A Structural Model for Fluctuations in Financial Markets
aa r X i v : . [ q -f i n . S T ] S e p A Structural Model for Fluctuations in Financial Markets
Kartik Anand , Jonathan Khedair , and Reimer K¨uhn Deutsche Bundesbank, Wilhelm-Epstein-Strasse 14, 60431 Frankfurt am Main, Germany Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK (Dated: October 2, 2017)In this paper ∗ we provide a comprehensive analysis of a structural model for the dynamics ofprices of assets traded in a market originally proposed in [1]. The model takes the form of aninteracting generalization of the geometric Brownian motion model. It is formally equivalent to amodel describing the stochastic dynamics of a system of analogue neurons, which is expected toexhibit glassy properties and thus many meta-stable states in a large portion of its parameter space.We perform a generating functional analysis, introducing a slow driving of the dynamics to mimic theeffect of slowly varying macro-economic conditions. Distributions of asset returns over various timeseparations are evaluated analytically and are found to be fat-tailed in a manner broadly in line withempirical observations. Our model also allows to identify collective, interaction mediated propertiesof pricing distributions and it predicts pricing distributions which are significantly broader than theirnon-interacting counterparts, if interactions between prices in the model contain a ferro-magneticbias. Using simulations, we are able to substantiate one of the main hypotheses underlying theoriginal modelling, viz. that the phenomenon of volatility clustering can be rationalised in termsof an interplay between the dynamics within meta-stable states and the dynamics of occasionaltransitions between them. ∗ The opinions expressed in this paper are those of the authors and do not necessarily reflect theviews of the Deutsche Bundesbank, the Eurosystem, or their staff.
PACS numbers: 02.50.r, 05.40.a, 89.65.Gh, 89.75.Da
I. INTRODUCTION
Predicting and measuring the risk that the value ofan investment portfolio will depreciate is a mainstay offinancial mathematics. Integral to the success of theseendeavours is identifying the various market risk (MR)factors and developing models for their evolution. TheseMRs include, amongst others, fluctuations in stock in-dices, changes in interest rates, foreign exchange paritiesor commodity (e.g., gold, oil, etc.) prices.In recognition of the importance of MRs, the BaselCommittee for Banking Supervision (BCBS) stipulatesthat banks must explicitly reserve a portion of equitycapital against MR. The Basel III accord [2] defines MRas the “the risk of losses arising from movements in mar-ket prices”. It proposes two approaches to measure MRs,a so-called Standardized Approach, and an Internal Mod-els Approach.Within the standardized approach the market risk ofa set of trading positions is defined by their exposure toa standardized set of risk factors and measured in termsof sensitivities of the market values of these positions tomovements of the risk factors. To determine the capitalto be held against MRs, the risks of various positions heldby a bank are aggregated, using prescribed risk weightsand prescribed correlations between risk factors.Under the Internal Models Approach, banks are per-mitted to design their own measurement model, whichmust adhere to strict guidelines. Guidelines cover a hostof qualitative and quantitative standards. At a mini-mum, internal models must encompass the positions cov-ered by the standard model, be regularly back-tested against historical market data to demonstrate their ad-equacy and accuracy, and be supplemented by a regularand rigorous regime of stress testing.The collapse of the US based hedge fund Long-TermCapital Management (LTCM) in 2000, following theEast-Asian financial crisis of mid 1997 [3] is a poignantexample of adverse MRs spreading across wide geo-graphic regions. The crisis was instigated by a deval-uation of Thailand’s currency, the Thai baht. This movesent shock waves through the economies of East-Asiancountries, thereby triggering recessions. The economicdownturns led to a sharp decline in the demand and priceof oil. Russia, as a major oil producing country, was ad-versely hit and defaulted on its’ public debt. The cul-mination of all these interlinked shocks resulted in hugelosses for LTCM and, in early 2000, its liquidation. Itcan fairly be argued that chains of events such as thiscall for a more interconnected approach to market risksthan stipulated by the BCBS, even today.The credit crunch of 2007-09 provides further evidenceto support the idea of interactions between risky marketpositions. While financial derivatives and globalization,which allow for greater portfolio diversification, may havehelped mitigate MRs, adverse feedback loops between fi-nancial markets and the real economy may be responsi-ble for propagating asset price shocks across borders andeven commodity classes. A more thorough understandingof the dynamics of asset prices would thus be welcome.A model of MRs should reproduce a set of stylizedfacts found from empirical analysis of time-series data ofreturns on investment [4–8], i.e., (i) return distributionsare “fat-tailed”, (ii) the variance of returns is time depen-dent, and (iii) there are long-range correlations betweenvariance of returns in time, a phenomenon referred to as“volatility clustering”.Over the years a variety of descriptive models havebeen developed for the returns in financial time-series.These models do not attempt to advance theories ofthe mechanisms underlying price processes, but concen-trate on capturing their statistics. Examples includeauto-regressive [9, 10] and stochastic volatility models[11, 12]. Other models assume that the statistics forreturn-increments follow symmetric [4] or asymmetric[13] stable Paretian distributions.In an alternative structural model approach one at-tempts to model the mechanisms behind market dynam-ics. One possible formulation in this regard, is to considerthe collective results of actions performed by agents op-erating in the market. Models of this type include theminority game [14, 15] and percolation models [16, 17].In [1] the authors take an intermediate approach andpropose an interacting variant of the geometric Brown-ian motion model (henceforth referred to as the iGBM)as a structural model of asset price dynamics. Theysuggest that the structure of such a model should fol-low from very generic considerations concerning marketmechanisms, arguing in particular that the dynamicalevolution of a market, when reduced to a descriptionin terms of asset price dynamics, quite generally must exhibit “. . . interaction[s] between prices, which may bethought of as arising effectively through the collection ofagents, each acting on the basis of his or her own, moreor less rational perception of the underlying economy andmarket mechanisms [1]”.Using simulations, the authors go on to demonstratethat such a model is capable of reproducing the main styl-ized facts for asset returns. Moreover, analytic investiga-tions revealed that, in a significant portion of the spaceof model parameters, the system is “glassy” and is there-fore expected to exhibit a large number of meta-stablestates. The authors argue that it is above all the in-terplay between dynamics within meta-stable states and occasional transitions between them — whether sponta-neous or induced by external stimuli — which accountsfor the phenomenon of volatility clustering.The purpose of the present paper is to provide a morethorough analysis of the iGBM proposed in [1]. Specifi-cally, we perform a generating functional analysis of themodel in the limit of large system size. We introduce aslow driving of the dynamics to mimic the effect of slowlyvarying macro-economic conditions, and investigate sta-tistical properties of asset returns by recourse to a sepa-ration of time-scales argument, assuming that the systemequilibrates at given values of the slow variable describ-ing macro-economic conditions. This analysis allows oneto compute distributions of asset returns on various time-scales, and it also exposes interesting collective effects onthe pricing of assets, which are driven by a combinationof the macro-economic driving and the effects of imita-tion as encoded in the couplings. The remainder of this paper is organized as follows.In Sec. II we introduce the model. Sec. III providesa solution based on a Generating Functional Analysis(GFA), with technical details of that analysis relegatedto an appendix. Phase diagrams are provided in Sec IValong with results of return distributions predicted bythe model at various time-scales. By looking at a variantof the model which has meta-stable states of a knownstructure embedded in its couplings, we are in a positionto elucidate in some detail the relation between meta-stable states with a dynamics switching between them atlonger time scales on the one hand side, and volatilityclustering on the other hand side. Finally, in Sec V weprovide a summary and a concluding discussion.
