Unstable Price Dynamics as a Result of Information Absorption in Speculative Markets

In speculative markets, risk-free profit opportunities are eliminated by traders exploiting them. Markets are therefore often described as"informationally efficient", rapidly removing predictable price changes, and leaving only residual unpredictable fluctuations. This classical view of markets absorbing information and otherwise operating close to an equilibrium is challenged by extreme price fluctuations, in particular since they occur far more frequently than can be accounted for by external news. Here we show that speculative markets which absorb mainly self-generated information can exhibit both: evolution towards efficient equilibrium states as well as their subsequent destabilization. This peculiar dynamics, a generic instability arising from an adaptive control which annihilates predictable information, is realized in a minimal agent-based market model where the impacts of agents' strategies adapt according to their trading success. This adaptation implements a learning rule for the market as a whole minimizing predictable price changes. The model reproduces stylized statistical properties of price changes in quantitative detail, including heavy tailed log return distributions and volatility clusters. Our results demonstrate that the perpetual occurrence of market instabilities can be a direct consequence of the very mechanisms that lead to market efficiency.


Abstract
In speculative markets, risk-free profit opportunities are eliminated by traders exploiting them. Markets are therefore often described as "informationally efficient", rapidly removing predictable price changes, and leaving only residual unpredictable fluctuations. This classical view of markets absorbing information and otherwise operating close to an equilibrium is challenged by extreme price fluctuations, in particular since they occur far more frequently than can be accounted for by external news. Here we show that speculative markets which absorb mainly self-generated information can exhibit both: evolution towards efficient equilibrium states as well as their subsequent destabilization. This peculiar dynamics, a generic instability arising from an adaptive control which annihilates predictable information, is realized in a minimal agent-based market model where the impacts of agents' strategies adapt according to their trading success. This adaptation implements a learning rule for the market as a whole minimizing predictable price changes. The model reproduces stylized statistical properties of price changes in quantitative detail, including heavy tailed log return distributions and volatility clusters. Our results demonstrate that the perpetual occurrence of market instabilities can be a direct consequence of the very mechanisms that lead to market efficiency.
Social systems self-organize. In consequence, collective behaviors can emerge that appear to pursue a common goal which is actually not present in the behavior of the single agents (1).
The view that markets in fact operate in distinguished equilibrium states became influential in economics (2). Here, a fundamental hypothesis is that markets operate informationally optimal. That is, prices are assumed to "fully reflect available information" (3), or at least come close to this limit (4), such that risk-free (arbitrage) profits cannot be made by (re-)using said information. If true, one of the implications of this "Efficient Market Hypothesis" (EMH) is that resulting prices fluctuate randomly (5).
Empirical findings in favor of the Efficient Market Hypothesis include the general absence of exploitable autocorrelations among price changes in financial markets (6). The magnitudes of price changes ("volatilities"), however, are correlated for long periods of time. That is, large price changes are typically followed by large ones and small changes by small ones, indicating that markets are complex dynamical systems involving long memories (7,8).
Furthermore, logarithmic price changes (log returns) exhibit large fluctuations even in the absence of external news (9,10), contradicting naive expectations for systems in stable equilibria. More precisely, log return distributions exhibit heavy tails where events that are many times bigger than the standard deviation occur at a much higher frequency than what would be expected if they were Gaussian distributed (7,11) (Fig. 1 (c)). These so-called stylized facts of price time series hint at underlying dynamics that are substantially selfreferential (12). They were associated either with herding effects (13), "bubbles", or with the interactions of heterogeneous traders with limited rationality (14) in a market exactly at a critical point (15).
Here we investigate whether the apparent antinomy of stabilizing information efficient control and dynamics resembling systems operating close to criticality can be resolved by a recent non-economic theory (16): It was shown that adaptive control of a dynamical system can generically lead to an instability where the susceptibility to noise dramatically increases close to the point of perfect balance. This principle applies to markets if two requirements are fulfilled: First, markets have to absorb information about predictable price changes.
Ideally, this property should hold independently of the rationality of the individual traders, which cannot be guaranteed. Second, a self-referential market needs to become susceptible to residual noise once all locally relevant information has been exploited.
The first property is a rather common view in economics. If a profit opportunity is present, investors will increasingly exploit it until their price impacts cancel said opportunity. For example, if a certain stock is priced too low, traders will increase buy orders and thereby raise its price. The second property is also intuitive. As traders try to detect trends or patterns in the price dynamics, they effectively predict how the market reacts to available information.
However, once the agents' actions have led to a balanced equilibrium, it becomes increasingly difficult to distinguish predictable price fluctuations from random noise. If traders then act upon the random fluctuations as if they would hold meaningful new information, their actions will not be balanced anymore. That is, it may be impossible to predict whether a group of traders overreact to the supposed new information and to attenuate the resulting price jump by exploiting it. Therefore, atypically large price movements may become much more likely than expected for a Gaussian distribution.
As a concrete example of the fundamental dynamical instability arising from information absorption as it may be realized in financial markets, we introduce a minimal agent-based trading model. Each agent is representative of one trading strategy and possesses two types of assets which are called money and stocks in the following. For simplicity, we consider a coarse-grained price time series where one step could be considered as e.g. a day. In each time step, every agent either buys or sells an amount of stocks. The fraction of an agent's money that is used to buy stocks or vice versa is denoted by the use parameter γ. At each time t, all trades are performed at the same price p(t) which is determined from the ratio of total demand and supply. Hence, trading per se conserves the total amounts of each asset, money and stocks.
Agents base their decisions on public information states. In each time step t one of D possible states, which is denoted by an index µ(t) ∈ {1, . . . , D}, is conveyed to the agents.
Each agent's decision to either buy or sell is fixed in time for each µ and assigned at random.
We consider two different methods for the generation of these information states: For endogenous information, agents possess a memory of the most recent K signs of the log returns which indicate whether the prices p(t − K), . . . , p(t) decreased or increased with respect to their predecessor. This information can take one out of D = 2 K possible states.
For exogenous information, µ(t) are drawn randomly and independently with probability P ext (µ). Unless stated otherwise all µ have equal probabilities P ext (µ) = 1/D. We also investigated mixed information and obtained results similar to the endogenous case (see supplement).
We focus on markets that are dominated by speculators who can win or loose assets only by betting on price changes within the market. To investigate the effect of a small number of traders that convey new assets to the market or draw out their profits, we divide the agents into N s speculators and N p producers. Producers' resources by definition stay constant.
Thus, speculators only redistribute their wealth and are competitive whereas producers provide a predictable supply of liquidity and stocks. All agents are initially provided with equal amounts of assets.

