Market shape formation, statistical equilibrium and neutral evolution theory
Abstract
Mathematical methods of population genetics and framework of exchangeability provide a Markov chain model for analysis and interpretation of stochastic behaviour of equity markets, explaining, in particular, market shape formation, statistical equilibrium and temporal stability of market weights.
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1 Introduction
Loglog plot of normalized stock market capitalizations ranked in descending order is called capital distribution curve. For example, figures below display distribution of capital on the NASDAQ market on three dates in 2014 (data source is http://www.google.com/finance#stockscreener). Ranked market weights experienced relatively small fluctuations, despite significant changes in overall capitalization of the NASDAQ market during that period of time.
One of the aims of this paper is to provide an example of a possible mechanism explaining temporal stability and statistical equilibrium of normalized stock capitalizations by means of the PolyaDirichlet Markov chain, analogous to the WrightFisher model of neutral theory of evolution. Classic and neutral evolution theory. Classic form of Darwinian theory suggests that forces of natural selection play central role in evolution of species. Theory of neutral evolution, proposed by Kimura, complements the classic theory by adding genetic dimension. Kimura observed that discrepancies in traits, such as small variations in colouring of beaks or feathers in a population of birds, occur at moleculargenetic level due to random effects in reproduction and majority of these variations are neutral with respect to fitness. According to the neutral theory, force of natural selection is still be important since it purges deleterious mutations. However, majority of surviving mutations are neutral, and possibly only few are advantageous.
Mutations and random combinations of genes in new generations lead to fluctuations of allelic frequencies or genetic drift. The WrightFisher and Moran models describe stochastic evolution of genetic frequencies as statistical equilibrium fluctuations, modelled by diffusion process with stationary Dirichlet distribution.
Evolution theory and finance. Application of evolutionary ideas in finance has a long history dating back to Malthus, Marshall and many others. Recently Evstigneev, Hens, and SchenkHoppé [evstigneev2008evolutionary] developed descriptive model of Evolutionary Finance, which employs principle of natural selection for modeling dynamics of asset prices and analysis of investment strategies.
Kirman [kirman1993ants] considered version of the WrightFisher model with mutation in a context of economic interpretation of behavior of ants searching for a food source. He observed that proportion of ants choosing one of the possible food channels is better described by stationary distribution of a Markov chain, rather than by single point of equilibrium. He proposed that the ’herding’ behaviour on financial markets as well is better described by means of stochastic equilibrium, rather than by single or multiple equilibria.
Formation of market limit shape. Standard and nonlinear versions of the Polya process have been used by Arthur et al. [arthur1994increasing] for illustration of appearance of market structure. Polya scheme has the following interesting property: proportions of balls converge to some limiting values, but these limits are random and described by the Dirichlet distribution.
Markov lattice and reversibility. PolyaDirichlet Markov chain with state space defined on lattice of ordered integer partitions provides a framework for analysis and modeling of stochastic equilibrium of market weights. Transitions on the lattice of partitions are described in terms of random up and dn operators proposed by Kerov [kerov2003asymptotic], Fulman[fulman2005stein], Borodin and Olshanski [borodin2009infinite] and Petrov [petrov2007two]. Historically, Markov chains with dn/uptransitions in a context of Polya model were first studied by Costantini, Garibaldi, et al. in [costantini2000purely], [garibaldi2004finitary].
Exchangeability and random fluctuations. Infinite exchangeability implies existence of up and dn random transitions, connecting adjacent levels of integer compositions. It is shown in Section LABEL:stocheq that probabilities of these transitions satisfy reversibility conditions and therefore induce a lattice of Markov chains. Random transitions on this lattice correspond to statistical equilibrium behaviour of market weights or allelic frequencies not only for fixed, but also for varying market or population sizes.
Neutral theory and financial markets. The PolyaDirichlet Markov lattice corresponds to the discrete version of the WrightFisher process with mutations and provides a toy model of equilibrium markets behaviour.

After initial phase of rapid expansion, in a same way as proportions of balls converge to random limits in Polya scheme, market weights settle down and form capital distribution curve.

Up and down changes in overall market capitalization lead to random drift of market weights fluctuating in stochastic equilibrium around limiting values, given by the capital distribution curve. The stationary distribution of market weights can be modeled by means of the up and dn Markov chain.

In general, increase of market capitalization enforces market structure and decrease of capitalization leads to weakening of the structure and higher volatility, which creates an opportunity for market reshaping. This mechanism is analogous to the socalled nearly neutral theory of evolution proposed by Ohta [ohta1992nearly], in which smaller populations have faster moleculargenetic evolution and adaptation rate.

This theory provides interpretation of market crises as markets selfadaption to changing economic conditions, where capitalization decrease leads to market reshaping and faster adjustment to new econofinancial landscape.

Arbitrage opportunities can be considered as corresponding to deleterious mutations, eliminated by forces of natural selection.
Mechanics, economics and reversible equilibrium. As pointed out by Garibaldi and Scalas [garibaldiscalas2010], equilibrium modeling in economics and finance was developed under strong influence of ideas of static or classical mechanics. Alternative approach is provided by framework of stochastic equilibrium and reversibility conditions, which have roots in Boltzmann’s work on statistical mechanics. Exhaustive treatment of econophysics from the point of view of exchangeability is contained in the book of Garibaldi and Scalas [garibaldi2010finitary].
Excellent explanation of the framework of reversible equilibrium is contained in the classic book of Kelly [kelly2011reversibility].
2 Polya process with down/up transitions
In a classic form of Polya process colored balls are placed into a box with probabilities proportional to weights of balls of existing colors. The process provides a discrete counterpart of the Dirichlet distribution, since if vector represents initial/prior weights of balls of each color, limiting values of proportions of weights in Polya scheme have Dirichlet distribution with the same vector of parameters .
Modified Polya process, in which balls can also be removed illustrates important ideas of

appearance and temporal stability of ranked proportions, and

stochastic equilibrium of these weights.
Let us consider an artificial stock market with finite number of stocks represented by different colours. Initially in the box there are ’prior’ balls of each colour and the same weight , such that total weight of all balls is . In other words, all stocks start with the same initial conditions and colours (or tickers) are used only to distinguish the stocks. Vector represents stock capitalizations equal to number of placed balls of each color at stage , so at initial stage this vector is . All possible market configurations with overall capitalization are represented by compositions (ordered partitions) in the integervalued simplex
At the first step one of the prior balls is drawn with probability . The ball is placed back into the box together with a ball of the same color and unit weight. At stage probability to add a ball of color is
where denotes number of balls of color in the box. For instance, with colors, say red, green and blue, probability of drawing 3 red balls, 2 green ones and 1 blue ball in this particular sequence is
with raising or ascending factorial power defined as