1.1 The Market Impact Function

The Dynamics of the Linear Random FarmerModelRui CarvalhoLaborat¶o rio de Mecatr¶o nica,Instituto Superior T¶e cnicoAv.Rovisco Pais,1049-001Lisboa Portugalrui@Version1.121th of July2000AbstractUnder the assumption that agents trade at each time step with prob-ability p(i),we establish conditions for price to re°ect value.This variation of Farmer's model belongs to a family of Stochastic Di®erence Equations known for displaying Pareto tails.When positionbased value investors dominate,we¯nd conditions for the distributionof returns to be consistent with a power-law asymptotic behavior andcharacterize its exponent.1The Model1.1The Market Impact FunctionConsider a stock market with N agents,labeled by an integer1·i·N, trading on a single asset(measured in units of shares),whose price at time t will be denoted by the continuous1variable P t.Let p t=log P t be the log 1See the ordered probit model in[1]for an approach to discrete prices.See also[2]for a discussion on the uniqueness of prices,which is equivalent to a zero spread.1price,r t+1=p t+1¡p t be the one-period return and!(i)(p t;p t¡1;:::;I t)be the order size for trader i at time step t;where I t represents any additional external information.The trade protocol in Farmer's model involves a two-step process.On the¯rst step,the asset's true value(º)is realized and each trader posts their orders based on this fundamental value,the current(p t)and past(p t¡1;:::) prices.On the second stage,the market maker observes the aggregate net order°ow!=N X i=1!(i)and sets a single price(p t+1)according to the market impact function(also called the price impact function).Orders are then¯lled at the new price, p t+1.Farmer deduces the market impact function[2]to bep t+1=p t+1¸NX i=1!(i)t+1(p t;p t¡1;:::;I t)+»t+1(1)where i indexes the trader,¸is the market maker liquidity and»t accounts for exogenous in°uences on the dynamics2.Under equation(1),the price is manipulated in the direction of the net of incoming orders[2].The simplest model is where the trader order sizes,!(i)(p t;p t¡1;:::;I t), are i.i.d.gaussian processes.This is the limit where traders are heteroge-neous,uncorrelated and the information°ow occurs stochastically with no memory.The solution of(1)retrieves the celebrated random walk for p t. 1.2The Market MetaphorThe dynamics of(1)depends on the collection of strategies,which Farmer divides into two classes[2].The¯rst type are fundamentalists,who believe that the price of an asset is determined solely by its e±cient market hypoth-esis(EMH)fundamental value.The second typical trader type are noise traders,sometimes called chartists or technical analysts,believing that asset 2On the following analysis,we will assume»t´0.2prices are not completely determined by fundamentals,but that they may be predicted by simple technical trading rules,extrapolation of trends and other patterns observed in past prices[3].The model builds upon a vision of the market based on interaction be-tween micro and macro dynamics.The former is the dynamics of individual agents,who invest according to fundamental or technical strategies.The later is the dynamics of fundamental value,a random walk for the scope of this text,and prices.Traders do not communicate with each other;in-stead they interact with the market maker through equation(1)and base their decisions upon present and past values of the price or fundamental value.Therefore,traders'decisions(the¯negrained variables)are in°uenced by prices and fundamental values(the coarse grained variables),and the fol-lowing price is determined by traders'investments.The market is born out of this feedback loop between¯negrained and coarsegrained variables.1.3The StrategiesA¯rst class of agents,the value investors,have pure position based value strategy which can be approximated by(see[2]for the details)!(i)t+1'¡c(i)¢m t=¡c(i)(r t¡´t)(2) where m t=p t¡ºt is the di®erence between(log)price and(log)fundamental value,¢is the di®erence operator¢m t=m t¡m t¡1,r t=p t¡p t¡1is the one-period return and´t=¢ºt is the random increment to value.The second class of agents attempt to identify price trends as a source of information,instead of focusing on fundamentals.Farmer calls these the trend following strategies.An example of a position based trend following strategy on timescaleµis!