Imre Kondor

• Retired at Eötvös Loránd University

We study the $\pm J$ SK model for small $N$'s up to $N=9$. We sort the $2^{N(N-1)/2}$ possible realizations of the coupling matrix into equivalence classes according to the gauge symmetry and permutation symmetry and determine the energy spectra for each of these classes. We also study the energy landscape in these small systems and find that the e...

This work contains a review of the theoretical understanding for the Almeida-Thouless transition. The effective field theoretic model for this transition to the replica symmetry broken phase is extended to low temperatures, making it possible to compute perturbative corrections to the mean field phase boundary even close to zero temperature. Nonper...

A phase transition in high-dimensional random geometry is analyzed as it arises in a variety of problems. A prominent example is the feasibility of a minimax problem that represents the extremal case of a class of financial risk measures, among them the current regulatory market risk measure Expected Shortfall. Others include portfolio optimization...

A phase transition in high-dimensional random geometry is analyzed as it arises in a variety of problems. A prominent example is the feasibility of a minimax problem that represents the extremal case of a class of financial risk measures, among them the current regulatory market risk measure Expected Shortfall. Others include portfolio optimization...

Expected Shortfall (ES), the average loss above a high quantile, is the current financial regulatory market risk measure. Its estimation and optimization are highly unstable against sample fluctuations and become impossible above a critical ratio r=N/T, where N is the number of different assets in the portfolio, and T is the length of the available...

Expected Shortfall (ES), the average loss above a high quantile, is the current financial regulatory market risk measure. Its estimation and optimization are highly unstable against sample fluctuations and become impossible above a critical ratio $r=N/T$, where $N$ is the number of different assets in the portfolio, and $T$ is the length of the ava...

The optimization of a large random portfolio under the expected shortfall risk measure with an ℓ 2 regularizer is carried out by analytical calculation for the case of uncorrelated Gaussian returns. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this e...

We consider the variance portfolio optimization problem with a ban on short selling. We provide an analytical solution by means of the replica method for the case of a portfolio of independent, but not identically distributed, assets. We study the behavior of the solution as a function of the ratio r between the number N of assets and the length T...

The optimization of the variance of a portfolio of N independent but not identically distributed assets, supplemented by a budget constraint and an asymmetric ℓ1 regularizer, is carried out analytically by the replica method borrowed from the theory of disordered systems. The asymmetric regularizer allows us to penalize short and long positions dif...

The optimization of the variance supplemented by a budget constraint and an asymmetric $\ell_1$ regularizer is carried out analytically by the replica method borrowed from the theory of disordered systems. The asymmetric regularizer allows us to penalize short and long positions differently, so the present treatment includes the no-short-constraine...

A large portfolio of independent returns is optimized under the variance risk measure with a ban on short positions. The no-short selling constraint acts as an asymmetric $\ell_1$ regularizer, setting some portfolio weights to zero and keeping the estimation error bounded, avoiding the divergence present in the non-regularized case. However, the su...

A large portfolio of independent returns is optimized under the variance risk measure with a ban on short positions. The no-short selling constraint acts as an asymmetric $\ell_1$ regularizer, setting some of the portfolio weights to zero and keeping the out of sample estimator for the variance bounded, avoiding the divergence present in the non-re...

In a recent paper, using data from Forbes Global 2000, we have observed that the upper tail of the firm size distribution (by assets) falls off much faster than a Pareto distribution. The missing mass was suggested as an indicator of the size of the Shadow Banking (SB) sector. This short note provides the latest figures of the missing assets for 20...

In a recent paper, using data from Forbes Global 2000, we have observed that the upper tail of the firm size distribution (by assets) falls off much faster than a Pareto distribution. The missing mass was suggested as an indicator of the size of the Shadow Banking (SB) sector. This short note provides the latest figures of the missing assets for 20...

We show that including a term which accounts for finite liquidity in portfolio optimization naturally mitigates the instabilities that arise in the estimation of coherent risk measures on finite samples. This is because taking into account the impact of trading in the market is mathematically equivalent to introducing a regularization on the risk m...

We consider the problem of mean-variance portfolio optimization for a generic covariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be ass...

We consider the problem of mean-variance portfolio optimization for a generic covariance matrix subject to the budget constraint and the constraint for the expected return, with the application of the replica method borrowed from the statistical physics of disordered systems. We find that the replica symmetry of the solution does not need to be ass...

The optimization of a large random portfolio under the Expected Shortfall risk measure with an $\ell_2$ regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data...

