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Heat baths and computational agent-based models
Physica A 391 (2012) 5512–5520
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Heat baths and computational agent-based models Andrew Clark ∗ Lipper, A Thomson Reuters Company, Andrew Clark/Lipper, Thomson Reuters, 707 17th Street, 80202 Denver, CO, United States
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Article history: Received 14 November 2009 Received in revised form 1 June 2012 Available online 15 June 2012 Keywords: Mean field models Agent based models Volatility Power laws Bifurcations
abstract In this paper, we examine an agent-based model, and an equation-based model in the form of a mean field model. We show how the mean field model is a small, fast model that identifies the high level properties of a subject, in this case financial time series’ stylized facts. The agent based model generates the granularity needed to understand the conditions and factors that generate the stylized financial facts. We conclude with the recommendation that both models be used in sequence so a complete description of a process be established or approximated. © 2012 Elsevier B.V. All rights reserved.
1. Computational agent-based models 1.1. The evolutionary dynamics of adaptive belief systems For agent-based models this paper focuses on the adaptive rational equilibrium modeling first developed by Brock and Hommes [1,2] and used by Chiarella and He [3]. Brock and Hommes (hereafter, B & H) developed what they call ‘‘adaptive belief systems’’ to model heterogeneous expectations in a way that naturally mimics real market activity. In adaptive belief systems agents (investors) adapt their prediction of asset prices by choosing a finite number of predictors or expectation functions, which are a function of past price performance. Each predictor has a performance measure that is visible to all participants. Based on the performance measure, agents make a (bounded) rational choice between price predictors. This assumption of bounded rational choice results in an adaptive rational equilibrium dynamic (ARED), which generates a dynamic across predictor choices that is coupled to the dynamics of the endogenous variables. B & H [1] show that ARED incorporates a general mechanism that can generate local instability in the equilibrium steady state and complicated global equilibrium dynamics. B & H apply the concept of adaptive belief systems to a simple asset-pricing model, where traders in a financial market use different types of predictors for the price forecasts of risky assets. Chiarella and He [3] extend B & H’s work to include agents with different risk aversion as well as expected dividend payments with noise. What follows in this section is the initial setup of an adaptive belief system as used by Chiarella and He (hereafter, C & H). To define investor wealth it is assumed the risk-free asset is perfectly elastic and supplied at a gross return R > 1. Let pt be the price (ex dividend) per share of the risky asset at time t, and let (yt ) be the stochastic dividend process of the risky asset. Then, investor wealth at t + 1 is defined as: Wt +1 = RWt + (pt +1 + yt +1 − Rpt ) zt
(1.1)
where Rpt is the return of the risk-free asset at time t, and zt is the number of shares of the risky asset purchased at time t.
∗
Tel.: +1 3039416017. E-mail address: [email protected].
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.06.011
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In both B & H’s and C & H’s work a Walrasian auctioneer is used to derive the demand equation, i.e., each trader is viewed as a price taker. The market is viewed as finding the price pt that equates the sum of the demand schedules to the supply. That is, the price pt at time t is formed by using information available at time t − 1 and the expected utility for time t + 1. Denote by: Ft = (pt , pt −1 , . . . , yt , yt −1 , . . .)
(1.2)
the information set formed at time t, where p is the price that equates the sum of the demand schedule to the supply of the asset and y is the dividend yield. Now, denote by Rt +1 the excess return on the risky asset at time t + 1: Rt +1 = (pt +1 + yt +1 − Rpt ) .
(1.3)
It follows from Eqs. (1.2) and (1.3) that the conditional expectation Et and conditional variance Vt can be defined as: Eht (Wt +1 ) = RWt + Eht (pt +1 + yt +1 − Rpt ) zt = RWt + Eht (Rt +1 )zt Vht (Wt +1 ) =
zt2 Vht
(pt +1 + yt +1 − Rpt ) =
zt2 Vht
(1.4)
(Rt +1 )
(1.5)
where the subscript h refers to the beliefs of investor type h about the conditional mean and variance. Note that in Eq. (1.5) the conditional variance of wealth Wt +1 equals zt2 times the conditional variance of excess return per share pt +1 + yt +1 − Rpt . This differs from B & H [2], who assume the beliefs about conditional variance of excess returns are constant and the same for everyone, i.e., Vht (pt +1 + yt +1 − Rpt ) ≡ σ 2 . C & H assume each type of investor is a mean-variance maximizer with different attitudes toward risk. Each investor type h has a risk aversion coefficient ah . Given ah , for each investor type the demand for shares zht solves:
max = Eh,t (Wt +1 ) − z
ah 2
Vh,t (Wt +1 )
(1.6)
or zh,t =
Eh,t (Rt +1 ) ah Vh,t (Rt +1 )
.
(1.7)
Let zst be the supply of shares and nht the fraction of type h investors at t. Using Eq. (1.7), the equilibrium state of supply equaling demand is described by:
nh , t − 1
h
Eh,t (Rt +1 ) ah Vh,t (Rt +1 )
= zst .
(1.8)
Now, assume a zero supply of outside shares, zst = 0, then (1.8) leads to:
nh , t − 1
h
Eh,t (Rt +1 ) ah Vh,t (Rt +1 )
= 0.