II. MODEL DEFINITIONS
In this section we describe the model for asset pricedynamics as introduced in [1]. One considers a systemconsisting of N assets, labelled i = 1 , . . . , N . To eachasset i , one associates a time dependent price S i ( t ) > S i ( t ) dd t S i ( t ) = µ i + σ i ξ i ( t ) , (1)where ξ i ( t ) ∈ R denotes a Gaussian white noise with zeromean and unit variance. The factor σ i ≥ µ i ≥ u i ( t ) = log[ S i ( t ) /S i ], in which S i is a reference price(needed to non-dimensionalize the argument of the loga-rithm) we obtain a new stochastic differential equationdd t u i ( t ) = I i + σ i ξ i ( t ) , (2)where I i = µ i − σ i / u i ( t ), isnow constructed by introducing three extra terms intoEq. (2), to give usdd t u i ( t ) = − κ i u i ( t ) + N X j =1 J ij g ( u j ( t )) + σ u ( t )+ I i + σ i ξ i ( t ) . (3)The first additional term describes what might bethought of as an effect of fundamentalist traders in themarket, creating a mean reversion effect with reversioncoefficients κ i >
0. The natural interpretation of thenormalizing factors S i introduced above would in thatcase be that of ‘rational prices’ of traded assets. Thesecond additional term in Eq. (3) describes the interac-tions between log-prices of mutually dependent assets.We choose the interaction to be most sensitive in thevicinity of the rational prices, by taking g ( u ) to be a non-linear, sigmoid function, describing the feedback mecha-nism. Possible choices for this function include, the errorfunction or hyperbolic-tangent. The strength of this in-fluence is given by J ij ∈ R . The sign of J ij depends onthe nature of mutual interactions. If, for example, assets i and j refer to firms with mutually beneficial economicrelations, one would have J ij >
0. Conversely if theyrefer to two competing firms, a negative shock on asset j , i.e., g ( u j ) <
0, may positively affect asset i , implying J ij <
0. Finally, the u term is introduced to act asa global risk component mimicking slowly evolving eco-nomic conditions affecting prices of all traded assets. Inthe present paper we will model the u term as a (slow)Ornstein-Uhlenbeck process,˙ u ( t ) = − γu ( t ) + p γ ξ ( t ) , (4)where we take γ ≪ u i ( t ) playing therole of a post-synaptic potentials, with g ( u ) describingthe neuronal input-output relation, the κ i representingtrans-membrane conductances and the J ij the synapticefficacies. The I i finally represent external (sensory) in-puts, and the u term — not typically included in theoriginal neural modelling [18] — could describe the ef-fects of neuro-modulators.A lot is know about systems of this type [1, 18–25].For the purpose of the present paper, the most impor-tant feature is that iGBM type models as described byEqs. (3) and (4) are — in a large part of their parameterspace — expected to exhibit glassy phases [1, 20, 23, 25]characterized by the existence of a very large numberof long-lived meta-stable states [21, 22]. The hypothe-sis investigated in [1] was that it would be the interplaybetween dynamics within meta-stable states and the dy-namics of (occasional) transitions between them , whichcould be held responsible for the intermittent dynamicsof financial markets.For the purposes of the present analytic study we willkeep a synthetic stochastic setting by taking the J ij tobe of the form J ij = c ij ˜ J ij , (5)where the c ij ∈ { , } are connectivity coefficients de-scribing whether or not an interaction between the pricesof assets i and j exists, and the ˜ J ij ∈ R describe thestrengths of the interactions. We assume that C = ( c ij )is the adjacency matrix of an Erd˝os-R´enyi random graphof mean degree c , but will specialize to the regime of sparse yet large connectivity by taking the limits N → ∞ and c → ∞ , with c/N → J ij are taken to be quenched random quantitieswith mean and variance scaling with the mean connec-tivity c to ensure the existence of the large system, i.e.we put ˜ J ij = J c + J √ c x ij , (6)in which the x ij are zero mean and unit variance ran-dom variables chosen to be independent in pairs with x ij x ji = α . The parameter α ∈ [ − ,
1] thus describesthe degree of correlations between ˜ J ij and ˜ J ji , with fullysymmetric interactions given by α = 1. It turns outthat the collective properties of such a dilute system inthe large mean connectivity limit are actually indistin-guishable from those of a fully connected system. III. MODEL SOLUTION
In this section we investigate the dynamics and sta-tionary states for the model introduced in Sec. II. Ananalysis of the collective properties of the system in thenoiseless limit σ = σ i = 0 was presented in [1], and usedto identify parameter ranges, viz. the regions of small κ i and sufficiently large J , where the system would exhibit alarge number of meta-stable states. We will demonstratein Sec. IV D below that our microscopic model describedby Eq. (3) does indeed produce intermittent dynamics inthis parameter range.An exact and formal treatment of the dynamics is pos-sible using a generating functional analysis (GFA) [26–29], to which we now turn. For systems of the type con-sidered here the analysis closely follows [28]. A. Generating Functional Analysis
In what follows we present a solution of the modelbased on the generating functional formalism, which pro-vides tools for the evaluation of correlation and responsefunctions in terms of a characteristic functional of pathprobabilities. Performing the average over bond disor-der in the sum over dynamical trajectories, details ofwhich are found in Appendix A, one obtains a familyof continuous-time effective single site processes, eachparameterized by a specific combination of single nodeparameters ϑ ≡ ( I, κ, σ ),˙ u ϑ ( t ) = − κ u ϑ ( t ) + I + J m ( t ) + σ u ( t )+ α J Z t d s G ( t, s ) n ϑ ( s ) + φ ( t ) (7)where n ϑ ( s ) = g ( u ϑ ( s )). The noise φ ( t ) in Eq. (7) iscoloured Gaussian noise, with h φ ( t ) i = 0 , (8) h φ ( t ) φ ( s ) i = σ δ ( t − s ) + J q ( t, s ) . (9)The order parameters m ( t ) and q ( t, s ) appearing in theequation of motion (7) and in the specification (8), (9) ofthe noise statistics must be determined self-consistentlyto satisfy m ( t ) = hh n ϑ ( t ) ii ϑ , (10) q ( t, s ) = hh n ϑ ( t ) n ϑ ( s ) ii ϑ , (11) G ( t, s ) = (cid:28) δ h n ϑ ( t ) i δφ ( s ) (cid:29) ϑ . (12)Here, the inner average h . . . i refers to an average overcoloured-noise φ for a given member of the single-site en-semble. The outer average h . . . i ϑ refers to an averageover the ensemble as characterised by the ϑ distribution.Further details for this calculation are provided in Ap-pendix A. It should also be noted that this formalism isexact in the N → ∞ limit.We highlight the following: (i) there is a dependenceof the single site dynamics on the overall ‘magnetiza-tion’ m ( t ); (ii) for any degree of symmetry of the in-teractions, α = 0, the effective single node dynamicsis non-Markovian, with memory given by the responsefunction G ( t, s ); (iii) the noise appearing in the singlesite dynamics is coloured, with correlations determinedby the average temporal correlation q ( t, s ) of single nodesas described by Eq. (9).We have thus reduced our system of of equations de-scribing the dynamics of prices of N interacting assetsto an ensemble of dynamical evolution equations self-consistently coupled via a set of order parameters, whichbecomes exact in the thermodynamic limit. As is usuallythe case with the GFA, the resulting effective equation ofmotion is highly non trivial and usually relies on sensibleassumptions to be analysed further. B. Separation of Time Scales — Quasi-StationaryRegime