The contour maps of the error of historical resp. parametric estimates for large random portfolios optimized under the risk measure Expected Shortfall (ES) are constructed. Similar maps for the sensitivity of the portfolio weights to small changes in the returns as well as the VaR of the ES-optimized portfolio are also presented, along with results...

The contour map of estimation error of Expected Shortfall (ES) is constructed. It allows one to quantitatively determine the sample size (the length of the time series) required by the optimization under ES of large institutional portfolios for a given size of the portfolio, at a given confidence level and a given estimation error.

Investors who optimize their portfolios under any of the coherent risk measures are naturally led to regularized portfolio optimization when they take into account the impact their trades make on the market. We show here that the impact function determines which regularizer is used. We also show that any regularizer based on the norm $L_p$ with $p>...

Using public data (Forbes Global 2000) we show that the asset sizes for the largest global firms follow a Pareto distribution in an intermediate range, that is "interrupted" by a sharp cut-off in its upper tail, where it is totally dominated by financial firms. This flattening of the distribution contrasts with a large body of empirical literature...

The problem of estimation error of Expected Shortfall is analyzed, with a view of its introduction as a global regulatory risk measure.

Correlations and other collective phenomena in a schematic model of heterogeneous binary agents (individual spin-glass samples) are considered on the complete graph and also on 2d and 3d regular lattices. The system's stochastic dynamics is studied by numerical simulations. The dynamics is so slow that one can meaningfully speak of quasi-equilibriu...

We consider the problem of portfolio optimization in the presence of market impact, and derive optimal liquidation strategies. We discuss in detail the problem of finding the optimal portfolio under Expected Shortfall (ES) in the case of linear market impact. We show that, once market impact is taken into account, a regularized version of the usual...

Using public data (Forbes Global 2000) we show that the asset sizes for the largest global firms follow a Pareto distribution in an intermediate range, that is ``interrupted'' by a sharp cut-off in its upper tail, where it is totally dominated by financial firms. This flattening of the distribution contrasts with a large body of empirical literatur...

A loop expansion around Parisi's replica symmetry breaking mean field theory is constructed, in zero field. We obtain the equation of state (and associated Parisi's solution) below the upper critical dimension d u =6, and, in particular, explicit corrections in εlnt and (εlnt) ² with t=(T c −T)/T c and ε=6−d. This allows us to verify that standard...

The standard model defined in the Capital Adequacy Directive issued by the EEC in 1993 imposes nonlinear constraints on certain parts of the trading portfolios of financial institutions. It is shown that an institution that complies with the rules of the standard model but wants to optimize its portfolio according to some internal criteria, such as...

We argue that complex systems must possess long range correlations and illustrate this idea on the example of the mean field spin glass model. Defined on the complete graph, this model has no genuine concept of distance, but the long range character of correlations is translated into a broad distribution of the spin-spin correlation coefficients fo...

The optimization of large portfolios displays an inherent instability to estimation error. This poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. In this paper, we approach the problem from the...

It is shown that the axioms for coherent risk measures imply that whenever there is a pair of portfolios such that one of them dominates the other in a given sample (which happens with finite probability even for large samples), then there is no optimal portfolio under any coherent measure on that sample, and the risk measure diverges to minus infi...

We consider the problem of portfolio optimization in the presence of market impact, and derive optimal liquidation strategies. We discuss in detail the problem of finding the optimal portfolio under Expected Shortfall (ES) in the case of linear market impact. We show that, once market impact is taken into account, a regularized version of the usual...

The optimization of large portfolios displays an inherent instability to estimation error. This poses a fundamental problem, because solutions that are not stable under sample fluctuations may look optimal for a given sample, but are, in effect, very far from optimal with respect to the average risk. In this paper, we approach the problem from the...

We study the feasibility and noise sensitivity of portfolio optimization under some downside risk measures (value-at-risk, expected shortfall, and semivariance) when they are estimated by fitting a parametric distribution on a finite sample of asset returns. We find that the existence of the optimum is a probabilistic issue, depending on the partic...

We study the feasibility and noise sensitivity of portfolio optimization under some downside risk measures (Value-at-Risk, Expected Shortfall, and semivariance) when they are estimated by fitting a parametric distribution on a finite sample of asset returns. We find that the existence of the optimum is a probabilistic issue, depending on the partic...

The problem of estimation error in portfolio optimization is discussed, in the limit where the portfolio size N and the sample size T go to infinity such that their ratio is fixed. The estimation error strongly depends on the ratio N/T and diverges for a critical value of this parameter. This divergence is the manifestation of an algorithmic phase...