(1.9)
Eq. (1.9) makes it appear that the effect of the risk-aversion coefficients is to rescale the nh,t −1 functions. This will be found to not be the case once Eq. (1.12) below is introduced. To get a notion of the rational expectation (RE) fundamental solution p∗t , consider the equation: Rp∗t = Et (p∗t +1 + yt +1 ) where Et is the expectation conditional on the information set Ft (see Eq. (1.2) above). To satisfy the ‘‘no-bubbles’’ version of the rational expectations B & H and C & H are using, the only solution can be: p¯ =
y¯
.
R+1 If we let xt denote the deviation of pt from the RE fundamental p∗t , then: xt = pt − p∗t . After establishing this relationship C & H give the equations for heterogeneous beliefs for returns and variance, i.e., the different classes of beliefs that are deviations from the fundamental. Both B & H and C & H use a fitness function, which is defined by the realized profits of investor type h:
πh,t = Rt +1 zh,t zh,t =
Eh,t (Rt +1 ) ah Vh,t (Rt +1 )
.
(1.10)
Memory can be introduced to the fitness function by considering a weighted average of past profits: Uh,t = πh,t + ηUh,t −1 where η represents the memory strength.
(1.11)
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Let the updating fractions be given by the discrete choice probability: nh,t = exp(β Uh,t −1 /Zt ) Zt =
(1.12)
exp(β Uh,t −1 )
h
where β > 0 is the intensity of choice, measuring how fast agents switch between different prediction strategies. In particular, β = +∞ means that in each period all the investors switch to the strategy with the highest fitness in the previous period, while β = 0 means most investors distribute themselves evenly across all the available strategies. For 0 < β < ∞ a fraction of each group of investors switches to the strategy with the highest fitness in the previous period. To summarize, the evolutionary dynamics described by the adaptive belief system are:
h
nh,t −1 fh,t ah (σ 2 +gh,t )
h
nh,t −1 ah (σ 2 +gh,t )
RXt =
(1.13)
where fht is the agent-specific predictive component of the mean for investor type h, and ght is the agent-specific predictive component of the variance for investor type h. σ 2 is the constant predicted variance used in B & H, while in C & H it is the fundamental or underlying variance that is modified by ght . The updating of fractions, fitness function, and memory strength are given by: nh,t = exp(β Uh,t −1 )/Zt Zt =
exp(β Uh,t −1 )
h
(1.14)
Uh,t = πh,t + ηUh,t −1
πh,t = Rt +1 zh,t = (xt −1 − Rxt + δt +1 )
fh,t − Rxt ah (σ 2 + gh,t )
.
1.2. Dynamics of fundamentalist, trend, and contrarian investors To investigate the role of heterogeneous belief types (investors with different beliefs) assume all beliefs follow a linear return and a nonlinear variance learning process: xt =
σ2 =
L 1
L i=1
xt −i (1.15)
L 1
L i=1
(xt −i − xt )2
and let:
υh (x) = µ 1 −
1
(1 + x) − ξ
(1.16)
where L (the number of lags or the length of the investor’s memory) is a positive integer and µ, ξ > 0 are constants. Assume further that: fh,t = dh,t xt
(1.17)
where dh is the trend of investor type h. Call h a pure trend chaser if dh > 0 (a strong trend chaser if dh > R) and a contrarian if dh < 0 (and a strong contrarian if dh < −R). When dh = 0, Eq. (1.17) yields the belief of fundamentalists, i.e., investors who believe prices return to their fundamental value. Using Eqs. (1.15)–(1.17), letting η = 0, δt = 0, a = a2 /a1 and mt = n1,t − n2,t , C & H derive the following equations1 :
(1 + υh (σ¯ 2 ))1 − mt −1 xt R a(1 + mt −1 ) + (1 + υh (σ¯ t2−1 ))(1 − mt −1 ) β Rxt −1 dxt −1 Rxt −1 βC mt = tanh + − . (Rxt −1 − xt ) 2a1 σ 2 a 2 1 + υh (σ¯ t2 ) xt =