For sufficiently small values of γ in Eq. (4) one expectsa separation of dynamical time scales to occur, entailingthat the fast u ϑ ( t ) processes become statistically station-ary on timescales on which the slow u ( t ) process can betreated as non-varying. In what follows we shall thusanalyse the u ϑ ( t ) dynamics under the assumption that itis stationary at a given value u of the slow process.To assist our analysis further, we approximate Eq. (7)by neglecting fluctuations in the memory term, rewritingit as ˙ u ϑ ( t ) = − κ u ϑ ( t ) + I + J m ( t ) + σ u ( t )+ α J Z t d s G ( t, s ) h n ϑ ( s ) i + φ ( t ) (13)in which, averages over the effective single process dy-namics at given ϑ appear in the retarded interaction.We are thereby discarding one source of noise in the dy-namics, and so are likely to overestimate values of macro-scopic order parameters. The important qualitative as- pects of the collective properties of the system are, how-ever, expected to remain intact as we shall verify throughsimulations later on.Assuming stationarity and time translational invari-ance for a given u , we introduce the integrated response χ = Z t d s G ( t, s ) , (14)and assume it to remain finite. This allows us to rewritethe effective dynamics in the stationary regime as˙ u ϑ ( t ) = − κ u ϑ ( t ) + I + J m + σ u + α J χm ϑ + φ ( t ) (15)where m ϑ ( s ) = h n ϑ ( s ) i can be regarded as independentof s for s sufficiently close to t for the response functionto be non-negligible.Anticipating that the correlation q ( t, s ) might developa time-persistent value q , q ( t, s ) → q , as | t − s | → ∞ , (16)we decompose the coloured noise φ into a static (frozen)and an independent time-varying component φ ( t ) = J √ qz + η ( t ) , (17)in which z ∼ N (0 , h η ( t ) i = 0 , h η ( t ) η ( s ) i = σ δ ( t − s ) + J C ( t, s ) , (18)with C ( t, s ) = q ( t, s ) − q → , as | t − s | → ∞ . (19)The effective single-process dynamics within the sta-tionary regime can then be rewritten in a more suggestiveform as ˙ u ϑ ( t ) = − κ ( u ϑ ( t ) − u ϑ ) + η ( t ) , (20)in which we have introduced the (long-term) average u ϑ = 1 κ (cid:2) I + J m + J √ qz + α J χ m ϑ + σ u (cid:3) . (21)We note that m ϑ = h g ( u ϑ ( t )) i , where the average is overthe stationary u ϑ distribution that has u ϑ as its long-term mean; thus Eq. (21) is a self-consistency equationfor the value of this long-term mean. Note also that ϑ now includes z , i.e., ϑ = ( I, κ, σ, z ). The solution toEq. (20) is now easily written down as u ϑ ( t ) = u ϑ + (cid:16) u ϑ (0) − u ϑ (cid:17) e − κt + Z t e − κ ( t − s ) η ( s ) d s , (22)implying that u ϑ ( t ) is a Gaussian process with expecta-tion h u ϑ ( t ) i = u ϑ + (cid:16) u ϑ (0) − u ϑ (cid:17) e − κt . (23)For the auto-covariance C u ϑ ( t, t ′ ) = D(cid:0) u ϑ ( t ) − h u ϑ ( t ) (cid:1)(cid:0) u ϑ ( t ′ ) h u ϑ ( t ′ ) (cid:1) i E of the u ϑ ( t ) in the large time limit we get a stationary lawdepending only on time differences, C u ϑ ( t, t ′ ) = C u ϑ ( t − t ′ )with C u ϑ ( t − t ′ ) = 12 κ (cid:16) σ e − κ | t − t ′ | + J ˆ C (0) (cid:17) , (24)in which ˆ C (0) is the zero-frequency limit of the Fouriertransform ˆ C ( ω ) = R ∞−∞ d s e − i ω s C ( s ). It is useful tospecifically record the equal time limit of C u ϑ , C u ϑ (0) = 12 κ (cid:16) σ + J ˆ C (0) (cid:17) ≡ σ u ϑ . (25) C. Self-Consistency Equations for theQuasi-Stationary Regime
With full knowledge of the statistics of the u ϑ ( t ) we canreformulate the self-consistent equations for the order pa-rameters describing the stationary regime. They are (i)the stationary magnetization m , (ii) the time persistentpart q of the node auto-correlations, (iii) the integratedresponse χ , and (iv) the zero-frequency limit ˆ C (0) of theFourier transform of the (non time-persistent) part C ( τ )of the node auto-correlations in the stationary regime. Tocompute the latter, we also have to evaluate the averagenode auto-correlations q ( τ ).To formulate the self-consistency equation for m = h m ϑ i ϑ , we recall that m ϑ = h g ( u ϑ ( t )) i where the averageis over the stationary u ϑ distribution, and hence can berewritten as m ϑ = h g ( u ϑ + σ u ϑ x ) i x (26)with σ u ϑ defined in Eq. (25), and h . . . i x denoting anaverage over a N (0 ,
1) Gaussian x . By definition, anaverage over the distribution of the set of parameters ϑ then gives m = h m ϑ i ϑ . Following the same logic for thetwo-point function q ( τ ), we obtain the following full setof self-consistency equations for the order parameters m = DD g ( u ϑ + σ u ϑ x ) E x E ϑ , (27) q ( τ ) = DD g (cid:0) u ϑ + σ u ϑ x (cid:1) g (cid:0) u ϑ + σ u ϑ y (cid:1) E x y E ϑ , (28) χ = DD g ′ ( u ϑ + σ u ϑ x ) E x E ϑ , (29)ˆ C (0) = Z + ∞−∞ d τ [ q ( τ ) − q ] , (30)In Eq. (28) the average h . . . i x y is over correlated normalrandom variables x, y ∼ N (0 ,
1) with correlation coeffi-cient given by ρ u ϑ ( τ ) = C u ϑ ( τ ) C u ϑ (0) = σ e − κ | τ | + J ˆ C (0) σ + J ˆ C (0) . (31) The u -dependent order parameters of our system arenow given by the solution of Eqs. (27)-(31) supplementedby the self-consistency equation (21) defining the u ϑ . Ananalytical characterization of the fixed-points is not read-ily available and we have to resort to numerical analysis. D. Analysis of the Self-Consistency Equations
In this section, we present our analysis of the fixedpoint equations describing the stationary dynamics ofthe system. In particular, we will be taking the errorfunction, g ( x ) = erf( x ) = 2 √ π Z x d y e − y (32)as the sigmoid function that governs the non-linear feed-back in the dynamics. This choice of feedback functionhas the advantage that it simplifies some of the Gaus-sian averages needed in the evaluation of Eqs. (27)-(30)To fully exploit this feature, we further assume that I ∼ N ( I , σ I ), so that one can combine the two Gaus-sians z and I in Eq. (21) into one. Likewise, we keep σ constant across the ensemble of effective single nodeproblems. These choices allow for some simplificationsin evaluating the averages appearing in the original fixedpoint equations. E.g., evaluating m ϑ gives m ϑ = h erf( u ϑ + σ u ϑ x ) i x = erf u ϑ p σ u ϑ ! , (33)with now u ϑ = 1 κ h J m + I + q σ I + J q z + αJ χm ϑ + σ u i . (34)The same simplifications can be made for the other orderparameters, allowing us to rewrite the set of fixed pointequations as m = * erf u ϑ p σ u ϑ ! + ϑ , (35) q ( τ ) = ** erf (cid:16) u ϑ + σ u ϑ x (cid:17) × erf u ϑ + ρ u ϑ ( τ ) σ u ϑ x p − ρ u ϑ ( τ )) σ u ϑ ! + x + ϑ , (36) χ = 1 p σ I + J q * z erf u ϑ p σ u ϑ !! + ϑ , (37)ˆ C (0) = Z + ∞−∞ d τ [ q ( τ ) − q ] , (38)where, given our current system specifications, the aver-age h . . . i ϑ now corresponds to an average over the Gaus-sian z and the κ distribution.To further accelerate the numerics we follow [25] andavoid solving the self-consistency problem Eq. (34) for u ϑ for every member of the ϑ ensemble, by using monotonic-ity of the self-consistent solution u ϑ = u ϑ ( z ) of Eq. (34)to replace the z average by u ϑ integrations instead. To doso we require the Jacobian of the transformation, whichfrom the z -derivative of Eq. (34), one obtains asd z d u ϑ = 1 p σ I + J q κ − αJ χ exp (cid:16) − u ϑ σ uϑ (cid:17)p π (1 + 2 σ u ϑ ) . (39)Following the reasoning in [25] we realise that for largevalues of αJ χ/κ the u ϑ distribution will have a gap cor-responding to a jump in the self consistent solution of m ϑ . The critical condition for a jump in the distributionis given by 2 αJ χκ p π (1 + 2 σ u ϑ ) = 1 (40)with u ϑ ( z ) jumping from the negative to the positive so-lutions of u ϑ = 1 κ αJ χ erf u ϑ p σ u ϑ ! . (41)This concludes the analysis of the general theoreticalframework. We now turn to results. IV. RESULTS
