but God would ever be able to apply it. The implications of Heisenberg’s Uncertainty Principle were recognized immediately, and it became a simple metaphor reaching beyond quantum mechanics to the broader world.

This metaphor extends neatly into the world of financial markets. In the purely mechanistic universe of classical physics, we could apply Newtonian laws to project the future course of nature, if only we knew the location and velocity of every particle. In the world of finance, the elementary particles are the financial assets. In a purely mechanistic financial world, if we knew the position each investor has in each asset and the ability and willingness of liquidity providers to take on those assets in the event of a forced liquidation, we would be able to understand the market’s vulnerability. We would have an early-warning system for crises. We would know which firms are subject to a liquidity cycle, and which events might trigger that cycle. We would know which markets are being overrun by speculative traders, and thereby anticipate tactical correlations and shifts in the financial habitat. The randomness of nature and economic cycles might remain beyond our grasp, but the primary cause of market crisis, and the part of market crisis that is of our own making, would be firmly in hand.

The first step toward the Laplacean goal of complete knowledge is the advocacy by certain financial market regulators to increase the transparency of positions. Politically, that would be a difficult sell – as would any kind of increase in regulatory control. Practically, it wouldn’t work. Just as the atomic world turned out to be more complex than Laplace conceived, the financial world may be similarly complex and not reducible to a simple causality. The problems with position disclosure are many. Some financial instruments are complex and difficult to price, so it is impossible to measure precisely the risk exposure. Similarly, in hedge positions a slight error in the transmission of one part, or asynchronous pricing of the various legs of the strategy, will grossly misstate the total exposure. Indeed, the problems and inaccuracies in using position information to assess risk are exemplified by the fact that major investment banking firms choose to use summary statistics rather than position-by-position analysis for their firmwide risk management despite having enormous resources and computational power at their disposal.

Perhaps more importantly, position transparency also has implications for the efficient functioning of the financial markets beyond the practical problems involved in its implementation. The problems in the examination of elementary particles in the financial world are the same as in the physical world: Beyond the inherent randomness and complexity of the systems, there are simply limits to what we can know. To say that we do not know something is as much a challenge as it is a statement of the state of our knowledge. If we do not know something, that presumes that either it is not worth knowing or it is something that will be studied and eventually revealed. It is the hubris of man that all things are discoverable. But for all the progress that has been made, perhaps even more exciting than the rolling back of the boundaries of our knowledge is the identification of realms that can never be explored. A sign in Einstein’s Princeton office read, “Not everything that counts can be counted, and not everything that can be counted counts.”

The behavioral analogue to the Uncertainty Principle is obvious. There are many psychological inhibitions that lead people to behave differently when they are observed than when they are not. For traders it is a simple matter of dollars and cents that will lead them to behave differently when their trades are open to scrutiny. Beneficial though it may be for the liquidity demander and the investor, for the liquidity supplier trans- parency is bad. The liquidity supplier does not intend to hold the position for a long time, like the typical liquidity demander might. Like a market maker, the liquidity supplier will come back to the market to sell off the position – ideally when there is another investor who needs liquidity on the other side of the market. If other traders know the liquidity supplier’s positions, they will logically infer that there is a good likelihood these positions shortly will be put into the market. The other traders will be loath to be the first ones on the other side of these trades, or will demand more of a price concession if they do trade, knowing the overhang that remains in the market.

This means that increased transparency will reduce the amount of liquidity provided for any given change in prices. This is by no means a hypothetical argument. Frequently, even in the most liquid markets, broker-dealer market makers (liquidity providers) use brokers to enter their market bids rather than entering the market directly in order to preserve their anonymity.

The more information we extract to divine the behavior of traders and the resulting implications for the markets, the more the traders will alter their behavior. The paradox is that to understand and anticipate market crises, we must know positions, but knowing and acting on positions will itself generate a feedback into the market. This feedback often will reduce liquidity, making our observations less valuable and possibly contributing to a market crisis. Or, in rare instances, the observer/feedback loop could be manipulated to amass fortunes.

One might argue that the physical limits of knowledge asserted by Heisenberg’s Uncertainty Principle are critical for subatomic physics, but perhaps they are really just a curiosity for those dwelling in the macroscopic realm of the financial markets. We cannot measure an electron precisely, but certainly we still can “kind of know” the present, and if so, then we should be able to “pretty much” predict the future. Causality might be approximate, but if we can get it right to within a few wavelengths of light, that still ought to do the trick. The mathematical system may be demonstrably incomplete, and the world might not be pinned down on the fringes, but for all practical purposes the world can be known.