It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that sample, and the risk measure diverges to minus infinity. Th...

This paper investigates the efficiency of minimum variance portfolio optimization for stock price movements following the Constant Conditional Correlation GARCH process proposed by Bollerslev. Simulations show that the quality of portfolio selection can be improved substantially by computing optimal portfolio weights from conditional covariances in...

The problem of estimation error in portfolio optimization is discussed, in the limit where the portfolio size N and the sample size T go to infinity such that their ratio is fixed. The estimation error strongly depends on the ratio N/T and diverges for a critical value of this parameter. This divergence is the manifestation of an algorithmic phase...

The talk reviews Parisi's mean field theory for an Ising spin glass, with emphasis on its stability properties and physical interpretation. Next, as a first step towards physical, shortrange systems, quadratic fluctuations around this mean field solution are considered and shown to be finite, despite the presence of an infinity of soft modes in the...

We address the problem of portfolio optimization under the simplest coherent risk measure, i.e. the expected shortfall. As is well known, one can map this problem into a linear programming setting. For some values of the external parameters, when the available time series is too short, portfolio optimization is ill-posed because it leads to unbound...

The mass spectrum of the fully dressed propagators in the condensed phase of the short-range Ising spin glass is investigated. It is shown that zero modes are present in the spectrum so long as the dimension of the system is high enough to support an ultrametric, Parisi-type order.

We study the sensitivity to estimation error of portfolios optimized under various risk measures, including variance, absolute deviation, expected shortfall and maximal loss. We introduce a measure of portfolio sensitivity and test the various risk measures by considering simulated portfolios of varying sizes N and for different lengths T of the ti...

It is well known that portfolio optimization is unstable: the out-of-sample variance and the weights of the optimal portfolio show large sample to sample fluctuations. Moreover, it has been shown recently that the estimation error in the minimum risk portfolio diverges at a critical value of the ratio of the portfolio size N and the length T of the...

We study the sensitivity to estimation error of portfolios optimized under various risk measures, including variance, absolute deviation, expected shortfall and maximal loss. We introduce a measure of portfolio sensitivity and test the various risk measures by considering simulated portfolios of varying sizes N and for different lengths T of the ti...

We address the problem of portfolio optimization under the simplest coherent risk measure, i.e. the expected shortfall. As it is well known, one can map this problem into a linear programming setting. For some values of the external parameters, when the available time series is too short, the portfolio optimization is ill posed because it leads to...

The dynamics of many social, technological and economic phenomena are driven by individual human actions, turning the quantitative understanding of human behavior into a central question of modern science. Current models of human dynamics, used from risk assessment to communications, assume that human actions are randomly distributed in time and th...

Portfolio selection has a central role in finance theory and practical applications. The classical approach uses the standard deviation as risk measure, but a couple of alternatives also exist in the literature. Due to its computational advantages, portfolio optimization based on absolute deviation looks particularly interesting and it is widely us...

We study empirical covariance matrices in finance. Due to the limited amount of available input information, these objects incorporate a huge amount of noise, so their naive use in optimization procedures, such as portfolio selection, may be misleading. In this paper we investigate a recently introduced filtering procedure, and demonstrate the appl...

Current models of human dynamics, used from risk assessment to communications, assume that human actions are randomly distributed in time and thus well approximated by Poisson processes. We provide direct evidence that for five human activity patterns the timing of individual human actions follow non-Poisson statistics, characterized by bursts of r...

A model-based approach was applied for a systematic investigation of the performance of various noise reduction procedures applied in portfolio selection and risk management. To demonstrate the usefulness of this approach, several toy models were developed for the structure of financial correlations. By considering only the noise arising from the f...

We introduce a covariance matrix estimator that both takes into account the heteroskedasticity of financial returns (by using an exponentially weighted moving average) and reduces the effective dimensionality of the estimation (and hence measurement noise) via techniques borrowed from random matrix theory. We calculate the spectrum of large exponen...

We show that some specific market risk measures implied by current international capital regulation (the Basel Accords and the Capital Adequacy Directive of the European Union) violate the obvious requirement of convexity in some regions in the space of portfolio weights.

Financial correlations play a central role in financial theory and also in many practical applications. From theoretical point of view, the key interest is in a proper description of the structure and dynamics of correlations. From practical point of view, the emphasis is on the ability of the developed models to provide the adequate input for the...

Recent studies inspired by results from random matrix theory [1,2,3] found that covariance matrices determined from empirical financial time series appear to contain such a high amount of noise that their structure can essentially be regarded as random. This seems, however, to be in contradiction with the fundamental role played by covariance matri...