d
1 See C & H, Section 3, for the full development.
(1.18)
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For the special case L = 1 Eqs. (1.15)–(1.16) generate: xt = xt −1
σ2 = 0 νh σ¯ 2 = 0
(1.19)
and the following system holds: xt =
1 − m t −1
d
R a + 1 + (a − 1)mt −1
mt = tanh
x t −1
β dxt −2 − Rxt −1 βC ( Rx − x ) Rx + − t − 1 t t − 1 2a1 σ 2 a 2
(1.20)
where a = a2 /a1 and mt = n1,t − n2,t . The lemma below (Lemma 3.1 in C & H and proved in their appendix) is used to establish the existence and the stability of equilibrium:
Lemma. Let meq = tanh −
tanh
βC 2
, m∗ = 1 −
2aR , d+R(a−1)
and x∗ be the positive solution (if it exists) of
β d−R βC ∗ 2 ( R − 1 ) R + ( x ) − = m∗ . 2σ 2 a1 a 2
(1.21)
• For 0 < d < R, E1 = (0, meq ) is the unique globally steady state of Eq. (1.18). • For R < d < (a + 1)R there are two possibilities: ◦ If m∗ < meq , then E1 is the unique globally steady state of Eq. (1.18). ◦ If m∗ < meq , then (1.18) has three steady states E1 , E2 , E3 ; E1 is unstable. • For d > (a + 1)R, Eq. (1.18) has three steady states E1 , E2 = (x∗ , m∗ ), E3 ; E1 is unstable. The lemma indicates that when trend chasers extrapolate only weakly (0 < d < R), the fundamental steady state E1 is globally stable no matter what risk aversion investors have. When d > R, the stability of the fundamental steady state E1 depends on the ratio a, which measures the relative risk aversion. When trend chasers extrapolate strongly, (d > (a + 1) R), the fundamental steady state E1 becomes unstable and bifurcates into two additional nonzero steady states E2 , and E3 . In the case where R < d < (a + 1) r the fundamental steady state is stable when a is large, i.e., when trend chasers become more risk averse than fundamentalists. (Statements made about trend chasers and fundamentalists hold for contrarians and fundamentalists simply by reversing the inequality signs.) And when a = 1, the results of B & H [2] are recovered. 2. Heat baths Before we develop the heat bath model, we will present a brief introduction to mean field models. Discussing these types of models will deepen the reader’s understanding of the heat bath model as well as make it easier in Section 3 to compare agent-based and heat bath models. In statistical physics systems of many interacting particles – such as the agents described in Section 1 – are analytically intractable because they tend not to have or do not have closed-form solutions. To achieve a closed-form solution, physicists use a single particle and estimate the influence of all the other particles on it as an average influence. The resulting model is called a mean field model because it replaces many individual entities with their average. Ising models are a canonical example of mean fields. In an Ising model there are discrete variables (spins) that can be in one of two states, more often than not ±1. The spins are arranged in a lattice, and each spin interacts at most with its nearest neighbors. Mean field theory makes possible the solution of the Ising model analytically by replacing all of its interactions with an average or effective interaction. This reduces the multi-body (the collection of individual agents) problem to a onebody problem. The relative ease of solving mean field problems means that insights into the behavior of the system can be obtained at a reasonable cost, especially when compared to the computational cost of agent-based models. A mean field model can be characterized as centralized because variables require global information. Because they require global information, mean field models provide valuable information from a global perspective, e.g., the volatility clustering often seen in financial return series. In contrast, the natural tendency of the agent-based models is to define agent behavior in terms of observables accessible to the individual agent, thereby favoring a local versus global viewpoint. The evolution of system-level observables does emerge from both C & H’s and B & H’s work, but they do not use these observables to drive their models’ dynamics. Therefore, the B & H and C & H models do not generate global stylized facts such as volatility clustering.2
2 In Section 3 the author will discuss how agent-based models can reproduce volatility clusters.
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The heat bath model of Krawiecki et al. [4] has I = 1, 2, . . . , N agents (spins) with orientations σi (t ) = ±1, which correspond to the decision to sell (−1) or to buy (+1) a share of stock at discrete time steps t. The orientation of agent i at time t depends on the local field: Ii ( t ) =
N 1
N j =1
Aij (t )σj (t ) + hi (t )
(2.1)
where Aij are time-dependent interaction strengths among agents and hi (t ) is an external field reflecting the effect of the environment (e.g., access to external information, which can differ between agents3 ). The interaction strengths and external fields change randomly in time. In Eq. (2.2): Aij (t ) = Aξ (t ) + aηij (t ) hi (t ) = hςi (t )
(2.2)
where ξ (t ), ηij′ (t ) and ζi (t ) are random variables with no correlation in either space or time and are uniformly distributed along the interval (−1, 1). A is a measure of the randomly varying average interaction strength between agents, a is a measure of the random deviations of the individual interaction strengths from the average, and h is a measure of the random influence of the environment. The strengths Aij (t ) can assume both positive and negative values corresponding to a tendency to imitate or avoid the orientation of the players in the market. Krawiecki et al. demonstrate that the noises ξ (t ) and ζi (t ) are necessary to get real market results such as volatility clustering. The orientation of all the agents are updated synchronously according to the discrete choice probability σi (t + 1) = 1 with probability p or −1 with probability 1 − p. The uncertainty in the investment decision (where p = 1/{1 + exp[−2Ii (t )]}) is analogous to heat bath dynamics. In terms of modeling the time series and distribution of financial returns, Krawiecki et al. make the (common) assumption that the relative change in a security’s price is proportional to the difference between the number of buying and selling decisions and use the following formula where the security’s returns are the forward changes in the logarithms of prices P (t ) over a time scale ∆t : G∆t (t ) = ln P (t ) − ln P (t − ∆t ). N By introducing an average orientation x(t ) = N −1 τ =0 x(t − τ ) of the agents, they get dP /dt ∝ xt. Discretizing time in dP /dt ∝ xt yields G∆t (t ) ∝
∆ t −1
x(t − τ )
(2.3)
τ
where G∆t (t ) is the return of the security. G∆t (t ) allows us to interpret Eqs. (2.1) and (2.2) in the following manner: since the average interaction strength Aξ (t ) is common to all connections, it is a measure of the average reaction of agents to price changes, influencing their decisions via the term Aξ (t )x(t ) in the mean field Ii (t ). The term aηij (t ) describes the fluctuating interaction network and hζ (t ) the fluctuating environment. The local field in Eq. (2.1) describes how agent i anticipates price changes by averaging the information of the opinions of his interaction partners and the external information accessible to him. A mean field approximation exists for the Krawiecki et al. model, where x(t ) can be reduced to a 1-d map. This mean field model allows Krawiecki et al. to generate time series that resemble the stylized facts of empirical price returns, such as large bursts of volatility and their corresponding volatility clustering, as well as power laws. The mean field approximation sets are A ̸= 0, a = 0, and ςi (t ) = ς (t ) for all i. The dynamics of x(t ) become: x(t + 1) = tanh[Aξ (t ) + hς (t )] ≈ Aξ (t )x(t ) + hς (t )
(2.4)