Unfortunately, while “almost” might work for horseshoes and hand grenades, 30 years after Gödel and Heisenberg yet a third limitation of our knowledge was in the wings, a limitation that would close the door on any attempt to block out the implications of microscopic uncertainty on predictability in our macroscopic world. Based on observations made by Edward Lorenz in the early 1960s and popularized by the so-called butterfly effect – the fanciful notion that the beating wings of a butterfly could change the predictions of an otherwise perfect weather forecasting system – this limitation arises because in some important cases immeasurably small errors can compound over time to limit prediction in the larger scale. Half a century after the limits of measurement and thus of physical knowledge were demonstrated by Heisenberg in the world of quantum mechanics, Lorenz piled on a result that showed how microscopic errors could propagate to have a stultifying impact in nonlinear dynamic systems. This limitation could come into the forefront only with the dawning of the computer age, because it is manifested in the subtle errors of computational accuracy.

The essence of the butterfly effect is that small perturbations can have large repercussions in massive, random forces such as weather. Edward Lorenz was testing and tweaking a model of weather dynamics on a rudimentary vacuum-tube computer. The program was based on a small system of simultaneous equations, but seemed to provide an inkling into the variability of weather patterns. At one point in his work, Lorenz decided to examine in more detail one of the solutions he had generated. To save time, rather than starting the run over from the beginning, he picked some intermediate conditions that had been printed out by the computer and used those as the new starting point. The values he typed in were the same as the values held in the original simulation at that point, so the results the simulation generated from that point forward should have been the same as in the original; after all, the computer was doing exactly the same operations. What he found was that as the simulated weather pattern progressed, the results of the new run diverged, first very slightly and then more and more markedly, from those of the first run. After a point, the new path followed a course that appeared totally unrelated to the original one, even though they had started at the same place.

Lorenz at first thought there was a computer glitch, but as he investigated further, he discovered the basis of a limit to knowledge that rivaled that of Heisenberg and Gödel. The problem was that the numbers he had used to restart the simulation had been reentered based on his printout from the earlier run, and the printout rounded the values to three decimal places while the computer carried the values to six decimal places. This rounding, clearly insignificant at first, promulgated a slight error in the next-round results, and this error grew with each new iteration of the program as it moved the simulation of the weather forward in time. The error doubled every four simulated days, so that after a few months the solutions were going their own separate ways. The slightest of changes in the initial conditions had traced out a wholly different pattern of weather.

Intrigued by his chance observation, Lorenz wrote an article entitled “Deterministic Nonperiodic Flow,” which stated that “nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states.” Translation: Long-range weather forecasting is worthless. For his application in the narrow scientific discipline of weather prediction, this meant that no matter how precise the starting measurements of weather conditions, there was a limit after which the residual imprecision would lead to unpredictable results, so that “long-range forecasting of specific weather conditions would be impossible.” And since this occurred in a very simple laboratory model of weather dynamics, it could only be worse in the more complex equations that would be needed to properly reflect the weather. Lorenz discovered the principle that would emerge over time into the field of chaos theory, where a deterministic system generated with simple nonlinear dynamics unravels into an unrepeated and apparently random path.

The simplicity of the dynamic system Lorenz had used suggests a far-reaching result: Because we cannot measure without some error (harking back to Heisenberg), for many dynamic systems our forecast errors will grow to the point that even an approximation will be out of our hands. We can run a purely mechanistic system that is designed with well-defined and apparently well-behaved equations, and it will move over time in ways that cannot be predicted and, indeed, that appear to be random. This gets us to Santa Fe.

The principal conceptual thread running through the Santa Fe research asks how apparently simple systems, like that discovered by Lorenz, can produce rich and complex results. Its method of analysis in some respects runs in the opposite direction of the usual path of scientific inquiry. Rather than taking the complexity of the world and distilling simplifying truths from it, the Santa Fe Institute builds a virtual world governed by simple equations that when unleashed explode into results that generate unexpected levels of complexity.

