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Financial regulation and complexity. Imre Kondor Collegium Budapest and Department of Physics of Complex Systems, Eötvös University, Budapest Invited talk given at the Bolyai János Memorial Conference, Budapest, August 30 - September 4, 2010.
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Financial regulation and complexity Imre Kondor Collegium Budapest and Department of Physics of Complex Systems, Eötvös University, Budapest Invited talk given at the Bolyai János Memorial Conference, Budapest, August 30 - September 4, 2010 This work has been supported by the National Office for Research and Technology under grant No. KCKHA005
This is work in progress.Collaborators: Alain Billoire, Saclay István Csabai, Budapest Jovanka Lukic, Rome Enys Mones, Budapest Enzo Marinari, Rome
Contents • Finance as a complex system • Long range, random correlations • The problem of the systemically important institutions
The financial system consists of a large number of very heterogeneous agents, distributed in wealth or impact on a scale spanning 12 orders of magnitude. There is no symmetry, the model of the representative agent is completely misleading. • It is not true that the players are all equally informed, equally rational, can process the available information in the same way, all are price takers. • It is also not true that, except for some significant exogeneous impacts, the system produces only small fluctuations about equilibrium. Instead, there are recurrent huge endogenous fluctuations (e.g., Black Monday in 1987).
There are complicated strong interactions between the components of the system (e.g. contractual, credit, ownership, or personal relationships, information exchange, „sentiment”, same news, etc.) • Collective effects (bubbles, herding, exuberance, panic, etc.) • No separation of scales, the distribution of wealth, or the market capitalization of firms is power-law-like • The system is global, interwoven by strong, seemingly random correlations, it cannot naturally be cut into parts. • It has a multiattractor structure („fitness landscape”) with several metastable states (various financial, economic, or political regimes) • It is sensitive to initial and boundary conditions, all kinds of control parameters
The number of relevant variables is huge • Financial time series are ‘irreducible’, cannot be compressed into a simple formula, or reduced to a small number of variables, we cannot make an AI system learn and predict them. (But some players possess IT machinery that grossly abuses the system.) They constitute a non-stationary, multivariate stochastic process that exists in a single copy. • Long (sometimes very short) range memory
„Mode slaving”: habits, traditions, laws, institutions • Emergence: all these coming about; bubbles, crises and their consequences. • Adaptation (e.g. to regulation), learning, self-organization. • The system is conditioned by the information about it (the impact of the appearence of the Black-Scholes formula on liquidity), the system reflects upon itself • Self-fulfilling prophecies (e.g. panics, Santa Claus rally, etc.)
Regulating complex systems • Is it possible? • On which scale and which time horizon shall we optimize? • Optimization is value driven
In principle, financial regulation is meant to - protect the interest of „orphans and widows”, - ensure a level playing field for the players, - protect society as a whole from the collaps of the financial system, prevent or at least mitigate systemic risk. The main tool has been an obligatory minimum of own capital, a rule applied to the individual institutions.
In its present form the system is opaque, unknowable and uncontrollable. • The complexity of the system must be reduced, if regulation is expected to become more than an illusion. • When a firm, or an activity reaches systemic significance, it must come under strict regulation.
How do we know that an institution is of systemic importance? • What are the relevant criteria? • Too big to fail? • Too interconnected to fail? • The system is not the “sum of its parts”, its stability does not solely depend on the components, but rather on their interrelations. • The logic of law cannot easily grasp such collective effects.
Vague analogies • When a crystal melts, hardly anything is happening to its atoms • Fitness is not an attribute of the individual creature or species, but a collectively created property that depends on the relative fitness of all the other species in the ecosystem • According to the Hebbian rule, associative memory is not carried by the individual neurons, but the whole network of their synaptic links
The rationale Simple, homogeneous systems in thermal equilibrium (a gas, a simple liquid or solid, etc.) display short range, exponentially decaying correlations. Their parts become independent beyond the correlation length. This allows the limit theorems to take over on large scales, and leads to the normal distribution of fluctuations, standard scaling (mean ~ N, standard deviation ~ N^1/2), hence well defined averages. Note that this picture of thermodynamic equilibrium greatly influenced thinking about equilibrium in economics. If complex systems are „more than the sum of their parts”, it must mean that the above picture does not hold for them. In particular, their correlations must be long ranged, connecting distant parts of the system, and preventing the usual simplifications brought about by the limit theorems. This also means that they are irreducible (incompressible), depend on a large number of variables in a significant way. (This is analogous to the Kolmogorov – Chaitin measure of the complexity of a string. Also Jorge Luis Borges’ map.) To construct models for such systems is difficult: High dimensional models are hard to calibrate or validate, we often face data scarcity or information deficit and large estimation errors. Simulation of such systems is also a delicate issue, the result depends on tiny details, initial and boundary conditions.