where the approximate equality holds for ⌊Aξ (t )x(t ) + hζ (t )⌋ ≪ 1. Eq. (2.4) is a generic model for on–off intermittency if h = 0 and for attractor bubbling if h > 0. The approximate linear form of Eq. (2.4) belongs to a more general class of stochastic processes with multiplicative noise. Although the Krawiecki et al.-generated time series results given Eq. (2.4) could be either on–off intermittency or attractor bubbling, there is a qualitative difference between these phenomena, write Krawiecki et al. In the case of on–off intermittency the bursts are caused solely by the multiplicative noise ξ (t ) when its amplitude A exceeds the blowout bifurcation4 threshold. In the case of attractor bubbling the multiplicative noise amplifies only to macroscopic sizes the small deviations from the fixed point caused by the additive noise. Krawiecki et al. suggest – by visual inspection – the attractor bubbling scheme bears the best resemblance to the empirical stylized facts. This is because it has frequent large bursts of volatility and volatility clustering, separated by long-lasting laminar phases when the map of Eq. (2.3) is governed mainly by the additive noise hζ (t ).
3 This can be the presence of asymmetric information or uncertainty about future dividend yields. 4 A blowout bifurcation takes place when the Lyapunov exponent in the transversal direction of the invariant subspace changes from negative to positive.
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Krawiecki et al.’s mean field results, the laminar phases with x ∼ 0, correspond to the disordered phase with no preferred orientation (the system is near equilibrium). x ∝ O(1) bursts occur and these correspond to the ordered phase with a majority of agents sharing one orientation, such as when herding occurs and financial arbitrage becomes possible. To summarize the Krawiecki et al. heat bath model, the mean field model quantitatively reflects the multi-agent dynamics. The necessary conditions for the applicability of the mean field model are (1) random fluctuations in time of the average interaction strength between agents, corresponding to their average reaction to price changes; (2) an uncertainty in decision making analogous to heat bath dynamics; and (3) in the thermodynamic limit small additive noise, simulating the effect of the external environment. When these conditions are used to simulate financial prices, stylized facts such as volatility clustering and the power-law distribution of returns appear. The volatility clustering, power-law distributions, and other phenomena are shown to be the result of attractor bubbling. 3. Mean field and agent-based modeling A difference between agent-based modeling and mean field models is that agent-based modeling directs the modeler’s attention to the individual entities in the domain, while mean field models focus on variables. These variables may be extensive (e.g., the total population or agent density) or intensive (e.g., parameters of individual agents). Intensive variables may reflect individual agents or (as averages) across the system as a whole. The Krawiecki et al. mean field model works with averages, while the C & H model focuses on the activity of individuals in the system. The heat bath model can be characterized as centralized, since its variables require global information. It provides valuable information from a global perspective, e.g., volatility clustering, but misses the information contained in agent-based entities. The Krawiecki et al. model neglects the interaction between spins. To quote from Krawiecki et al., ‘‘The details of the interactions among pairs of agents, and the details of the variation of the mean interaction strength seem unimportant for the qualitative result’’. The heat bath model replaces the average of the interactions with the interaction of the averages. This replacement corresponds to the assumption that the spins are statistically independent. But, in real markets spins are not independent, and it is this lack of independence that makes modeling financial data interesting. It is what can give agent-based models an upper hand over mean field models. There are ways mean field models can work with agents as entities, and there are ways agent-based models can generate the global features of markets. For mean field models increasing the dimensionality of the mean field allows it to (reasonably) reproduce the granularity of agent-based models. When the dimensions of the mean field model are extended, however, a significant problem arises. The mean field model, be it in a planar or nonplanar5 form, is NP-complete.6 As [5] writes, ‘‘New results show that the computational barrier lies not so much in the extra dimension as in the nonplanarity7 of the essential underlying graph,8 which explains why physicists have been stymied even in certain two-dimensional generalizations of the Ising model. Although it does not completely put the kibosh on the search for exact solutions (for one thing, the P-versus-NP question is still famously open), recent work sheds new light on the likely limitations of techniques that, because of their success in the plane, had theorists chasing wild geese into the third dimension’’. Onsager did derive a closed-form solution of the 2-d model when there is no external field, but there is no closed-form solution when the external field does exist in the 2-d planar model or when the model is 3-d or higher. For the 2-d planar problem it has been shown that arriving at a solution is tantamount to solving the NP-complete problem of finding the largest set of vertices in a planar, degree-3 graph with no two vertices in the set connected by an edge. So, while dimensions can be added to a mean field, the computational time could rise significantly. NP-completeness, however, does not mean things are completely hopeless. The complexity result Cipra discusses bars algorithms only from solving all instances of the problem in polynomial time. To add granularity to the heat-bath model the first dimension could be fundamentalists and the second technical traders. Each would be operating in an environment where an external field is present, and that field would be the same for each investor. Fundamentalists’ orientation, at least to start with, will always generate a near-equilibrium state, i.e., for them any change in price will always correspond to x ∼ 0, reflecting their belief that prices will always return to their fundamental value. For technical traders the average reaction to price changes will vary. If x(t ) ∝ O(h), technical traders will have no preferred orientation. If x(t ) ∝ O(1), technical traders will exhibit an ordered state, with a majority of agents sharing one orientation, such as in the case of herding. This ordered state will conflict with the equilibrium state of the fundamentalists
5 In a planar model interactions occur only with nearest neighbors. In a nonplanar model interactions occur (or bonds are formed) between nearest and next nearest neighbors. 6 A problem is NP-complete, non-polynomial complete, if it is in the set of NP problems where any solution to the problem can be verified in polynomial time. The most notable characteristic of NP-complete problems is that no fast solution is known for them. The time required to solve an NP-complete problem using any currently known algorithm can increase quickly as the size of the problem grows. For example, the time to solve a moderately large version of an NP-Complete problem would require years. 7 This criterion includes 2-d models with next nearest neighbor interaction in addition to the nearest neighbor kind. 8 This graph can be characterized as one where vertices represent particles or atoms in a crystal and edges represent the bonds between adjacent atoms. An external field can be represented graphically as an extra vertex with edges to all the vertices.