Recent studies inspired by results from random matrix theory [1,2,3] found that covariance matrices determined from empirical financial time series appear to contain such a high amount of noise that their structure can essentially be regarded as random. This seems, however, to be in contradiction with the fundamental role played by covariance matri...

According to recent findings [1,2], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [1], e.g., it is reported that about 94% of the spectrum of these matrices c...

According to recent findings [1,2], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [1], e.g., it is reported that about 94% of the spectrum of these matrices c...

We analyze the performance of RiskMetrics, a widely used methodology for measuring market risk. Based on the assumption of normally distributed returns, the RiskMetrics model completely ignores the presence of fat tails in the distribution function, which is an important feature of financial data. Nevertheless, it was commonly found that RiskMetric...

We analyze the performance of RiskMetrics, a widely used methodology for measuring market risk. Based on the assumption of normally distributed returns, the RiskMetrics model completely ignores the presence of fat tails in the distribution function, which is an important feature of financial data. Nevertheless, it was commonly found that RiskMetric...

On the basis of a generalised star-triangle transformation and of existing conjectures concerning the critical point of some planar Potts models, the phase boundary is determined for the combined site-bond percolation problem on a number of lattices, including the triangular and the honeycomb lattice. As a special case, the critical probability for...

The exact phase boundary for a site-diluted q-state Potts model on the honeycomb lattice with vacancies appearing on one of the sublattices only is derived from a duality argument. While the result offers a partial test of recent renormalisation group calculations, it is pointed out that the RG flow structure in this model must be different from th...

The reparametrization transformation between ultrametrically organised states of replicated disordered systems is explicitly defined. The invariance of the longitudinal free energy under this transformation, i.e. reparametrization invariance, is shown to be a direct consequence of the higher level symmetry of replica equivalence. The double limit o...

The reparametrization transformation between ultrametrically organised states of replicated disordered systems is explicitly defined. The invariance of the longitudinal free energy under this transformation, i.e. reparametrization invariance, is shown to be a direct consequence of the higher level symmetry of replica equivalence. The double limit o...

In a recent paper Galluccio, Bouchaud and Potters demonstrated that a certain portfolio problem with a nonlinear constraint maps exactly onto finding the ground states of a long-range spin glass, with the concomitant nonuniqueness and instability of the optimal portfolios. Here we put forward geometric arguments that lead to qualitatively similar c...

A statistical analysis of the Budapest Stock Index (BUX) is presented. The high time resolution (5 s sampling) makes it possible to extract information on market functioning which does not emerge from daily data. The main results are as follows: from a statistical point of view the large drop in October 1997 was a “normal” event. Strong autocorrela...

The expansion of the order parameter q(x) of an Ising spin glass in the inverse number 1/z of interacting neighbours, corresponding to the standard field theoretic loop expansion, is considered above d = 6 in the vicinity of the critical temperature. At first-loop order and in dimensions d > 8, q(x) is linear with an essentially temperature-indepen...

Field theory has been held back from being able to describe the spin-glass condensed phase in physical dimensions by the strong infrared divergences of its (bare) propagators with small overlaps, e.g. the zero overlap replicon propagator G00R(p) approximately p-4. Here the authors examine the effect of fluctuations at the one-loop level on the equa...

Parisi's replica symmetry breaking solution for spin glasses is extended to finite replica number n. The free energy Fp(n) obtained this way, as well as its first two derivatives with respect to n, are shown to join the corresponding values in the Sherrington-Kirkpatrick (SK) solution at a characteristic value ns(T), where stability breaks down in...

The spectrum of the stability matrix associated with Sompolinsky's solution (1981) for a long-range spin glass is studied near Tc in a magnetic field. It is shown that the reparametrisation (or gauge) invariance of the theory locks together not only the order parameters q(x) and Delta (x) but also their fluctuations, and gives rise to a gauge-invar...

By a loop-expansion around Parisi's mean-field theory for an Ising spin-glass it is shown that the overlap of the magnetization patterns belonging to two different temperatures, T and T', vanishes to any order, (si)T(si)T=0, while the correlation overlap (sisj)T(sisj)T calculated to first loop order (and, for technical reasons, for dimensions d>8 o...

The effect of field or temperature changes on spin correlations is calculated for the finite-range Ising spin glass in the Gaussian approximation. Correlation overlaps are found to decay exponentially, with characteristic lengths xi H approximately H-2/3 and xi Delta H approximately ( Delta H)-12/H-1/6 near Tc, when the field is changed from 0 to H...