Stylized Facts
Log-return time series for the model with exogenous information are shown in Fig. 1 (a).
A strong reduction of initial fluctuations is observed leaving only a narrow band of Gaussian distributed returns after the transient. Fig. 1 Fig. 1 (d) and is also consistent with the DJIA.

Dynamics with Exogenous Information
To understand the model dynamics, we first consider the exogenous case which is fully analytically tractable. The rules of asset redistribution by trading are equivalent to a learning rule related to gradient descent, allowing for the market as a whole to minimize predictable price changes. The reason for this stabilizing control is that trading success increases the impact of agents whose actions contribute to a reduction of price fluctuations. A phase transition with respect to the critical parameter α = D/N s is identified at α = 1/2, the point where random binary vectors (the agents) with positive weights (the assets) form a complete basis in the D-dimensional strategy space in the limit N s → ∞. Beyond this point, a speculative market without producers evolves the distribution of assets onto a manifold where the price is invariant to trading. That is, agents still trade and exchange assets, but the price remains constant. Markets that include producers still exhibit finite returns also for α < 1/2. Otherwise, for N p ≪ N s , return distributions only depend weakly on N p . See

Information Annihilation Instability
When the µ(t) are endogenously generated, the same mechanism of information absorption present in the exogenous case ensures that the system relaxes towards local price equilibria and returns vanish, but only transiently. Fig. 2 (a) shows the price time series for a simulation with a very small use γ which formally is a learning rate. At any point in time, the system moves towards a certain price which characterizes a local equilibrium. As the system approaches this equilibrium, price fluctuations are reduced. These fluctuations generally consist of complex oscillations like the one shown in Fig. 2 (b). The equilibria become unstable once all predictable information is exploited by the speculators. Then, even little overshooting of the adaptation dynamics or noise can lead to price changes corresponding to information states that were not predicted by patterns in the immediate past. Because the market is not well-adapted to these new states, the possibility of large price changes increases dramatically. Compared to (16), we here discovered an instability due to information annihilation in a mathematically different way, which demonstrates that this concept is even more general.
For larger γ, this behavior is not as obvious: time series appear random and distinct oscillations are rarely visually recognizable ( Fig. 1 (b)). Still, phase diagrams from extensive simulations demonstrate that return distributions are largely unaffected by these effects over wide ranges of γ: Fig. 3 (a) shows how much initial mean log return magnitudes are reduced for different α and γ; Fig. 3 (b) shows the impacts that infrequent extreme returns have on the remaining variances, which are measured by kurtoses after transients. The stronger the reduction of return magnitudes ( Fig. 3 (a)), the heavier tailed the return distributions are ( Fig. 3 (b)). Therefore, a clear link between information annihilation and extreme returns is established in our model for the whole parameter space. This relation between return reduction and kurtosis is not found for exogenous information drawn from a uniform distribution ( Fig. 1 (b), and supplement).

Surprising Information Causes Large Returns
We find that large returns are caused by information states that have not occurred for a long time. Intuitively, such states are more surprising and therefore carry more information than the ones visited more recently -a concept that is closely related to Shannon information. The more surprising an information state is, the higher the corresponding log return.
The correlation between log return magnitudes and the times τ that have passed since the respective information states occurred last is shown in Fig. 4  extreme price fluctuations. We tested this hypothesis by using inhomogeneously distributed exogenous information states leading to similarly distributed τ . Then, as in the endogenous case, return magnitudes are strongly correlated with τ (Fig. 4, dashed line).

Discussion