(j)t+1=c(j)(r t¡r t¡µ)(3) We consider a system with N v agents with position based value strategies and N t agents with position based trend following strategies,so that N= N v+N t is the total number of agents.Replacing(2)and(3)on(1),de¯ning the sums of the agents capi-tal3for each class of strategies as®vi=1P N v i=1c(i)for value investors and 3To simplify the language,we refer to an agent's capital,with the meaning of c i=¸.3®tf=1P N j=N v+1c(j)for trend followers,and the di®erence between the value investors'and trend followers'total capital as¢®=®vi¡®tf,yields:½r t+1=¡¢®r t¡®tf r t¡µ+®vi´tp t+1=p t+r t+1(4) Although r t is the relevant variable,p t+1must not be unbounded.2Results2.1Conditions for Price to Re°ect ValueTo better understand(4),we notice thatr t+1=4m t+1+´t+1so that the dynamics of the mispricing is4m t+1=¡4®4m t¡®tf4m t¡µ¡´t+1+®tf´t¡®tf´t¡µ(5) Equation(5)is a discrete dynamical system with delay structure.By making the substitution(X1t;:::;Xµt;Xµ+1t)´(4m t¡µ;:::;4m t¡1;4m t)(6) the dynamics of4m t becomesX t+1=J m X t+´t(7) where J m is a(µ+1)£(µ+1)companion matrix,J m=2666664011...1¡®tf0:::0¡¢®3777775(8) 4and´t;µ=2640...¡´t+1+®tf´t¡®tf´t¡µ375(9)Farmer allows the trend follower'sµto be agent dependent4,that isµ´µj on(3).On our model,there is a probability p(i)that agent i will trade(either buy or sell)at time step t and a probability1¡p(i)that he remains inactive for that time step.This probability is independent of t:As a consequence of these assumptions,J m´J m(t)and the number of active agents varies with time.De¯ne±k;(µi;t)=1if trend follower k has time lagµi and is active at time step t and0otherwise;±k;t=1if trader k is active at time step t and0otherwise(10)Then the sum of the capital of active trend followers with time lagµi at time step t is®tf(µi;t)=N tX k=1®(k)tf±k;(µi;t)and the di®erence between the(active)value investors and trend followers total capital at time step t is¢®(t)=N vX i=1®(i)vi(t)±k;t¡N X j=N v+1®(j)tf(t)±k;tThus,matrix J m(t)can be written as:4Farmer makes further assumptions,e.g.that each trader perceives a distinct value, etc.,but we will only consider these two here.5J m(t)=2666664010:::0001:::0...00:::01¡®tf(µmax;t)¡®tf(µmax¡1;t):::¡®tf(µ1;t)¢®(t)3777775(11)and7¯nally becomes:X t+1=J m(t)X t+´t(12) where´t=2640...¡´t+1+®tf´t¡Pµmaxµ=1®tf(µi;t)´t¡µ375(13)Equation(12),together with equation(11)and(13),de¯ne an A±ne Random Dynamical System.A±ne Random Dynamical Systems verify Oseledets's multiplicative er-godic theorem[4,134]and thus the Lyapunov numbers are the eigenvalues oflimn!1[(J¤m)n(J m)n]1=2nIf the largest Lyapunov number of(12)is less than one and agents trade w.p. 1,then(12)is contractive and´t is a stationary process.This means that the mispricing is a random walk and the returns follow a stationary process. Notice that if p i=1,´t is not stationary and the mispricing will not follow a random walk,as®tf(µi;t)in(13)will be time dependent.We simulate a market with N v=N tf=750agents each trading at every time step w.p.1,where the market maker liquidity is unitary(¸=1).Trend followers look back in time with a n agent dependent lagµ(i),whereµ(i)is unifomly distributed between1and max(µ(i))=100.The value process is a random walk with h´i=0and h´2i1=2=0:1.The agents'capitals are uniformly distributed on the interval]0;c max[where c max=3¢10¡3.6We verify that the Lyapunov number of(12)is less than1,as can beobserved on Fig. 3.The mispricing is a random walk and thus price and value do not cointegrate,as shown on Fig.1,but price tends to re°ect valueif a lerge enough time scale is considered.Trend followers seem to introduceno signi¯cant correlations on the returns as patent on Fig.4.We propose that the insertion of order based value investors who shallhave a low probability for being active,has the e®ect of an attractive potential, moving price towards value,but not being enough for a rigorous cointegrationof the two.Indeed,on a previous version of[2],Farmer introduces the pure order based value strategies which can be approximated by(see[2]for the details)!(i)t+1'¡c(i)m t=¡c(i)(p t¡ºt)(14) Pure order based strategies are dismissed as being irrealistic from a behaviour point of view as their positions can grow without bound.This undesired e®ect can be avoided if order based value investors areactive with a probability much lower than other agents and the dynamics of m t is slower than the dynamics of p t andºt.In this situation,the(fast)dynamics of the order based value investors(even when a very low percentageof order based value investors is active),has the e®ect of a restoring force on the(slow)dynamics of the mispricing,approaching it from zero.Therefore,even if order based value investors are rarely active,their e®ect is enough for price to follow value.We have simulated a market with N v=N tf=675,but added150or-der based value investors with the same maximum capital as the remaining strategies.The trading probabilities are now p(i)=1for position basedagents and p(i)=10¡2for order based value investors.We verify that even a weak presence of order based value investors is enough for price to re-°ect value on a large enough time scale,as can be observed from Fig2.Wehave veri¯ed that under these circumstances,the maximum position attained among the order value investors for a simulation of105iteractions was0:7916, much lower than the market maker liquidity.2.2Dominance of Position Based Value Investors7Consider®viÀ®tf,in other words,position based value investors dominate the market.Then equation(4)can be approximated byr n+1=¡®vi(t)r n+®vi(t)´n(15) where®vi(t)=NvX i=1®(i)vi(t)±i;n(16)for±i;n=½1w.p.p(i)0w.p.