These ideas are illustrated on a toy model incorporating the elements of cooperation and competition, called a spin glass
Spin glass: a set of N agents („spins”) having a binary choice and linked by competing interactions The objective function to minimize: where the couplings are randomly scattered over a regular lattice or some more general graph.
The underlying geometry Such a model can be implemented on a regular lattice, like the 2d square lattice shown here
or on a complete graph: Full circles mean spins +1, empty ones -1.
On a small complete graph, e.g. The red edges represent negative („antiferromagnetic”) couplings. Spins linked by such a negative coupling would like to point in opposite directions.
The optimal arrangement of the spins is a complicated distribution of plus-minus ones, correlated with the distribution of couplings in a complicated manner. Even the optimal arrangement can contain a lot of tension: not all the couplings can be satisfied simultaneously. Finite temperature: coupling to an external heat bath kicks some of the agents out of their optimal arrangement.
Frustration The presence of negative couplings leads to „frustration”: one may have two friends who hate each other. Such a trio cannot be made happy. In the little example the triangles containing an odd number of red edges are frustrated.
For large N, the low temperature structure of such a model can be extremely complicated, with an exponentially large number of nearly degenerate minima and their basins of attraction cutting up the set of microscopic states into a set of „pure states” or „phase space valleys”.
Correlations in spin glasses Due to the random structure of the model, the correlations in a given sample behave in a chaotic, random manner as function of the distance. For that reason they had not been regarded as legitimate objects of study, it was rather their sample average that had been investigated. When averaged over the random distribution of the couplings these correlations turn out to be long-ranged, power-law-like, monotonically decaying functions all through the low temperature phase. This implies that correlations must be long ranged in some sense also in the individual samples.
It may be of interest to look into the distribution of correlations as random variables in the individual samples. For that purpose we measured all the N(N-1)/2 correlations and ranked them according to magnitude. Exact enumeration on small systems (up to N=20) and numerical simulations up to N = 2048 indicate that the correlations are anomalously large in the low temperature spin glass. Some raw data follow.
Two samples on a complete graph of size N=128, at low temperature (T=0.4), averaging over all microstates
Two samples on a complete graph of size N= 2048, at T=0.4, averaging over all microstates
The sorted distribution is the inverse of the cumulative distribution function. On the complete graph the sorted distribution seems to tend to a straight line! That would mean that the probability density of the correlations is a constant, every value appearing with the same probability! Note the apparent symmetry of the sorted distribution which does not correspond to any exact symmetry of the system.
N=128 complete graph, T=1.3 (above the critical temperature T=1) Note the change of scale! Most correlations are very small now.
N=2048, complete graph, T=1.3 For this large system the correlations are even smaller. Clearly, for N large the number of large correlations is O(N), which is negligible on the scale of the figure, O(N²)
3d, 10x10x10 lattice, abs. values of correlations, free boundary conditions
Same, with absolute values of correlations measured from an other point
Further findings • The distribution of correlations depends on the actual arrangement of the random couplings. • Correlations strongly depend on boundary conditions and other constraints. • Dependence on distance is non-monotonic, random • Averaging over random couplings washes away all this structure, yields nice, monotonic power law decay • Similar results are obtained on random graphs
Implications for identifying the systemically important banks • A systemically important node must be strongly correlated with most of the system • Correlations depend on the whole distribution of links • Focusing on the individual shop will never reveal the true source of systemic risk
Which kind of law will impose a capital requirement that depends not only on the particular bank’s assets and liabilities, risk management, etc., not even on its links to the other institutions, but also on all the other banks’ parameters and links?