The overlaps of spin-spin correlations inside one phase space valley and between different valleys are calculated in the Gaussian approximation around Parisi's mean-field solution for the Ising spin glass. The large mass of the theory is shown to be related to the coherence length of the longitudinal fluctuations of the order parameter inside a sin...

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d = 6. A scaling theory of the spin glass phase, based on Parisi’s ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that ampli...

The introduction of ``small permutations'' allows us to derive Ward-Takahashi identities for the spin-glass, in the Parisi limit of an infinite number of steps of replica symmetry breaking. The first identities express the emergence of a band of Goldstone modes. The next identities relate components of (the Replica Fourier Transformed) 3-point func...

The state of art in spin glass field theory is reviewed.

The problem of diagonalizing a class of complicated matrices, to be called ultrametric matrices, is investigated. These matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique of replica symmetry breaking. The residual symmetry, remaining after the breaking of permutation symmetry bet...

A complex problem is solved here; we show how to write Dyson's equations for the (Ising) spin-glass that relate the propagator G to the mass operator M. In other words we are able to reduce the inversion of an ultrametric matrix M to the solution of a Dyson's equation in all sectors (for the replicon sector the result had already been derived). It...

A loop expansion around Parisi's replica symmetry breaking mean field theory is constructed in zero field. We obtain the equation of state (and associated Parisi's solution) below the upper critical dimension d(u) = 6, and, in particular, explicit corrections in epsilonlnt and (epsilonlnt)2 with t = (T(c) - T)/T(c) and epsilon = 6 - d. This allows...

A consistent loop expansion around Parisi's replica symmetry breaking mean field theory for the Ising spin glass is constructed. Above the upper critical dimension d = 6 the short range corrections can be absorbed into the parameters of a Ginzburg-Landau type free energy functional whereby the d < ∞ dimensional theory can be mapped back onto mean f...

Strong infrared divergences prevent standard spin glass field theory to be meaningful, at least below d = 6. A mechanism that softens infrared divergences is displayed, that, via the existence of a sea of zero or near zero modes, relates softening in infrared divergences and in the small x behavior of Parisi order parameter q(x). This mechanism may...

A particular class of massless fluctuations around Parisi’s mean field solution for an Ising spin glass is identified as inducing reparametrization of the order parameter function.

As building blocks of a field theory to be constructed upon Parisi's mean field solution for an Ising spin glass, the complete set of the Gaussian propagators is given. Comme élément d'une théorie des champs à construire autour de la solution de champ moyen de Parisi pour un verre de spin d'Ising, nous donnons l'ensemble complet des propagateurs au...

Goltsev's recent stability analysis for Parisi's solution of the Sherrington-Kirkpatrick model is shown to be incorrect near Tc, and also in disagreement with the analysis of the present authors.

We study, near Tc, the stability of Parisi's solution for the long-range spin-glass. In addition to the discrete, "longitudinal" spectrum found by Thouless, de Almeida, and Kosterlitz, we find "transverse" bands depending on one or two continuous parameters, and a host of zero modes occupying most of the parameter space. All eigenvalues are non-neg...

A certain relationship between a recent conjecture by F.Y. Wu (1979) and one by W. Klein et al. (1978) is established through considering a three-colour site percolation problem, where the three sublattices of a triangular lattice are populated with probability s1, s2 and s3, respectively. Both conjectures imply the same critical condition for vari...

Bray's self-consistent screening approximation for calculating critical exponents is extended to higher orders, resulting in a scheme equivalent to a resummation of the 1/n expansion. While the integration techniques borrowed from conformal invariant field theory reduce the task of evaluating Feynman integrals appearing in this scheme to about the...

Details of the calculation to O(1/n2) of the polarization part and of the vertex function with one ϕ2 insertion are given for an n-component, three-dimensional system with O(n) symmetric, short-ranged interaction. The critical indices λ and μ are deduced and are found to conform to other indices available to the same order. An analysis of how the r...

O(1/n2) corrections to the critical exponents λ and μ associated with the energy propagator and the two point vertex with one ϕ2 insertion, respectively, are calculated for a d = 3 dimensional system with short range forces. Combined with Abe's previous result for η they yield the other indices and confirm a scaling law to the same order. Les expos...

We compute the bare correlation function < Qαβ p Qαβ-p> for the short range spin glass, around Parisi mean field solution, in field near Tc. We evaluate both the correlation function associated with a given overlap x (of replicas α, β) and the x-average i.e. the Fourier transform of the spin correlation < σi σj >2, (p conjugate to |i - j|). In fiel...