Our model demonstrates that even a simple yet plausible mechanism of order size adaptation suffices to coordinate the impacts of diverse strategies such that information becomes completely absorbed in the price. The proof that the trading rules in our model correspond to an efficient gradient based learning rule that minimizes predictable return magnitudes provides a rigorous link of a fundamental market mechanism to adaptive control. We tested variations of this mechanism including adaptations of individual agent use parameters, and order-book based pricing (17); information absorption proved to be a robust property. We therefore consider it highly plausible to prevail in real markets despite the complexities of real pricing mechanisms and order size adaptations, as long as the latter correlate with trading success.
"Rationality", in the sense of an efficient adjustment of prices to new information, in our model emerges from the collective behavior of many traders. We find that a minority rule with respect to the returns is a dominating factor in its dynamics. That is, traders whose actions counteract those of the majority profit most.
In the well known minority game (15)  Each agent i = 1, . . . , N possesses two types of assets which without loss of generality are called money M i (t) and stocks S i (t). N is the total number of agents. In each round, every agent places a market order to either buy or to sell an amount of stocks. Since trading should conserve the amount of traded assets, the price in each round is determined by: with demand δ and supply of stocks ς. This is a fair rule that could be used in a real market with only market orders.
For a market including stochastic limit orders gathered over some period of time, consider the hypothetical price p * (t) at which trades would take place if all agents scaled their orders by a common factor. Then the volume would change, but to preserve market clearing the price is not affected; that is p(t) * = p(t). Therefore, the price is a function of the ratio of demand and supply. After linearization of this function for small small excess demands the price is proportional to the aforementioned ratio which justifies this choice of the pricing rule also as an approximation of the mean prices obtained from limit orders.
Agents base their decisions on a public information state. In each time step one of D possible states, which is denoted by an index µ(t), is conveyed to the agents. We distinguish two different methods for the generation of these information states at each time step t: For exogenous information, µ(t) are independent identically distributed random variates drawn from a distribution P ext (µ).
For endogenous information, agents possess a memory of the most recent K signs of the log returns. To eliminate the possibility of a lock, signs for vanishing returns are chosen at random. This information can take one out of D = 2 K possible states: where Θ denotes the Heaviside step function and η is an arbitrarily small symmetric random variable with zero mean. Simulation results do not depend on Var(η) as long as it is small enough.
ii Each agent's i consequent decision is now determined by a strategy vector whose elements σ µ i are drawn randomly out of {0, 1} once and then kept constant. These two possible decisions correspond to trading an amount m i (t) of money or an amount s i (t) of stocks for the respective other asset in the next round. Orders are placed with a constant use parameter γ: Case σ µ(t) i = 1 (agent i buys stocks): Demand and supply are the sums of all buy and sell orders respectively where ǫ is a small positive number. This ensures that prices and returns are always well defined. The cases with zero demand or supply are, however, irrelevant for all practical purposes. A sufficiently small ǫ ≪ 10 −3 does not influence simulation results to a meaningful degree. All figures were generated using ǫ = 10 −10 .
We investigate the effect of a market ecology by dividing the agents into N s speculators and N p = N − N s producers. We focus on markets that are dominated by speculators whose resources are redistributed due to trading: Producers' resources stay constant throughout the game: iii Thus, only speculators redistribute their wealth and are competitive whereas producers provide a predictable supply of liquidity and stocks. All agents are initially provided with equal amounts of resources M i (0) = S i (0) = 1.