In economics and finance, institute’s agenda was to create artificial markets with traders and investors who followed simple and reasonable rules of behavior and to see what would happen. Some of the traders built into the model were trend followers, others bought or sold based on the difference between the market price and perceived value, and yet others traded at random times in response to liquidity needs. The simulations then printed out the paths of prices for the various market instruments. Qualitatively, these paths displayed all the richness and variation we observe in actual markets, replete with occasional bubbles and crashes. The exercises did not produce positive results for predicting or explaining market behavior, but they did illustrate that it is not hard to create a market that looks on the surface an awful lot like a real one, and to do so with actors who are following very simple rules. The mantra is that simple systems can give rise to complex, even unpredictable dynamics, an interesting converse to the point that much of the complexity of our world can – with suitable assumptions – be made to appear simple, summarized with concise physical laws and equations.

The systems explored by Lorenz were deterministic. They were governed definitively and exclusively by a set of equations where the value in every period could be unambiguously and precisely determined based on the values of the previous period. And the systems were not very complex. By contrast, whatever the set of equations are that might be divined to govern the financial world, they are not simple and, furthermore, they are not deterministic. There are random shocks from political and economic events and from the shifting preferences and attitudes of the actors. If we cannot hope to know the course of the deterministic systems like fluid mechanics, then no level of detail will allow us to forecast the long-term course of the financial world, buffeted as it is by the vagaries of the economy and the whims of psychology.


Econophysics: Financial White Noise Switch. Thought of the Day 115.0


What is the cause of large market fluctuation? Some economists blame irrationality behind the fat-tail distribution. Some economists observed that social psychology might create market fad and panic, which can be modeled by collective behavior in statistical mechanics. For example, the bi-modular distribution was discovered from empirical data in option prices. One possible mechanism of polarized behavior is collective action studied in physics and social psychology. Sudden regime switch or phase transition may occur between uni-modular and bi-modular distribution when field parameter changes across some threshold. The Ising model in equilibrium statistical mechanics was borrowed to study social psychology. Its phase transition from uni-modular to bi-modular distribution describes statistical features when a stable society turns into a divided society. The problem of the Ising model is that its key parameter, the social temperature, has no operational definition in social system. A better alternative parameter is the intensity of social interaction in collective action.

A difficult issue in business cycle theory is how to explain the recurrent feature of business cycles that is widely observed from macro and financial indexes. The problem is: business cycles are not strictly periodic and not truly random. Their correlations are not short like random walk and have multiple frequencies that changing over time. Therefore, all kinds of math models are tried in business cycle theory, including deterministic, stochastic, linear and nonlinear models. We outline economic models in terms of their base function, including white noise with short correlations, persistent cycles with long correlations, and color chaos model with erratic amplitude and narrow frequency band like biological clock.



The steady state of probability distribution function in the Ising Model of Collective Behavior with h = 0 (without central propaganda field). a. Uni-modular distribution with low social stress (k = 0). Moderate stable behavior with weak interaction and high social temperature. b. Marginal distribution at the phase transition with medium social stress (k = 2). Behavioral phase transition occurs between stable and unstable society induced by collective behavior. c. Bi-modular distribution with high social stress (k = 2.5). The society splits into two opposing groups under low social temperature and strong social interactions in unstable society. 

Deterministic models are used by Keynesian economists for endogenous mechanism of business cycles, such as the case of the accelerator-multiplier model. The stochastic models are used by the Frisch model of noise-driven cycles that attributes external shocks as the driving force of business fluctuations. Since 1980s, the discovery of economic chaos and the application of statistical mechanics provide more advanced models for describing business cycles. Graphically,


The steady state of probability distribution function in socio-psychological model of collective choice. Here, “a” is the independent parameter; “b” is the interaction parameter. a Centered distribution with b < a (denoted by short dashed curve). It happens when independent decision rooted in individualistic orientation overcomes social pressure through mutual communication. b Horizontal flat distribution with b = a (denoted by long dashed line). Marginal case when individualistic orientation balances the social pressure. c Polarized distribution with b > a (denoted by solid line). It occurs when social pressure through mutual communication is stronger than independent judgment. 


Numerical 1 autocorrelations from time series generated by random noise and harmonic wave. The solid line is white noise. The broken line is a sine wave with period P = 1. 