In order to structure our presentation of results, itis useful to recall that — in the absence of symmetrybreaking fields, i.e. for I i + σ u ≡ Z symmetry u i ↔ − u i .Due to the presence of interactions, this symmetry canbe spontaneously broken, giving rise to ferro-magneticor spin-glass like phases [20, 23] at sufficiently low noiselevels (and for sufficiently small values of the κ i ). If cou-plings are symmetric and if their ferro-magnetic bias issufficiently small, the system may in fact exhibit expo-nentially (in system size) many meta-stable states in thezero noise limit [21, 22]. Recent work [30] has in factdemonstrated that a large number of stationary states ofthe noiseless dynamics continues to exist in a broad classof non-linearly interacting systems when constraints suchas symmetries of interactions are dropped.In the absence of symmetry breaking fields, phaseswith spontaneously broken symmetries are usually sepa-rated by sharp phase-boundaries from phases where thesesymmetries remain unbroken. In the context of modellingthe evolution of interacting prices, however, a situationwithout any symmetry breaking fields in the evolutionequations (3) would have to be regarded as highly atyp-ical . Transitions, if any, between phases of broken andunbroken symmetries would therefore typically appear to be rounded if described in terms of the order parameters m , q and χ appearing in the theory. It would there-fore not make too much sense to precisely locate phase-boundaries which wouldn’t exist as sharp boundaries forvirtually any realistic parameter setting. In such a situ-ation the primary interest would be to locate regions inparameter space where we expect the existence of ferro-magnetic or spin-glass like phases. We will endeavourto do this in Sect. IV A, taking properties of the phasestructure that exists in the absence of symmetry breakingfields as a guidance.Having identified interesting regions in parameterspace, we will in Sect. IV B analyse distributions of log-returns for representative parameter values within theseregions of interest, evaluating them for various time scalesdefined relative to the time scale γ − of the slow u pro-cess that mimics the effect of macro-economic conditions.In Sect. IV C we explore the phenomenon of collectivepricing by investigating the distribution of equilibrium(log-)prices and in particular the effect that interactionsbetween prices have on that distribution. In Sect. IV D,finally, we attempt to underpin our hypothesis concern-ing the relation between the existence of many long-livedstates in a system and the phenomenon of volatility clus-tering by setting up and simulating a system for whichwe know - at least partially - the structure of some of itsmeta-stable states. A. Phase Structure
Here, we briefly discuss the phase structure of themodel, with an eye mainly towards identifying regions inparameter space were we would expect phases with glassyproperties characterized by a large number of meta-stablestates. The authors of [1] went some way in that direc-tion by analysing macroscopic properties of attractors inthe noiseless ( σ i ≡
0) limit of the dynamics. In particularit was shown that the mean reversion constants κ i , takento be homogeneous across the system in [1], would playa role analogous to temperature.Continuing on the assumption of Gaussian I i made inSect. III D, and assuming that the σ i are homogeneousacross the network, σ i ≡ σ , we have 7 parameters charac-terizing the system, viz. J and J determining the meanand variance, and the parameter α quantifying the degreeof asymmetry of the couplings, as well as the mean I andvariance σ I of the distribution of the I i , the strength σ of the noise in the dynamics, and κ , the mean of the κ distributions that we will consider in this paper. Un-less stated otherwise, results presented in the figures be-low, were in fact obtained by choosing an exponential κ -distribution for the mean reversion constants κ i .There can be no question of exploring this 7-dimensional parameter space completely. Fortunately wefind that collective properties of the system are in the in-teresting region of parameter space fairly robust againstparameter changes, so we will restrict ourselves to high-lighting a few of the most important trends. FIG. 1. Magnetization as a function of u , shown for three dif-ferent values κ of the mean of the kappa distribution. Here,we take J = J = 0 .
5, with α = 0 .
5, wile I = 0, σ I = 0 . σ = 0 .
1. From top to bottom the curves correspond to κ = 0 . , .
7, and 1.2, respectively.
In Fig. 1 we show the behaviour of the stationarymacroscopic magnetization m as a function of the value u of the slow process, using an unbiased I i distributionwith I = 0, and σ I = 0 .
1. Note that parameters char-acterizing the distribution of couplings and the strength σ of the dynamic noise are chosen such that there is nospontaneous ‘ferro-magnetic’ order in the u → κ of the κ dis-tribution has an effect analogous to increasing the tem-perature, in that it reduces the degree of macroscopic(ferromagnetic) order.Fig. 2 illustrates that in the absence of global symme-try breaking fields, the system exhibits a sharp second-order phase transition to ferro-magnetic order as thevalue of the ferro-magnetic bias J in the coupling distri-butions is increased above a critical value J c . For valuesof the other parameters as given, the system is in a frozen‘spin-glass’ like phase for J < J c ≃ .
75. Transitions toferro-magnetic order could also be induced by reducingthe noise level σ at sufficiently large ratios of J /J andfor sufficiently low κ . In a similar vein transitions intothe spin-glass like phase could be induced by lowering thenoise level at sufficiently small J /J -ratio, again provided κ is sufficiently small. Alternatively one could chose tolower κ at sufficiently small value of σ to induce thesetransitions.Fig. 3 shows the phase boundary separating a spin-glass like phase at small values of J from a ferro-magnetic phase at larger values of J as a function of κ in the absence of global symmetry breaking fields. Forsuch a phase boundary to exist the noise level σ has, ofcourse, to be sufficiently low. It is expected that a spin-glass like phase will continue to exist even in the presenceof weak symmetry breaking fields, in analogy to what is FIG. 2. (Color online) Magnetisation m (blue full line), time-persistent correlation q (red dashed line) and integrated re-sponse χ (black dotted line) as functions of J in the absenceof any global symmetry breaking fields, i.e. for I = u = 0.Other parameters are σ I = 0 .
1, so there is a local randomfield, J = 0 . α = 0 . κ = 0 .
2, and σ = 0 .
1. The figureshows the appearance of a ferro-magnetic phase as J is in-creased beyond J c ≃ .
75. For J < J c the system is in afrozen ‘spin-glass’ like phase.FIG. 3. Phase boundary separating a spin-glass like phase atsmall values of J from a ferro-magnetic phase at larger valuesof J as a function of κ in the absence of global symmetrybreaking fields, i.e. for I = u = 0, but σ I = 0 .