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and will generate the same results as Eq. (1.19) because the orientation of the technical traders and potentially of the fundamentalists will vary. For the 2-d mean field model to generate results similar to Eq. (1.19) a nonplanar version of the mean field model with an external field is needed. In a planar model no bonds are formed between nearest and next nearest neighbors. But, as was written in Section 1, if β = +∞, in each period all the investors switch to the strategy with the highest fitness in the previous period. If β = 0, the mass of investors distributes itself evenly across all the available strategies. For 0 < β < ∞ a fraction of each group of investors switches to the strategy with the highest fitness in the previous period. This interaction between investor types could lead to bonds being formed between nearest and next nearest neighbors in the 2-d model. Approximate solutions for a 2-d model in the presence of an external field have been developed by Weiss [6], and the reader is referred to his work for details. Another way of untangling the problems related to multi-dimensional mean field models is knot theory. Interested readers are referred to Kauffman [7] for an introduction and Nelson [8] to read about recent advances. To show how an agent-based model can generate global properties of financial markets, we will briefly review the work of Gaunersdorf and Hommes [9]. Their solution involves going from the linear equations of C & H to nonlinear equations, in particular a nonlinear equation that is known to produce any given autocorrelations. Their nonlinear difference equation to generate autocorrelations is:
(xt , yt ) = (a1 xt −1 + · · · + aL xt −L + yt −1 , 1 − 2y2t −1 ).
(3.1)
The variable yt follows a chaotic process with zero mean and zero autocorrelations at all lags. Since yt is generated independently of all past values of xt , the series xt and yt are uncorrelated. The variable xt thus follows a linear AR(L) process driven by a chaotic series with zero autocorrelations at all lags and thus has the desired autocorrelation structure of volatility clustering. To quote Gaunersdorf and Hommes: In this paper we present a simple nonlinear structural model where prices are driven by a combination of exogenous random news about fundamentals and evolutionary forces underlying the trading process itself. Volatility clustering becomes an endogenous variable caused by the interaction between heterogeneous traders, fundamentalists and technical analysts, having different trading strategies and expectations about future prices and dividends [· · ·]. The heterogeneous market is characterized by irregular switching between phases of low volatility when price changes are small and phases of high volatility where small price changes due to random news are reinforced and become large due to trend following trading rules. Volatility clustering is thus driven by heterogeneity and conditional evolutionary learning. Due to the heterogeneity in expectations, our model is a nonlinear dynamical system [· · ·]. Nonlinear dynamic models can generate a wide variety of irregular patterns [· · ·] a nonlinear chaotic model buffeted by dynamic noise with almost no autocorrelation in returns but at the same time persistence in absolute and squared returns provides a structural explanation of the unpredictability of returns and at the same time preserving the structure of the second moment of the return series [volatility]. As Gaunersdorf and Hommes write, their model has technical traders responding to the global phenomenon of price changes, i.e., they have technical traders respond to the market’s own price history. By doing so, if particular price patterns occur, then successive price reactions are triggered that eventually build into an increase or decrease in volatility. Technical traders in Gaunersdorf and Hommes watch and learn from the pricing process – a global phenomenon – which is a pricing process they help create. The action of technical traders then can be compared to the multiplicative noise that amplifies the additive noise of the random news events. As noted by Krawiecki et al. (2000) and Nakao [10], in the presence of small but positive additive noise the basic process, in this case financial prices, will be amplified and attain large values by a positive value of the multiplicative noise. Both authors identify these large values as being nothing else but bursts. By incorporating global phenomena and a nonlinear reaction to the same, Gaunersdorf and Hommes generate volatility clusters with high persistence in absolute and squared returns—the same stylized global phenomenon generated by the heat bath model. The final topic in this section is a look at the bifurcations and the related bifurcation dynamics each model generates: The on–off intermittency examined in Krawiecki et al. (2000) is one that occurs when the system loses stability as a result of a blowout bifurcation. On–off intermittency exhibits dynamics that are confined to the phase space of the original basin of attraction and are characterized by a small positive Lyapunov exponent as well as by all the properties briefly described in Krawiecki et al. (2000). For attractor bubbling to arise a riddled basin is needed. A riddled basin is one where the domain of attraction is a fractal domain that can be observed on one side of the blowout bifurcation, where the transverse Lyapunov exponent is negative (versus positive for on–off intermittency). A riddled basin occurs through a riddled bifurcation.9 And after a supercritical riddling bifurcation occurs, attractor bubbling and local riddling arise.