Linear harmonic cycles with unique frequency are introduced in business cycle theory. The auto-correlations from harmonic cycle and white noise are shown in the above figure. Auto-correlation function from harmonic cycles is a cosine wave. The amplitude of cosine wave is slightly decayed because of limited data points in numerical experiment. Auto-correlations from a random series are an erratic series with rapid decade from one to residual fluctuations in numerical calculation. The auto-regressive (AR) model in discrete time is a combination of white noise term for simulating short-term auto-correlations from empirical data.

The deterministic model of chaos can be classified into white chaos and color chaos. White chaos is generated by nonlinear difference equation in discrete-time, such as one-dimensional logistic map and two-dimensional Henon map. Its autocorrelations and power spectra look like white noise. Its correlation dimension can be less than one. White noise model is simple in mathematical analysis but rarely used in empirical analysis, since it needs intrinsic time unit.

Color chaos is generated by nonlinear differential equations in continuous-time, such as three-dimensional Lorenz model and one-dimensional model with delay-differential model in biology and economics. Its autocorrelations looks like a decayed cosine wave, and its power spectra seem a combination of harmonic cycles and white noise. The correlation dimension is between one and two for 3D differential equations, and varying for delay-differential equation.


History shows the remarkable resilience of a market that experienced a series of wars and crises. The related issue is why the economy can recover from severe damage and out of equilibrium? Mathematically speaking, we may exam the regime stability under parameter change. One major weakness of the linear oscillator model is that the regime of periodic cycle is fragile or marginally stable under changing parameter. Only nonlinear oscillator model is capable of generating resilient cycles within a finite area under changing parameters. The typical example of linear models is the Samuelson model of multiplier-accelerator. Linear stochastic models have similar problem like linear deterministic models. For example, the so-called unit root solution occurs only at the borderline of the unit root. If a small parameter change leads to cross the unit circle, the stochastic solution will fall into damped (inside the unit circle) or explosive (outside the unit circle) solution.


Infinite Sequences and Halting Problem. Thought of the Day 76.0


In attempting to extend the notion of depth from finite strings to infinite sequences, one encounters a familiar phenomenon: the definitions become sharper (e.g. recursively invariant), but their intuitive meaning is less clear, because of distinctions (e.g. between infintely-often and almost-everywhere properties) that do not exist in the finite case.

An infinite sequence X is called strongly deep if at every significance level s, and for every recursive function f, all but finitely many initial segments Xn have depth exceeding f(n).

It is necessary to require the initial segments to be deep almost everywhere rather than infinitely often, because even the most trivial sequence has infinitely many deep initial segments Xn (viz. the segments whose lengths n are deep numbers).

It is not difficult to show that the property of strong depth is invariant under truth-table equivalence (this is the same as Turing equivalence in recursively bounded time, or via a total recursive operator), and that the same notion would result if the initial segments were required to be deep in the sense of receiving less than 2−s of their algorithmic probability from f(n)-fast programs. The characteristic sequence of the halting set K is an example of a strongly deep sequence.

A weaker definition of depth, also invariant under truth-table equivalence, is perhaps more analogous to that adopted for finite strings:

An infinite sequence X is weakly deep if it is not computable in recursively bounded time from any algorithmically random infinite sequence.

Computability in recursively bounded time is equivalent to two other properties, viz. truth-table reducibility and reducibility via a total recursive operator.

By contrast to the situation with truth-table reducibility, Péter Gacs has shown that every sequence is computable from (i.e. Turing reducible to) an algorithmically random sequence if no bound is imposed on the time. This is the infinite analog of far more obvious fact that every finite string is computable from an algorithmically random string (e.g. its minimal program).

Every strongly deep sequence is weakly deep, but by intermittently padding K with large blocks of zeros, one can construct a weakly deep sequence with infinitely many shallow initial segments.

Truth table reducibility to an algorithmically random sequence is equivalent to the property studied by Levin et. al. of being random with respect to some recursive measure. Levin calls sequences with this property “proper” or “complete” sequences, and views them as more realistic and interesting than other sequences because they are the typical outcomes of probabilistic or deterministic effective processes operating in recursively bounded time.

Weakly deep sequences arise with finite probability when a universal Turing machine (with one-way input and output tapes, so that it can act as a transducer of infinite sequences) is given an infinite coin toss sequence for input. These sequences are necessarily produced very slowly: the time to output the n’th digit being bounded by no recursive function, and the output sequence contains evidence of this slowness. Because they are produced with finite probability, such sequences can contain only finite information about the halting problem.