International Journal of Theoretical and Applied Finance
Vol.3,No.4(2000)675–702
c Worl
d Scientific Publishing Company
VOLATILITY CLUSTERING IN FINANCIAL MARKETS:
A MICROSIMULATION OF INTERACTING AGENTS
THOMAS LUX
Department of Economics,University of Kiel,Olshausenstr.40,24118Kiel,Germany
E-mail:lux@bwl.uni-kiel.de
MICHELE MARCHESI
Department of Electrical and Electronic Engineering,
University of Cagliari,piazza D’Armi,09123Cagliari,Italy
E-mail:michele@diee.unica.it
Received4October1999
Thefinding of clustered volatility and ARCH effects is ubiquitous infinancial data.This
paper presents a possible explanation for this phenomenon within a multi-agent frame-
work of speculative activity.In the model,both chartist and fundamentalist strategies
are considered with agents switching between both behavioural variants according to
observed differences in pay-offs.Price changes are brought about by a market maker re-
acting to imbalances between demand and supply.Most of the time,a stable and efficient
market results.However,its usual tranquil performance is interspersed by sudden tran-
sient phases of destabilisation.An outbreak of volatility occurs if the fraction of agents
using chartist techniques surpasses a certain threshold value,but such phases are quickly
brought to an end by stabilising tendencies.Formally,this pattern can be understood
as an example of a new type of dynamic behaviour known as“on-offintermittency”
in physics literature.Statistical analysis of simulated time series shows that the main
stylised facts(unit roots in levels together with heteroscedasticity and leptokurtosis of
returns)can be found in this“artificial”market.
Keywords:Volatility clustering;interacting agents;on-offintermittency.
1.Introduction
Both foreign exchange markets and national stock markets share a number of stylised facts for which a satisfactory explanation is still lacking in standard the-ories offinancial markets.Pagan[35]provides an authoritative survey of those salient features that appear to be common characteristics of allfinancial markets together with the econometric techniques for dealing with them.As concerns for-eign exchang
e markets,his description can be supplemented by recent reviews of their empirical regularities by de Vries[39]and Guillaume et al.[20].Comparing these papers,the main difference seems to be that a remarkable number of facts presented by de Vries concern negative results,like the rejection of uncovered in-terest rate parity or purchasing power parity as well as other theoretically sensible
675
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but empirically doubtful relationships between exchange rates and other economic variables.However,his positive results are mostly striking uni-variate statistical features of the data which also play a prominent role in the other surveys and ap-pear to be extremely uniform across various assets,nations and sampling horizons.a As it appears from the empirical literature,the features highlighted below are also routinely found in the prices and returns offinancial and commodity futures as well as in prices for precious metals.
In this paper,we will concentrate on those three uni-variate properties which ap-pear to be the most important and pervasive,and will try to provide an explanation using a multi-agent model of speculative activity.Beginning with the characteris-tics of share prices and foreign exchange rates themselves(or th
eir logarithms),we encounter the following empirical regularity:
Fact1.Unit root property of asset prices and spot exchange rates(or their logs).
More formally,denoting by p t the price at time t,Fact1in its most elemen-tary form implies that p t follows an autoregressive process:p t=ρp t−1+ t with stationary increments t,and that one is usually unable to reject the hypothesis ρ=1using standard statistical procedures such as the Dickey-Fuller test.Ex-pressed somewhat differently,one is unable to reject the hypothesis thatfinancial prices follow a random walk or martingale.b While the implied non-stationarity and lack of predictability of spot rates appears to be at odds with traditional models of exchange rate determination,it squares well with the efficient market view of stock price determination and served as a starting point for models that propose a view of forex markets as arbitrage-freefinancial markets.
If levels(or logs)obey a unit root dynamics,returns or differences of logs should be stationary.In fact,this has been confirmed throughout the literature.However, certain distributional characteristics of returns also count as well-established facts which—in the words of de Vries—“have a sound statistical basis but for which no convincing economic explanation has been established”.Thefirst of these is:
Fact2.Fat tail phenomenon.