9 In a riddling bifurcation: (1) the bifurcation is subcritical, (2) the attractor will start to have a riddled basin of attraction and (3) the existence of (1) and (2) will mean that small noisy perturbations will eventually drive the system away from its invariant subspace.
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Krawiecki et al. (2000) conclude that attractor bubbling is the more likely dynamic for the observed power laws and volatility bursts. However, the full dynamics of the heat path process such as phase diagrams and Lyapunov exponents are not available in Krawiecki et al., so we think a case can be made that the bursts and power laws of the heat bath model can be described via noisy on–off intermittency. We say this because if weak additive noise is present in a system, Platt et al. [11] observe that intermittency, which before their work had been observed only in a narrow supercritical region of the blowout bifurcation, can also be present in a wider region, including the subcritical side of the blowout bifurcation. This wider form of intermittency is called noisy on–off intermittency. Platt et al. propose a random multiplicative model (as used in the heat bath model) that incorporates the effect of the additive noise by introducing a lower bound to the standard on–off intermittency model. Nakao [10] demonstrates that noisy on–off intermittency generates power laws and bursts. Noisy on–off intermittency, then, is a possible second candidate that can account for the power laws and bursts observed in the heat bath model. In C & H volatility bursts occur in the presence of additive noise, modeled in their case by adding an IID noise to the existing constant dividend process. Volatility bursts occur in the range of the Hopf bifurcation, which occurs when a ∈ (1.1, 1.2), i.e., the risk aversion of the counterparties is approximately equal and the system is roughly stable.10 Noisy on–off intermittency can occur near the Hopf bifurcation as Hammer et al. [12] demonstrate, and the small positive Lyapunov exponents in the 1.1–1.2 interval (as appears in C & H) appears to confirm noisy on–off intermittency as a possible solution. As C & H write, ‘‘· · · with noise, the solution fluctuates in a quite irregular fashion and exhibits bursts of volatility. This simulation makes the point that non-linear models of the financial market dynamics (even when stable) ‘process’ external noise is far more complicated than is possible with linear models’’. The appeal to nonlinear models is confirmed in the work of Gaunersdorf and Hommes. 4. Conclusions We describe a mean field model as one that relies on system-level averages or agent variables. We examine the Krawiecki et al. heat bath model and show that it can easily reproduce such financial stylized facts as volatility clustering and power laws. And, while we propose another phenomenon to describe the results of Krawiecki et al. – noisy on–off intermittency versus attractor bubbling – this is an issue of interpretation and not one that affects the basic results of the Krawiecki et al. model. As noted above, mean field models have limited accuracy but they provide concise insight into the behavior of a system that is often obscured by discrete models. They are small, fast models that can quickly produce plausible market-pricing data for long simulation times, they more than likely are sufficient for many financial purposes, and they yield important analytical insights. We also discuss ways that mean field models can approach the granularity of agent-based models using graph theory. While the results of such a model would be approximate, they are approximate because the whole space cannot be searched in a reasonable amount of time, not because the answers in the completed search are approximate. Agent-based models, in particular those of B & H and C & H, which emphasize heterogeneous agent models, are attempts to derive the necessary and sufficient conditions that drive financial markets as well as to identify the factors that drive those markets. The models of C & H and B & H highlight the importance of differing attitudes of risk on securities prices. They also show that the dynamics of asset prices are also affected by the differing risk attitudes of different kinds of investors and the impact of different learning schemes. However, with the C & H and B & H results, because of their high dimensionality it is difficult to distill the salient features of the models into a simpler model based on easily understood principles. An attempt is made in this paper to distill part of the C & H model: the interaction of noise and the associated nonlinear dynamics. It is the author’s hypothesis that there is a closer relation relationship between mean field models and agent based models than some expect. As shown here, mean field models can potentially generate the necessary high level features one is looking for. The results of the mean field model can then be given some level of granularity – the sufficient conditions and factors – via agent-based models. In the author’s view, both models need to be presented in any complete attempt to understand markets or other phenomena where agents are at work and where averaging across agents is expected to produce high level insights. References [1] W. Brock, C. Hommes, A rational route to randomness, Econometrica 65 (5) (1997) 1059–1095. [2] W. Brock, C. Hommes, Heterogenous beliefs and routes to chaos in a simple asset pricing model, Journal of Economic Dynamics and Control (1998) 1235–1274. [3] C. Chiarella, X. He, Heterogeneous beliefs, risk and learning in a simple asset pricing model, Computational Economics (2002) 95–132. [4] A. Krawiecki, J.A. Holyst, D. Helbind, Volatility clustering and scaling of financial time series due to attractor bubbling, Physical Review Letters (2002). [5] B.A. Cipra, The Ising model is np-complete, SIAM News 33 (6) (2000).