1. Otherparameters are J = 0 . α = 0 .
5, and σ = 0 . known for the SK model [31]. B. Return Distributions
We now look to compute distributions of log-returnsacross the ensemble of interacting assets. (In what fol-lows, we will, somewhat loosely refer to them as returndistributions). To begin with, we consider the distri-bution of returns for an arbitrary member of the ϑ -ensemble, and so we need to consider the statistics ofdifferences, ∆ u ϑ ≡ u ϑ ( t ) − u ϑ ( t ′ ) , (42)omitting time-arguments on the l.h.s. for simplicity.We will always consider late times such that γt ≫ γt ′ ≫ γ | t − t ′ | ≪ γ | t − t ′ | = O (1). Explicitly, this involves looking at return distribu-tions for stationary fast processes paramaterized by u ( t )and u ( t ′ ), for which correlations between the two slowprocesses still exist.(iii) The long time scale, which we define as γ | t − t ′ | ≫
1, so that even the slow process has decorrelated.In any case, we will be interested in variations inducedby both the ϑ distribution and for generality, the u statistics too. It could, however, also be of interest toinspect return distributions conditioned on specific val-ues of the slow process.
1. Quasi-Stationary regime
Here, we look at the distribution of returns for the fastprocess in equilibrium for a given value u of the slowprocess. This implies we look at time differences suchthat u ϑ obeys the equation of motion Eq. (15) for alltimes of interest. Naturally, we are interested in the lim-its κt, κt ′ ≫ κ | t − t ′ | ≪ κ | t − t ′ | = O (1) ,(iii) long: κ | t − t ′ | ≫ κ dis-tributions (upper and lower cutoffs) may be needed tomake the definition of these time windows and some ofthe arguments below well defined for all members of the ϑ -ensemble; regularizations/cutoffs can then be removedat the end of each calculation in question. In order notto overburden the presentation, we will, however, not ex-plicitly retrace and document these steps in what follows.Using the solution given in Eq. (22), we see that∆ u ϑ = Z t e − κ ( t − s ) η ( s ) d s − Z t ′ e − κ ( t ′ − s ′ ) η ( s ′ ) d s ′ (43)As η is a Gaussian noise, we find that the returns fora single member of the ensemble of effective single site processes in the quasi-stationary regime are normally dis-tributed, i.e.∆ u ϑ ∼ N (cid:18) , σ κ (cid:16) − e − κ | t − t ′ | (cid:17)(cid:19) . (44)For the short time scale defined above, one may expandthe exponential appearing in the variance, entailing thatthe κ dependence vanishes (at first order in the expan-sion), ∆ u ϑ ∼ N (cid:0) , σ | t − t ′ | (cid:1) . (45)At these very short time separations the return distri-bution thus exhibits simple diffusive broadening. If σ istaken to be constant across the ensemble, this result re-mains true across any portfolio of assets traded in themarket.At the long and intermediate time scales within thequasi-stationary regime there will of course be a κ -dependence of individual returns. However, if we concernourselves with the distribution p (∆ u ) of returns acrossthe ensemble of processes, we can obtain it by averagingthe above over the ϑ distribution, p (∆ u ) = Z d ϑP ( ϑ ) p (∆ u ϑ ) (46)In general, this integral has to be done numerically.A simplification is possible for very large time separa-tions within the quasi-stationary regime, for which theexponential correction in the variance in Eq. (44) can beneglected, and ∆ u ϑ ∼ N (cid:16) , σ κ (cid:17) . Keeping σ constantacross the ensemble, the only ϑ component to averageover in this limit then is κ . An analytically closed formfor the distribution of returns across the ensemble canthen be obtained if we assume the κ to be Γ- distributed, P ( κ ) = 1 κ Γ( ν ) (cid:16) κκ (cid:17) ν − exp( − κ/κ ) . (47)Here κ is a scale parameter which also defines the meanof the κ distribution, while ν determines its actual shape.For this family of κ distributions we obtain p (∆ u ) = √ κ √ πσ Γ( ν + )Γ( ν ) (cid:18) κ (∆ u ) σ (cid:19) − ( ν +1 / . (48)Within this family of return distribution we observepower law tail behaviour, p (∆ u ) ∼ (∆ u ) − µ for | ∆ u | ≫ µ = 1 + 2 ν . We note that the case ν = 1 would cor-respond to an exponential κ distribution, which wouldbe the distribution naturally selected by the maximumentropy principle for a strictly positive random variablewith a prescribed mean, in which case the tail exponentwould be µ = 3. In Fig. 4 we compare this analyticalasymptotic result with that of a full numerical evalua-tion of the distribution of returns across the ensemble for κ | t − t ′ | = 20, observing excellent agreement betweenthe two already for moderate time separations.Although we are unable to evaluate the full distri-bution of returns across the ϑ -ensemble for interme-diate time-separations, a quantity that we can evalu-ate in closed form for all time-separations in the quasi-stationary regime is its variance h (∆ u ) i = hh (∆ u ϑ ) ii ϑ ,in which the inner average is the variance of the returndistribution for a given member of the ϑ -ensemble, spec-ified in Eq. (44), and the outer average is over the ϑ -distribution. With specifications as before, i.e. keeping σ constant across the ensemble, the only ϑ componentto average over is once more the κ -distribution. For Γ-distributed κ as specified above we obtain h (∆ u ) i = σ κ ( ν − " − κ | t − t ′ | ) ν − (49)for ν = 1. In the ν → h (∆ u ) i (cid:12)(cid:12) ν =1 = σ κ log (cid:0) κ | t − t ′ | (cid:1) (50)In the limit of very short time separations, this repro-duces a diffusive broadening h (∆ u ) i ∼ σ | t − t ′ | of thevariance which is independent of properties of the κ -distribution, as observed earlier.It is worth pointing out that the return distributions inthe quasi-stationary regime are independent of the global u process, and in fact independent also of other parame-ters characterizing the interactions, as the u -dependentmeans u ϑ , which do depend on the interaction parame-ters, cancel when taking differences. -20 -10 0 10 2010 -5 -2 FIG. 4. (Colour online) Return distribution in the quasi-stationary regime evaluated for κ | t − t ′ | = 20 (red dashedline) compared with the analytic prediction for its asymp-totic behaviour Eq. (48) (blue full line) for an exponential κ distribution with ν = 1.
2. Intermediate and Long Time Scales
We now turn our attention to the case where the fastprocess is in equilibrium at two different values for u .In particular, for a single member of the ensemble evalu-ation of Eq. (42) gives∆ u ϑ = u ϑ ( t ) − u ϑ ( t ′ ) + Z t e − κ ( t − s ) η ( s ) d s − Z t ′ e − κ ( t ′ − s ′ ) η ( s ′ ) d s ′ (51)In contrast to the quasi-stationary regime, the time-dependent mean values u ϑ determined by the values of u at two different times t and t ′ now explicitly appearin the returns. We also expect that on this timescale, thefast noise processes have decorrelated with one another.Therefore, the distribution of returns for a given memberof the ensemble and for given values of u ( t ) and u ( t ′ )now normal with non-zero mean and given by∆ u ϑ (cid:12)(cid:12) u ( t ) ,u ( t ′ ) ∼ N (cid:0) ∆ u ϑ ( t, t ′ ) , σ u ϑ ( t ) + σ u ϑ ( t ′ ) (cid:1) , (52)where∆ u ϑ ( t, t ′ ) = u ϑ ( t ) − u ϑ ( t ′ )= 1 κ h J ( m t − m t ′ ) + σ ( u ( t ) − u ( t ′ ))+ αJ ( χ t m ϑ,t − χ t ′ m ϑ,t ′ ) i . (53)Here we denote an order parameter A of interest by A t to denote its value in equilibrium for a given value u ( t ) of the slow process u at time t . As before, weare interested in the return distribution across the wholeensemble, which is obtained by averaging over the ϑ -distribution. In addition to this, we can either look atreturn distributions for a range of specific u values, or wemay choose to average over their distribution. Since the u term mimics the state of global economic behaviour,this average corresponds to return distributions across allmarket conditions. As the u statistics are Gaussian, thejoint distribution becomes easy to write down allowingus to perform this average in a straightforward manner.Finally, the return distribution across the ensemble ofprocesses, across all market conditions is written downas p (∆ u ) = Z d ϑ d u ( t ) d u ( t ′ ) P ( ϑ ) p ( u ( t ) , u ( t ′ )) × p (∆ u ϑ | u ( t ) , u ( t ′ )) (54)Differences between the intermediate and long timescales arise in through differences in the the joint dis-tribution p ( u ( t ) , u ( t ′ )) for the slow process. In the firstcase, the market conditions are still correlated whilst inthe long time limit these correlations no longer persist.In both cases, we find that returns across the portfoliomaintain their power-law distributed tails.0 -10 -5 0 5 1010 -3 -2 -1 FIG. 5. Distributions for long time log returns across the en-semble averaged over all market conditions. Again, power lawtails are observed and we expect that upon suitable normal-ization the distributions across timescales should scale verywell as is seen in microscopic simulations.