(1¡p(i))If the agents'capital and trading probability are constants,respectively ®(i)vi(t)=®and p(i)=p,expression(16)can be written in the following way(where N´Nv)5:®vi(t)=8>>>>>>>< >>>>>>>:0(1¡p)N®p(1¡p)N¡1N 2®p2(1¡p)N¡2¡N2¢...... (N¡1)®p N¡1(1¡p)¡N N¡1¢N®p NEquation(15)is therefore an A±ne Random Dynamical System and there is a unique invariant measure for r t if[4,134]Z+1¡1log+k®vi(x)k dx<1(17) where log+u=max(0;log u).Condition(17)is guaranteed if®vi(t)is always ¯nite,that is if the number of agents,Nv,is¯nite.Furthermore,the solution of(15)is then given by[6,208]r n=nX k=1Ãn Y j=k+1¡®vi(j)!®vi(k)´k+Ãn Y j=1¡®vi(j)!r05The procedure we follow was inspired on[5].8If the largest Lyapunov exponent of(15)is negative,thennY j=1¡®vi(j)!0exponentially fast and under very weak conditions on the product®vi(k)´k, the distribution of r n will converge independently of r0to that of the seriesnX k=1Ãn Y j=k+1¡®vi(j)!®vi(k)´kEquation(15)is well know to present Pareto tails.In fact,Kesten[7] states the followingTheorem:Consider a stochastic di®erence equation:Y t+1=M t Y t¡1+Q t t=1;2;:::where the pairs f M t;Q t g are i:i:d:real valued random variables.If E[log j M t j]<0then Y t converges in distribution and has a unique limiting distribution.If,additionally,Q t=(1¡M t)is non-degenerate and there exists some¹>0such that:i)0<E[j Q t j¹]<1ii)E[j M t j¹]=1iii)E£j M t j¹log+j M t j¤<1then the tails of the limiting distribution are asymptotic to a power law,i.e. they obey a law of the type P rob(j Y t j>x)¼c¢x¡¹De¯ne<®vi(t)¹>=®¹NX i=0i¹p i(1¡p)N¡iµN i¶(18)Convergence in probability of(15)is thus equivalent to the condition<®vi(t)>=Np®<1(19) and the above theorem requires the existence of a¹>0such that<®vi(t)¹>=1(20)9Although the estimation of¹from(20)is in general not possible,one canestimate an interval for¹,given N,p and®.Numerical calculations show that for N=103,p=9:5¢10¡3and®=0:1,<®vi(t)>=:95,<®vi(t)2>= 0:996598and<®vi(t)3>=1:13478.Due to the continuity of(18),thismeans that one should expect a¹very close to2.Figure5is the modi¯ed Hill plot[8]for r n which numerically con¯rms theabove calculations.3ConclusionsWe have shown that when the only strategies are(position based)trendfollowers and value investors,the mispricing is a random walk,independently of the maximum lag in time,µ(i)of the trend followers.Trend followers do not seem to introduce relevant correlations on the returns6.The introduction of a low percentage of active order based value investors,with a low probability of trading,seems to guarantee that price will follow7value on a large enough time scale.Rigorous conditions for this phenomenol-ogy deserve further study.We have shown that when the dominant strategies are position basedvalue investors,if there exist N;®;p and¹that satisfy(19)and(20)thenP rob(j r t j>x)¼c¢x¡¹(21) and the returns display a probability distribution asymptotic to a power tail of exponent¹.We have also shown that,if one¯xes N and p,one can determine heuristically®from(18)such that the probability distribution of r t is given by(21)with the desired¹.6Aparently,ifµmax is large,the correlations induced by trend followers on the returns seem to compensate each other.7but not cointegrate.10References[1]J.A.Hausman,A.W.Lo,and A.C.MacKinlay,Journal of FinancialEconomics31,319(1992).[2]D.Farmer,adap-org/9812005(1998).[3]W.A.Brock and C.H.Hommes,Journal of Economic Dynamics andControl22,1235(1998).[4]L.Arnold,Random Dynamical Systems,Springer-Verlag,1998.[5]T.Lux and D.Sornette,http://www.bwl.uni-kiel.de/vwlinstitute/gwrp/german/publ/papers/rebub.zip(2000).[6]L.d.Haan,S.Resnick,H.Rootz¶e n,and C.Vries,Stochastic Processesand their Applications32,213(1989).[7]H.Kesten,Acta Mathematica,207(1973).[8]H.Dress,L.d.Haan,and S.Resnick,http:==www:mi:uni¡koeln:de=Math¡Net=shadow f iles=preprints=990630:hdrees:1:html(1999).11Figure1:Simulation with N v=N tf=750,¸=1,max(µ(i))=100,h´i=0(i)and h´2i1=212Figure3:Lyapunov Number for the simulation on Fig.1.Figure4:The return ACF for the simulation of Figure113Figure5:Hill estimator for the tails of r n when N=103,p=9:5¢10¡3and ®=0:1.14。