B. Invariant Manifold
We show, that if one distribution of resources (M, S) = (M 1 , . . . , M N , S 1 , . . . , S N ) exists for which the price p(M , S, µ) = p is independent of the information µ, this price is invariant with respect to any resource redistribution due to trading in a purely speculative market.
for which the price is independent of µ and this manifold is closed with respect to trading according to Eq. (7). For the proof, assume that at some point in time the system is in a suitable state such that Then, denoting the distributions of stocks and money after trading by M ′ i and S ′ i we obtain: We can therefore iteratively calculate the probability distribution P (d Ns−1 ) of d after adding N s − 1 agents and the probability that one out of N s agents is linearly independent of the others. If a vector is linearly independent of the other agents in d dimensions, it cannot be canceled by a linear combination of the other agents for all µ. However, it may still be possible to cancel this agent's impact for a subset of all possible µ, i.e. for a smaller subspace. Therefore, the probability that an agent cannot be canceled in any given time step is The last term weights each summand with the fraction of dimensions in which the agent's impact is not canceled. Finally, to relate the fraction of not canceled agents to returns we need to consider the fluctuations prior to any resource redistribution. Since all strategies and µ are chosen randomly, agents initially contribute to the demand or the supply at random.
These fluctuations of demand and supply then follow a binomial distribution with N s trials and equal probability for buying or selling: Since Var(δ) = Var(ς) = N s /4 we can approximate the price for small deviations: vi Therefore, and finally where the logarithm stems from using the logarithm with base 10 in the log return.
Combining Eqns. (18) and (27), we obtain the expected variance of the return for an optimal superposition of agents without the positivity constraint on the resources Var(r n.c. ) = Var(r(0)) P c.c. .
Since resources cannot be negative, they form a positive cone. Each agent thatlocalized is linear independent of the others spans a half space. Therefore, 2N s agents are necessary to completely span the strategy space. Yet for small numbers of agents, each agent still represents a full degree of freedom since the probability that two agents lie on the same 1dimensional submanifold is vanishingly small. However, as the number of agents is increased such that α → 1, an increasingly large number of new agents just converts a halfspace into a full one. Therefore, Eq. (28) represents a lower limit for the variance of the log returns which is a good description for N s ≪ D. An upper limit is obtained by changing Eq. (18) such, that each agents increases d by 1/2. This is a good approximation for N s ≈ 2D. The area in between these limits is shown in Fig. S5 (gray shaded). The lower limit has a phase transition at α = 1 while the upper limit has a phase transition at α = 1/2. A phase transition at α = 1 is already present in Eq. (17). The gradual convergence for the true variance of the system from the lower to the upper limit is captured by a simple heuristic interpolation: For the dark gray line in Fig. S5, the probability for a new linearly independent agent to increase d by one is P 1 = min(1, N s /2 m+1 ) while the probability to increase d by 1/2 is P 1/2 = 1 − p 1 . The presented theory describes the numerical results (Fig. S5) for the model with endogenous information very well for α ≤ 1/2. For full markets, the residual error for simulations with only speculators is determined by the numerical precision. When producers are present, the residual error is noticeably higher. This is due to the fact, that producers push the system off vii the invariant manifold. This error is dependent on the agents' use and vanishes for small γ.
Still, predictable producers are canceled much better than random ones because speculators can successfully predict their choices. For endogenous information (D = 2 K ), the phase transition appears smother and slightly shifted towards larger α. A stronger reduction of average returns for α < 1/2 occurs due to the more localized adaptation: Agents do not adapt to all possible values of µ at the same time.