10 a, the ratio of fundamental risk aversion to trend followers’ risk aversion, is modeled by C & H more often than not in the range of 0.2–1.5.
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[6] Y. Weiss, Comparing the mean field method and belief propagation for approximate inference in MRF’s, in: D. Saad, M. Opper (Eds.), Advanced Mean Field Theory: Theory and Practice, MIT Press, 2001. [7] L.H. Kauffman, Knots and Physics, World Scientific, 2001. [8] S. Nelson, The combinatorial revolution in knot theory, Notices of the American Mathematical Society (2012). [9] A. Gaunersdorf, C. Hommes, A Nonlinear Structural Model for Volatility Clustering, SSRN, 2000. [10] H. Nakao, Asymptotic power law of moments in a random multiplicative process with additive noise, Physical Review E (1998). [11] N. Platt, S.M. Hammel, J.F. Heagy, Effects of additive noise on on–off intermittency, Physical Review Letters (1994). [12] P.W. Hammer, N. Platt, S.M. Hammel, J.F. Heagy, B.D. Lee, Experimental observation of on–off intermittency, Physical Review Letters (1994).
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Heat baths and computational agent-based models Andrew Clark ∗ Lipper, A Thomson Reuters Company, Andrew Clark/Lipper, Thomson Reuters, 707 17th Street, 80202 Denver, CO, United States
article
info
Article history: Received 14 November 2009 Received in revised form 1 June 2012 Available online 15 June 2012 Keywords: Mean field models Agent based models Volatility Power laws Bifurcations
abstract In this paper, we examine an agent-based model, and an equation-based model in the form of a mean field model. We show how the mean field model is a small, fast model that identifies the high level properties of a subject, in this case financial time series’ stylized facts. The agent based model generates the granularity needed to understand the conditions and factors that generate the stylized financial facts. We conclude with the recommendation that both models be used in sequence so a complete description of a process be established or approximated. © 2012 Elsevier B.V. All rights reserved.
1. Computational agent-based models 1.1. The evolutionary dynamics of adaptive belief systems For agent-based models this paper focuses on the adaptive rational equilibrium modeling first developed by Brock and Hommes [1,2] and used by Chiarella and He [3]. Brock and Hommes (hereafter, B & H) developed what they call ‘‘adaptive belief systems’’ to model heterogeneous expectations in a way that naturally mimics real market activity. In adaptive belief systems agents (investors) adapt their prediction of asset prices by choosing a finite number of predictors or expectation functions, which are a function of past price performance. Each predictor has a performance measure that is visible to all participants. Based on the performance measure, agents make a (bounded) rational choice between price predictors. This assumption of bounded rational choice results in an adaptive rational equilibrium dynamic (ARED), which generates a dynamic across predictor choices that is coupled to the dynamics of the endogenous variables. B & H [1] show that ARED incorporates a general mechanism that can generate local instability in the equilibrium steady state and complicated global equilibrium dynamics. B & H apply the concept of adaptive belief systems to a simple asset-pricing model, where traders in a financial market use different types of predictors for the price forecasts of risky assets. Chiarella and He [3] extend B & H’s work to include agents with different risk aversion as well as expected dividend payments with noise. What follows in this section is the initial setup of an adaptive belief system as used by Chiarella and He (hereafter, C & H). To define investor wealth it is assumed the risk-free asset is perfectly elastic and supplied at a gross return R > 1. Let pt be the price (ex dividend) per share of the risky asset at time t, and let (yt ) be the stochastic dividend process of the risky asset. Then, investor wealth at t + 1 is defined as: Wt +1 = RWt + (pt +1 + yt +1 − Rpt ) zt
(1.1)
where Rpt is the return of the risk-free asset at time t, and zt is the number of shares of the risky asset purchased at time t.
∗
Tel.: +1 3039416017. E-mail address: [email protected].
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.06.011
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In both B & H’s and C & H’s work a Walrasian auctioneer is used to derive the demand equation, i.e., each trader is viewed as a price taker. The market is viewed as finding the price pt that equates the sum of the demand schedules to the supply. That is, the price pt at time t is formed by using information available at time t − 1 and the expected utility for time t + 1. Denote by: Ft = (pt , pt −1 , . . . , yt , yt −1 , . . .)
(1.2)
the information set formed at time t, where p is the price that equates the sum of the demand schedule to the supply of the asset and y is the dividend yield. Now, denote by Rt +1 the excess return on the risky asset at time t + 1: Rt +1 = (pt +1 + yt +1 − Rpt ) .
(1.3)
It follows from Eqs. (1.2) and (1.3) that the conditional expectation Et and conditional variance Vt can be defined as: Eht (Wt +1 ) = RWt + Eht (pt +1 + yt +1 − Rpt ) zt = RWt + Eht (Rt +1 )zt Vht (Wt +1 ) =
zt2 Vht
(pt +1 + yt +1 − Rpt ) =
zt2 Vht
(1.4)
(Rt +1 )
(1.5)
where the subscript h refers to the beliefs of investor type h about the conditional mean and variance. Note that in Eq. (1.5) the conditional variance of wealth Wt +1 equals zt2 times the conditional variance of excess return per share pt +1 + yt +1 − Rpt . This differs from B & H [2], who assume the beliefs about conditional variance of excess returns are constant and the same for everyone, i.e., Vht (pt +1 + yt +1 − Rpt ) ≡ σ 2 . C & H assume each type of investor is a mean-variance maximizer with different attitudes toward risk. Each investor type h has a risk aversion coefficient ah . Given ah , for each investor type the demand for shares zht solves:
max = Eh,t (Wt +1 ) − z
ah 2
Vh,t (Wt +1 )
(1.6)
or zh,t =
Eh,t (Rt +1 ) ah Vh,t (Rt +1 )
.