Returns at weekly,daily and higher frequencies exhibit more probability mass in the tails and in the centre of the distribution than does the standard Normal.It is perhaps also remarkable that,besides this deviation from the Gaussian,the shape of the distribution usually appears well-behaved:namely,histograms of stock price or exchange rate returns mostly show a uni-modal bell shape with,in most cases, only modest levels of skewness.In the early literature,Fact2has been identified with excessive fourth moments(leptokurtosis).Kurtosis is,however,a very limited measure of deviations from Gaussian shape.Fortunately,recent literature provides a Harrison[21]shows that18th centuryfinancial data also share most of the characteristics of today’sfinancial markets.In particular,they also exhibit leptokurtosis and volatility clustering.
b The former notion applies if increments t are independently and identically distributed,while the latter only requires that E[ t]=0.
Volatility Clustering in Financial Markets677 a sharper characterisation of the behaviour in the distribution’s extreme parts:in particular,it could be established that the decline of probability mass in the outer parts follows a power law(whereas the Normal and a number of other often-used distributions have an exponential decline).As a consequence,the distribution in the tails can be approximated by a Pareto distribution:
F(x)=1−ax−α(1.1)
Furthermore,the tail shape parameterαhas mostly been found to hover between about2and4(with lowerαvalues indicating fatter tails!)for both foreign exchange rates and stock returns.
While Fact2concerns the unconditional distribution of returns,the third item focuses on salient features of their conditional distribution.Using again the words of de Vries,it may be stated as follows:
Fact3.Volatility Clustering:periods of quiescence and turbulence tend to cluster together.
Formally,this property can be identified with what is now known as ARCH effects:non-homogeneity of volatility together with highly significant autocorre-lation in all measures of volatility despite insignificant autocorrelation in raw re-turns.There is some relationship between Facts2and3as persistence after shocks to volatility tends to generate a relatively high concentration of large returns.In fact,it has been shown in the econometrics literature that the popular(G)ARCH family of time series models generates unconditional distributions with a limiting behaviour conforming to the Pareto law(1.1).On the other hand,it has also been found that residuals from(G)ARCH specifications usually still exhibit fat tails so that the frequency of large returns can not solely be traced back to autoregres-sive behaviour of volatility.Also in this case,recent literature provides a somewhat sharper view on this stylised fact showi
ng absolute returns rather than squared returns possess the highest degree of autocorrelation(cf.Ding et al.[12]).
Though properties1to3characterise the behaviour of almost allfinancial prices, they defy a straightforward explanation.It should be pointed out that the task of finding an explanation for these facts differs quite fundamentally from attempts to explain other economic phenomena.In particular,the economist’s interest is often in questions of the type:how does one economic he exchange rate) react to variations in some other y supply).While linear dynamic models or comparative static analysis as a formal tool might appear perfectly ade-quate for the latter purpose(at least in order to gainfirst insights into the structure of the problem),this is not so when dealing with the above statisticalfindings.The reason is that an elementary requirement for any adequate analytical approach is that it must have the potential for bringing about the required behaviour in the-oretical time series.Therefore,it seems rather obvious that one has to go beyond linear deterministic dynamics,which of course is insufficient to account for the phe-nomena under study.Furthermore,allowing for homogeneous(white)noise in some
678T.Lux&M.Marchesi
economic variables will also not achieve our goals simply because,in the view of Fact3,we are dealing with time-varying statistical behaviour.In fact,the situation one faces is more often encountered by natural scientists.What one wishes to ex-plain is a feature of the empirical time series as a whole.In the natural sciences, such characteristics of the data are often described by scaling laws and it is indeed possible to express Facts2and3in a similar fashion.First,Eq.(1.1)can be read-ily interpreted as a scaling law remaining probability in the tails,1−F(x),since the is scaling according to a power of x.Second,some formalisations of autore-gressive heteroscedasticity concretised this feature as a hyperbolic decline of the autocorrelations of absolute or squared returns(cf.Brock and de Lima[8]).