C. Collective Pricing
In this section we explore the phenomenon of collectivepricing mentioned at the beginning of this section. Morespecifically, we take a closer look at the role of u ϑ asdefined in Eq. (21); we know that this quantity takes therole of the equilibrium value of the associated asset priceunder given macro-economic conditions as parameterizedby the value of the slow u process.In order to identify the collective interaction-mediatedproperties of pricing distributions, we begin by lookingat the non-interacting baseline. Without interactions inthe system, the combined effect of mean reversion, drift,volatility and the value u of the slow process describingmacro-economic conditions is, according to Eq. (21), toproduce a mean log-price u ϑ = 1 κ (cid:2) I + σ u (cid:3) (55)that depends linearly on I and on the value of the u -process. Recall that I i = µ i − σ i , so I includes effectsof drift and volatility.Assuming a normal distribution for I as above, I ∼N ( I , σ I ) we get a normally distributed family of meanprices at given mean reversion, u κ ∼ N (cid:16) I + σ u κ , σ I κ (cid:17) , (56)which, upon averaging over κ which are Γ-distributedwith ν > − p ( u ) = νκ p π σ I exp n − σ I (cid:16) I + σ u (cid:17) o × β − (1+ ν ) / exp n γ β o D − (1+ ν ) (cid:16) γ √ β (cid:17) , (57) in which D ν ( z ) is a parabolic cylinder function [32], and β = (cid:16) κ uσ I (cid:17) , γ = 1 − κ uσ I ( I + σ u ) . (58)Note that γ /β → const . as | u | → ±∞ , so the tail-behaviour of the u distribution is governed by the β − (1+ ν ) / term in the above expression, giving p ( u ) ∼ u − (1+ ν ) for | u | ≫
1. Conversely, the singularities whichthe three terms in the second line of Eq. (57) exhibit as u → β →
0) cancel, so that p ( u ) remains finitein this limit.Having analyzed the non-interacting case, we now re-turn to Eq. (21), and more specifically to its version fornormally distributed I , Eq. (34), to study the effects ofinteractions on the u ϑ . As indicated at the end of SecIII D the distribution of the solution u ϑ of Eq. (34) isobtained by transforming the normal density of zp ( u ϑ ) = P ( z ) (cid:12)(cid:12)(cid:12)(cid:12) d z d u ϑ (cid:12)(cid:12)(cid:12)(cid:12) (59)in which P ( z ) = √ π e − z / , with z = z ( u ϑ ) obtained bysolving Eq. (34) for z , and the Jacobian d z d u ϑ of the trans-formation — for the error-function feedback (32) — givenby Eq. (39). This allows us to obtain the distribution ofequilibrium prices as induced by normal variable z for afixed κ . The distribution p ( u ) of equilibrium log-pricesover the ensemble is obtained by averaging over the ϑ distribution as before; the average once more reduces toan average over the κ distribution, if σ is kept constantacross the ensemble.We see in Fig. 6 that interactions lead to a consider-able broadening for the equilibrium distributions whencompared with their non-interacting counterparts. Alsothe degree of asymmetry of these distributions is sig-nificantly enhanced by the interactions. A highly non-trivial effect is the systematic suppression of equilibriumlog-prices which would be characterized as typical in thenon-interacting system. This effect is primarily inducedby the ferro-magnetic bias of the interaction, which couldbe induced by herding or imitation effects of agents actingin the market, or by economic fundamentals suggestingco-movement of asset prices. It can fairly be said thatthis mechanism creates an interaction mediated mecha-nism of a market to push prices of assets to more extreme,i.e. both to very high and to very low values. The effectappears to be stronger for members of the ensemble withsmall values of the mean reversion constant κ ; it weakensfor those with a larger mean reversion constant.We may also look at the global characteristics of pric-ing distribution across the entire ensemble. This isachieved by averaging over the κ -distribution. For theparameter settings used, one can see in Fig. 7 that thissmoothes out the bimodal nature observed for the sub-ensembles of assets with selected κ values shown in Fig. 6,but it shows once more a considerable broadening of thedistribution and a significant enhancement of the degree1 -5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.8 -10 -5 0 5 1000.10.20.3 FIG. 6. (Colour online) Distribution of equilibrium log-prices u ϑ for the non-interacting system (first panel), compared tothose of the corresponding interacting system (second panel)for selected values of the mean reversion κ and the slow pro-cess u ; note the different scales. System parameters are I = 0, σ I = 0 . κ = 0 . σ = 0 .
1; for the interactingsystem we chose J = J = 0 . α = 0 .
5. For the indi-vidual curves the parameters are ( κ = 0 . , u = 0 .
1) (bluenarrow pair distributions), ( κ = 0 . , u = 0 .
1) (black broadpair distributions), and ( κ = 0 . , u = 1) (red broad pairof distributions). The degree of asymmetry of the broaderdistributions at smaller values of κ increases with increasing u . of asymmetry when compared with the non-interactingcounterpart. D. Meta-Stable States and Volatility Clustering
We finally return to one of the central hypotheses un-derlying our modelling, namely that the complexity ofreal market dynamics, including in particular the phe-nomenon of volatility clustering could be rationalized interms of the interplay of dynamics within (meta-stable)market states and the dynamics of occasional transitions -10 -5 0 5 1000.050.10.150.20.25
FIG. 7. Equilibrium distribution of log-prices across the entireensemble for u = 1, both for the interacting system (full line)and for the non-interacting system. Parameters are the sameas in Fig 6. between them. In this respect we note recent work [33]which identified a set of distinct market states in histor-ical data.The assumption behind our modelling is that marketstates would indeed emerge naturally as attractors of thecollective (non-linear) dynamics of interacting prices. Forthe Gaussian couplings that we have been using in thepresent study, analogies with the SK spin-glass modelsuggest that we do in fact expect a very large numberof such attractors to exist in a large region of parameterspace. However, we have no way of a-priori knowing theirstructure, and thus no immediate means of testing ourhypothesis quantitatively.In order to make progress in elucidating this issue wepropose to look at a version of the market in which we em-bed a small number of known random attractors in thesystem, in order to analyze whether there is a relationbetween system state — measured in terms of similaritywith these known attractors — and the observed volatil-ity of the dynamics. For simplicity we take the systemto be fully connected, and introduce couplings with aGaussian and a Hebbian coupling component as follows, J ij = J (G) ij + J (H) ij , (60)with J (G) ij = J N + J √ N x ij (61)and J (H) ij = 1 N p X µ =1 ξ µi ξ µj , (62)in which the ξ µi are i.i.d. random variable taking values ξ µi = ± x ij are normally2 FIG. 8. Simulation of a market with of N = 50 tradedassets, exhibiting the relation between volatility and meta-stable state structure. The upper panel shows overlaps of thesystem state with three random attractors embedded in thecoupling matrix in a Hebbian form as explained in the maintext, while the lower panel shows returns on the index as afunction of time. The other system parameters are κ = 0 . I = 0, σ I = 0 . σ = 0 . J = J = 0 . α = 0 .