D. Gradient Descent
We now investigate, how the system evolves towards the invariant manifold. We focus on large numbers of agents and small γ. The resource redistribution due to subsequently trading the two assets for one another is found to be a special case of a learning rule which minimizes log-return magnitudes. Even more generally, we consider the error function and show that its gradient ∂e ∂X = 2r ∂r ∂X , X ∈ {M, S} with respect to the agents' resources is dominated by terms with the opposite sign as the change in the agents' resources. Therefore, any scaling of the agents' resources which keeps the sign of the return for money and the opposite sign for stocks corresponds to minimizing log return magnitudes similar to a gradient descent.
To begin with, consider two subsequent time steps where the information takes the values µ and µ ′ . We again consider a market consisting of speculators only. The derivative of the return with respect to the resources of an agent k is viii The change in resources after trading twice is We are interested in and continue only with leading terms in γ.
For now, we also assume that agents can only perform roundtrip trades (RT). The general case will be discussed later. Two cases are left: Above, we used p = δ/ς and p ′ = δ ′ /ς ′ . Then, Therefore, the change in the total error function can never be positive if agents only perform roundtrip trades.
On average, this result holds even for the general case. The reason why we have to consider averages is, that agents who buy or sell two times in a row always decrease the ix amount of money or stocks they own after two time steps. Therefore, these agents' resources are expected to change in the opposite direction of the gradient half of the time. That is, for every given pair of informations (µ, µ ′ ), a quarter of all agents' is expected to have their resources evolve such that future r(µ, µ ′ ) 2 increase. However, the actual influence of these agents is much lower and can be neglected for large systems. Then, demand and supply can be well described as binomial processes as shown above. Here, express demand and supply as: and the relative fluctuations around the mean demand N/2 are only √ N /2 and therefore small for large N. Thus we can expand Eq. (31) for small fluctuations: As the above equation shows, when agents perform roundtrip trades, they contribute a term of order N −1 to the gradient with respect to each asset. When agents buy or sell twice, they only contribute a term of order N −1.5 for one asset. Therefore, the influence of these agents vanishes for sufficiently large N. By a similar argument, approximately a quarter of all agents performs either one of the actions (buy, sell), (sell, buy), (buy, buy), and (sell, sell) while fluctuations can be neglected for large N. Summing up, the expected change in r(µ, µ ′ ) 2 over repeated trades with the same information is always negative given a sufficiently large number of agents. x

A. Phase Diagrams for Exogenous Information
For uniformly distributed exogenous information states, a reduction of fluctuations does not generally increase the kurtosis (Fig. S6). This is opposed to the endogenous case shown in Fig. 3 in the main paper. Only very large uses in overcomplete markets cause high kurtoses. Then, even uniformly distributed µ occasionally are not repeated for a sufficiently long time to be "forgotten" by the market.  Fig. S10 shows the phase transition with respect to α in more detail. As in Fig. S5, we take one parameter set as a reference to which we compare simulations after transients for different parameters. For orientation, the log-return variances (Fig. S10 (a)) and kurtoses( Fig. S10 (b)), which have been discussed earlier, are shown again.
Mean speculator capitals are not constant over time in markets that include producers. For empty markets, the ratio of average speculator and producer capitals quickly evolves towards an equilibrium.
The more agents are added, the longer it takes for C s to saturate. For critical or crowded markets a positive speculator income persists over long times. Then, average speculator capitals after transients can be well described as: The income factor a is shown in Fig. S10 (c) and quantifies how well the speculators can exploit the producers. a is found to be independent of the initial ratio between speculator and producer capitals. It becomes maximal close to the critical point which can be intuitively understood: For empty markets, there is a finite chance for a producer strategy to lie outside of the space spanned by the speculators. Therefore, increasing the number of speculators increases their average income. For crowded markets, producers are already optimally exploited. Then, adding more speculators just distributes the maximal total income over more of them. An analogous maximum can be found in Minority Games (see e.g. ref. 15 in the main paper).