Despite this somewhat unfamiliar research environment,some attempts have been made in recent economics literature towards an explanation of the above facts. Concerning the unit root property,both Kirman[24]and De Grauwe et al.[10]con-structed complex structural models of speculative activity generating time series which are not distinguishable from a unit root process using standard tests.Inter-estingly,both contributions have as their starting point the model of Frankel and Froot[14]which focuses on the interaction of chartist and fundamentalist traders. Kirman extends this approach by adding a stochastic mechanism for the formation of majority opinion among chartists(see also Kirman[25]).De Grauwe et al.[10], on the other hand,investigate the chaotic dynamics resulting fro
m an extended de-terministic version of the original model set-up.In both models,the data-generating mechanism is surely not a(pure)unit root process and onefinds the short-run dy-namics dominated by speculative deviations from a fundamental-oriented path.As a consequence,the usual interpretation of non-rejection of a unit root as evidence against existence of speculative bubbles is called into question by these results.As concerns the approach taken by De Grauwe et al.,one may,however,object that the empirical search for chaos infinancial time series has not been very successful —at least in the sense that no low-dimensional attractor could be identified.c Turning to Fact3,wefind evidence of volatility clustering in complex simula-tion studies offinancial markets by Grannan and Swindle[19]and in the artificial stock market of Arthur et al.[2].Ramsey[36],on the other hand,shows how a statistical description of individual behaviour may quite generally give rise to dy-namics with time varying second moments.Independently,Lux[29]made the same point when deriving the dynamics of second moments of a stochastic multi-agent model of speculation.However,while the model presented there seemed to yield results conforming with Fact3,it appears to be too simple to confront it with other regularities.In another recent paper(Lux[30]),a chaotic model of specula-tive dynamics constructed along the lines of Day and Huang[9]and Lux[27]was c However,such an approach may be justified by the recognition that an underlying chaotic dynam-ics may be concealed by additional noise which is surely present in economic data.It has indeed been shown,that rather small amounts of noise may strongly influence the results of certain tests for nonlinearity and chaotic dynamics.
Volatility Clustering in Financial Markets679 shown to give rise to well-behaved,uni-modal and leptokurtotic distributions of re-turns.The model presented in this paper will be close in economic content to the one analysed in Lux[30].However,in contrast to the earlier article,we will not be interested in the potential of cyclic or chaotic time paths but will concentrate on investigating the system’s dynamics in the presence of stable equilibria.Though this restriction may at afirst glance appear to be contradictory to the above outline of an appropriate research strategy and seems to constitute a refinement to a rather uninteresting case,it will turn out that this is not so.This seeming contradiction will be resolved by showing that an otherwise stable equilibrium can be subject to sudden transient phases of destabilisation.The characteristic features of these periods are bursts of severefluctuations around the equilibrium which,however, quickly die out in the course of events with the system returning to a stable and calm state again.This type of punctuated equilibrium generates time series with clusters of excessive volatility interspersed among long tranquil periods.Statistical analysis shows that the resulting time paths for returns share the basic character-istics of real-life markets as captured in Facts2and3above.The time series of prices(or exchange rates)itself,on the other hand,resembles a random walk,thus conforming to Fact1.
Similar dynamics have also been found in a somewhat different economic context recently by Youssef
mir and Huberman[40]who dealt with the evolution of resource utilisation by adaptive agents.d In their paper,they conjectured that the same mechanism may serve as an explanation for volatility clustering infinancial markets. The present paper confirms this conjecture.e
In natural science literature,a number of papers with qualitatively similar dy-namic behaviour can be found which may,however,result from very different types of models(Fujisaka and Yamada[15];Heagy et al.[22];Ott and Sommerer [34]).The phenomenon under study has been denoted on-offintermittency.Loosely speaking,the unifying feature of all examples of its occurrence is an attracting state (which may not always be afixed point)becoming temporarily unstable due to a local some key variable surpassing some stability threshold.This destabilisation may be generated in a deterministic hrough weak coupling to another dynamics)or may occur stochastically.In any case,there will be no lasting deviation from the equilibrium as the system is driven back to stability by some endogenous mechanism.
In our model of chartist/fundamentalist interaction,the bifurcation parame-ter is the time-varying fraction of traders pursuing a chartist strategy.In general, agents are allowed to switch between a chartist and a fundamentalist trading strat-egy after comparing the respective profits.However,in the vicinity of the equi-librium the price(on average)equals the fundamental value and capital gains are
characterised A similar behaviour has also been observed by Glance[17].
e In personal communication,Michael Youssefmir in fact conjectured that this effect may be found in the type o
f model presented in Lux[27].

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