5, and γ = 10 − . distributed x ij ∼ N (0 ,
1) and independent in pairs with x ij x ji = α as in the original set-up.Figure 8 presents results of a simulation of such a sys-tem of size N = 50, with p = 3 ‘patterns’ embeddedin the couplings, in which we simultaneously record thechanges of the index, and the values of the overlaps m µ ( t ) = 1 N X i ξ µi g ( u it ) (63)with the three random patterns { ξ µi } , for µ = 1 , V. SUMMARY AND DISCUSSION
In this paper, we have provided a comprehensive anal-ysis of the iGBM introduced in [1]. The line of reasoningleading to a model of interacting prices of this type isdescribed in detail in that paper. Suffice it to mentionhere that the structure of the model follows from verygeneric arguments concerning the description of marketmechanisms and of agents acting in a market within re-duced models based on the evolution of prices alone. Inthe present investigation we couple the dynamics of the system to a slow Ornstein-Uhlenbeck process, which weintroduce to mimic the effect of slowly evolving macro-economic conditionsWe have performed a generating functional analysisof the dynamics, which maps the dynamics of the inter-acting system onto an ensemble of systems exhibiting anon-Markovian dynamics which is self-consistently cou-pled through a set of dynamic order parameters. Using aseparation of time-scales argument, which assumes thatthe fast internal dynamics of the interacting system equi-librates at given values of the slow Ornstein-Uhlenbeckprocess, we are able to analyse the stationary dynamicsof the system. This then allows to identify regions in pa-rameter space where the system exhibits ferro-magneticor spin-glass like phases.Our analysis of the stationary dynamics (at given val-ues of the slow driving) allows us to evaluate the distri-bution of log-returns for the ensemble for various time-scales, both in the quasi-stationary regime and at largertime separations. For a broad class of distributions of themean reversion terms in the model, we find that distri-butions of log-returns across the ensemble are fat-tailed,exhibiting asymptotic power law behaviour broadly inline with empirical facts [6]. We note, however, that ourmodel, as it is currently set up, does not reproduce thesefat tails at the single asset level that were found empiri-cally in [7]. We will discuss the origin of that shortcom-ing, and thus possible ways to improve the model in thisrespect below. We are also able to evaluate the time de-pendent variance of the distribution of log-returns in thequasi-stationary regime, and find diffusive broadening inthe limit of small time separations, with the broadeningbecoming sub-diffusive at later times. These findings arebroadly in line with empirical observations.Interestingly, our model predicts the existence of equi-librium prices for assets, and we are able to explic-itly trace the influence of interactions on the distribu-tion of equilibrium prices across the ensemble. Thetwo main effects of collective pricing, as predicted bythe iGBM are to considerably broaden the distribu-tion of equilibrium prices in comparison with their non-interacting counterparts, as well as a significant enhance-ment of asymmetries characterizing such distributionsfor given (favourable or unfavourable) economic condi-tions as quantified by the value u of the slow noise pro-cess. More specifically, we also observe a pronouncedinteraction-induced preference for very high or very lowasset prices, which we think, deserves further study.Note that distributions of log-returns and pricing dis-tributions across the market are of collective origin, andso they can be expected to be to a certain extent inde-pendent of details of the model specifications. In partic-ular, collective properties of the system will not dependon specific realizations of inter-asset couplings, thoughthey may, and in general will depend of properties ofcoupling distributions . This aspect could indeed providean avenue to analysing market data within the presentmodelling framework which does not require to get indi-3vidual couplings correct. It is also the main aspect fromwhich the current modelling approach may eventually de-rive some predictive power, and might, for instance, beused to provide tools to assess market risk at a systemiclevel. It goes without saying that further investigationsusing real data will be required to get there.One of the principal motivations for constructing theiGBM was to explore whether some of the stylized factsof financial time series could be understood in terms ofeffective interactions between prices of assets traded ina market, given that effective interactions between as-set prices are a necessary feature of any model that at-tempts to describe market dynamics in a reduced formas a dynamics of prices alone. We have gone some wayto demonstrate that this is true at the level of returndistributions. Another important phenomenon is that ofvolatility clustering which in fact finds a quite naturalexplanation in terms of interacting prices. Due to the in-teractions, the system is expected to exhibit a large num-ber of (dynamic and static) attractors if there is a suffi-cient degree of disorder and frustration. In the presenceof noise, many of these attractors will survive as long-lived states, and volatility clustering is expected to arisenaturally through the interplay of the dynamics withinlong-lived states and the dynamics of occasional transi-tions between them. Such transitions can occur spon-taneously or be triggered by news or slowly changingmacro-economic conditions. Different long lived stateswill be characterised by different values of their suscep-tibilities and so the presence of noise in the dynamicsis expected to induce fluctuations with different degreesof volatility. Using simulations of a system with a par-tially known attractor structure, we have demonstratedin Sect. IV D above that our hypothesis about a rela-tion between meta-stable states and volatility clusteringis correct at least for models of the type considered here.In [1] the authors simulated the model using an ex-ternal perturbation which they argued could representthe effect of the arrival of unexpected news (e.g. inthe context quarterly reporting). The process used inthat paper is difficult to implement in analytically closedform, which was one of our reasons for adopting the slowOrnstein-Uhlenbeck process, which uniformly affects allprices in a market, as a mechanism to induce transitionsbetween meta-stable states. We believe that it is the ab-sence of a jump-process component of the noise in the ver-sion of the model investigated in the present paper whichis ultimately responsible for the fact that the model doesnot exhibit fat-tailed return distributions at the level ofsingle assets. This could easily be rectified in the modelformulation, by adding e.g. a Poisson jump process com-ponent to the noise, but it is likely to considerably com-plicate attempts at solving the model analytically. Webelieve it would be important to explore to what extenta model with a combination of continuous and discretenoise sources is amenable to analysis.Our last remark refers to the presence of mean revert-ing forces in the iGBM, given that the existence of such forces is debated in economic circles. Within our mod-elling, the existence of mean reverting forces is respon-sible for ensuring long-term stability of the market. Itwould be easier to motivate the existence such forces ifthe u i ( t ) were introduced as log-prices on a co-movingframe as u i ( t ) = log[ S i ( t ) /S i e ( µ i − σ i ) t ]. This modifica-tion would in the first instance eliminate the drift I i fromthe transformed equation (2), and it would suggest tointroduce an iGBM formally in the same manner as wasdone originally, albeit with the drift term I i missing fromthe interacting version Eq. (3) as well. Within this mod-ified interpretation of the u i ( t ), the mean reversion andthe interactions would have to be interpreted as meanreversion and interactions relative to an expected trendrather than relative to some fixed log-price, which mightbe easier to justify in economic terms. Long-term stabil-ity of the model would be saved, albeit on a co-movingframe. As an additional benefit the random symmetrybreaking field I i would also disappear from the equations,which could simplify the ensuing analysis. As a downsidethough, such a model would likely be harder to calibrateagainst real market data, which should indeed be one ofthe next natural steps to undertake within the presentproject. Acknowledgements:
J.K. is supported by theEPSRC Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems(CANES, EP/L015854/1).
Appendix A: Generating Functional Analysis
In this appendix we use Generating Functional Analy-sis (GFA) [26] to formally solve the model dynamics. Webegin by introducing the generating functional in termsof source fields, ℓ , Z [ ℓ | u ] = * exp n − i Z d t X i ℓ i ( t ) n i ( t ) o+ , (A1)in which the n i ( t ) = g ( u i ( t )) are the variables in termsof which the interaction between log-prices are defined,and we condition on a realization u of the path ofthe slow process representing the evolution of macro-economic conditions. The angled brackets refer to theaverage over all paths, which are trajectories of micro-scopic states. Explicitly, Z [ ℓ | u ] = Z D u P [ u ] exp n − i Z d t N X i =1 ℓ i ( t ) n i ( t ) o , (A2)where D u is the flat measure over a set of paths u = { u i ( t ) } , i = 1 , . . . , N over some finite risk horizon 0 ≤ t ≤ T , and P [ u ] denotes the probability of these paths.The generating functional can be used to compute expec-4tation values and correlation functions as h n i ( t ) i = i δZ [ ℓ | u ] δℓ i ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ℓ ≡ , (A3) h n j ( s ) n i ( t ) i = i δZ [ ℓ | u ] δℓ j ( s ) δℓ i ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ℓ ≡ . (A4)The evaluation of the generating functional followsstandard reasoning; see e.g. [26–29]. For stochastic pro-cesses described by a Langevin equation driven by Gaus-sian white noise, one uses δ -functionals and their Fourierrepresentations to enforce the equations of motion, whichallows to transform probabilities of noise-trajectories intopath probabilities. Assuming Ito-discretization for theLangevin-equation one can thus express the generatingfunctional as Z [ ℓ | u ] = Z D{ u , ˆ u } exp ( − Z d t X i " σ i u i ( t ) +iˆ u i ( t ) (cid:16) ˙ u i ( t ) + κ i u i ( t ) − I i − X j J ij n j ( t ) − σ u ( t ) (cid:17) − i ℓ i ( t ) n i ( t ) . (A5)We are interested in evaluating the generating functionalfor a typical realization of disorder. This is achieved byaveraging Eq. (A5) over the bond-disorder, i.e. over the c ij and x ij in terms of which the J ij are expressed. Thisdisorder average factors in pairs ( i, j ), D = Y i 1, we follow e.g. [34], expand-ing the exponential to perform the x average, keepingonly dominant terms in the expansion in terms of inversepowers of c , and then re-exponentiate to write D ≃ exp (cid:16) N h J Z d t k ( t ) m ( t )+ J Z d s d t h Q ( s, t ) q ( s, t ) + αG ( s, t ) G ( t, s ) ii(cid:17) , (A8) where we have introduced the set of one-time and two-time order parameters m ( t ) = 1 N N X i =1 n i ( t ) ,k ( t ) = 1 N N X i =1 i ˆ u i ( t ) ,q ( s, t ) = 1 N N X i =1 n i ( s ) n i ( t ) ,Q ( s, t ) = 1 N N X i =1 i ˆ u i ( s ) i ˆ u i ( t ) ,G ( t, s ) = 1 N N X i =1 i ˆ u i ( s ) n i ( t ) . One then enforces these definitions using Dirac δ -functions identities and their Fourier representations, totransform the disorder averaged generating functionalinto a functional integral, which to leading order in thesystem size N can be expressed in the following compactform, Z [ ℓ | u ] = Z D{ . . . } exp { N [Ξ + Ξ + Ξ ] } . (A9)Here, D{ . . . } represents the functional measure over theset of macroscopic order parameter functions and theirconjugates. The functionals Ξ , Ξ and Ξ , appearing inthe exponential of Eq. (A9), are defined asΞ = J Z d t k ( t ) m ( t ) + J Z d s d t (cid:16) Q ( s, t ) q ( s, t )+ αG ( s, t ) G ( t, s ) (cid:17) , (A10)Ξ = i Z d t (cid:16) m ( t ) ˆ m ( t ) + k ( t ) ˆ k ( t ) (cid:17) +i Z d s d t (cid:16) q ( s, t ) ˆ q ( s, t )+ Q ( s, t ) ˆ Q ( s, t ) + G ( t, s ) ˆ G ( t, s ) (cid:17) , (A11)Ξ = 1 N X i log Z D{ u, ˆ u } exp (cid:16) − S i − i Z d t ℓ i ( t ) n ( t ) (cid:17) (A12)Here S i denotes the effective local dynamic action of pro-cess i , S i = Z d t h − σ i u ( t )) + iˆ u ( t ) (cid:16) ˙ u ( t ) + κ i u ( t ) − I i − σ u ( t ) (cid:17) + i ˆ m ( t ) n ( t ) + iˆ k ( t )iˆ u ( t ) i +i Z d s d t h ˆ q ( s, t ) n ( s ) n ( t ) + ˆ Q ( s, t )iˆ u ( s )iˆ u ( t )+ ˆ G ( t, s ) n ( t )iˆ u ( s ) i . (A13)5It depends on i only through the locally varying param-eters ( I i , κ i , σ i ) ≡ ϑ i One now evaluates Eq. (A9) using the saddle pointtechnique, which requires the macroscopic order param-eters of interest to satisfy the following fixed point equa-tions: m ( t ) = 1 N X i h n ( t ) i ( i ) ,q ( s, t ) = 1 N X i h n ( s ) n ( t ) i ( i ) , (A14) G ( t, s ) = 1 N X i h n ( t )iˆ u ( s ) i ( i ) , t > s . All other order parameters are self-consistently zero dueto causality. In Eq. (A14), we use h . . . i ( i ) to represent anaverage over the dynamics of effective single site processes i which takes the form h . . . i ( i ) = R D{ u, ˆ u } ( . . . ) exp (cid:16) − S i (cid:17)R D{ u, ˆ u } exp (cid:16) − S i (cid:17) (A15)We note that due to causality the effective single siteaction simplifies to S i = Z d t " − σ i u ( t )) + iˆ u ( t ) (cid:16) ˙ u ( t ) + κ i u ( t ) − I i − J m ( t ) − αJ Z t d s G ( t, s ) n ( s ) − σ u ( t ) (cid:17) − J Z d s d t q ( s, t ) iˆ u ( s )iˆ u ( t ) . (A16)By the Law of Large numbers the saddle point equa-tions (A14) for the order parameters can be written asaverages over the distribution of the locally varying pa-rameters ϑ ≡ ( I, κ, σ ),1 N X i h . . . i ( i ) → hh . . . ii ϑ , as the large system limit N → ∞ is taken. Here in-ner averages correspond to those over the dynamics of asingle process with a particular parameter combination,while the outer average stands for an average over the ϑ distribution, i.e. h . . . i ϑ ≡ R d I d κ d σ p ( I, κ, σ )( . . . ).One finally notes that the appearance of a contributionin the effective single-site action (A16) which is non-localin time and quadratic in the conjugate dynamical vari-ables ˆ u ( t ) is a manifestation of the fact that the effec-tive single site processes are governed by coloured noise,while the non-local contribution involving the responsefunction G ( t, s ) expresses the effect that effective singlesite dynamics is non-Markovian. The equation of motionfor the effective single site dynamics can be inferred fromthe effective single site action (A16), giving˙ u ϑ ( t ) = − κ u ϑ ( t ) + I + J m ( t ) + σ u ( t )+ α J Z t d s G ( t, s ) n ϑ ( s ) + φ ( t ) , (A17)where we write u ( t ) = u ϑ ( t ) when referring to sin-gle site process with local parameters ϑ = ( I, κ, σ ), so n ϑ ( t ) = g ( u ϑ ( t )), and where the coloured noise φ ( t ) andthe dynamical order parameters appearing in Eq. (A17)must satisfy the self consistency equations h φ ( t ) i = 0 , (A18) h φ ( t ) φ ( s ) i = σ δ ( t − s ) + J q ( t, s ) , (A19)and m ( t ) = (cid:10) h n ϑ ( t ) i (cid:11) ϑ , (A20) q ( t, s ) = (cid:10) h n ϑ ( t ) n ϑ ( s ) i (cid:11) ϑ , (A21) G ( t, s ) = (cid:28) δ h n ϑ ( t ) i δh ( s ) (cid:29) ϑ , t > s . 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