(1.7)
Let zst be the supply of shares and nht the fraction of type h investors at t. Using Eq. (1.7), the equilibrium state of supply equaling demand is described by:
nh , t − 1
h
Eh,t (Rt +1 ) ah Vh,t (Rt +1 )
= zst .
(1.8)
Now, assume a zero supply of outside shares, zst = 0, then (1.8) leads to:
nh , t − 1
h
Eh,t (Rt +1 ) ah Vh,t (Rt +1 )
= 0.
(1.9)
Eq. (1.9) makes it appear that the effect of the risk-aversion coefficients is to rescale the nh,t −1 functions. This will be found to not be the case once Eq. (1.12) below is introduced. To get a notion of the rational expectation (RE) fundamental solution p∗t , consider the equation: Rp∗t = Et (p∗t +1 + yt +1 ) where Et is the expectation conditional on the information set Ft (see Eq. (1.2) above). To satisfy the ‘‘no-bubbles’’ version of the rational expectations B & H and C & H are using, the only solution can be: p¯ =
y¯
.
R+1 If we let xt denote the deviation of pt from the RE fundamental p∗t , then: xt = pt − p∗t . After establishing this relationship C & H give the equations for heterogeneous beliefs for returns and variance, i.e., the different classes of beliefs that are deviations from the fundamental. Both B & H and C & H use a fitness function, which is defined by the realized profits of investor type h:
πh,t = Rt +1 zh,t zh,t =
Eh,t (Rt +1 ) ah Vh,t (Rt +1 )
.
(1.10)
Memory can be introduced to the fitness function by considering a weighted average of past profits: Uh,t = πh,t + ηUh,t −1 where η represents the memory strength.
(1.11)
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Let the updating fractions be given by the discrete choice probability: nh,t = exp(β Uh,t −1 /Zt ) Zt =
(1.12)
exp(β Uh,t −1 )
h
where β > 0 is the intensity of choice, measuring how fast agents switch between different prediction strategies. In particular, β = +∞ means that in each period all the investors switch to the strategy with the highest fitness in the previous period, while β = 0 means most investors distribute themselves evenly across all the available strategies. For 0 < β < ∞ a fraction of each group of investors switches to the strategy with the highest fitness in the previous period. To summarize, the evolutionary dynamics described by the adaptive belief system are:
h
nh,t −1 fh,t ah (σ 2 +gh,t )
h
nh,t −1 ah (σ 2 +gh,t )
RXt =
(1.13)
where fht is the agent-specific predictive component of the mean for investor type h, and ght is the agent-specific predictive component of the variance for investor type h. σ 2 is the constant predicted variance used in B & H, while in C & H it is the fundamental or underlying variance that is modified by ght . The updating of fractions, fitness function, and memory strength are given by: nh,t = exp(β Uh,t −1 )/Zt Zt =
exp(β Uh,t −1 )
h
(1.14)
Uh,t = πh,t + ηUh,t −1
πh,t = Rt +1 zh,t = (xt −1 − Rxt + δt +1 )
fh,t − Rxt ah (σ 2 + gh,t )
.
1.2. Dynamics of fundamentalist, trend, and contrarian investors To investigate the role of heterogeneous belief types (investors with different beliefs) assume all beliefs follow a linear return and a nonlinear variance learning process: xt =
σ2 =
L 1
L i=1
xt −i (1.15)
L 1
L i=1
(xt −i − xt )2
and let:
υh (x) = µ 1 −
1
(1 + x) − ξ
(1.16)
where L (the number of lags or the length of the investor’s memory) is a positive integer and µ, ξ > 0 are constants. Assume further that: fh,t = dh,t xt
(1.17)
where dh is the trend of investor type h. Call h a pure trend chaser if dh > 0 (a strong trend chaser if dh > R) and a contrarian if dh < 0 (and a strong contrarian if dh < −R). When dh = 0, Eq. (1.17) yields the belief of fundamentalists, i.e., investors who believe prices return to their fundamental value. Using Eqs. (1.15)–(1.17), letting η = 0, δt = 0, a = a2 /a1 and mt = n1,t − n2,t , C & H derive the following equations1 :
(1 + υh (σ¯ 2 ))1 − mt −1 xt R a(1 + mt −1 ) + (1 + υh (σ¯ t2−1 ))(1 − mt −1 ) β Rxt −1 dxt −1 Rxt −1 βC mt = tanh + − . (Rxt −1 − xt ) 2a1 σ 2 a 2 1 + υh (σ¯ t2 ) xt =
d
1 See C & H, Section 3, for the full development.
(1.18)
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For the special case L = 1 Eqs. (1.15)–(1.16) generate: xt = xt −1
σ2 = 0 νh σ¯ 2 = 0
(1.19)
and the following system holds: xt =
1 − m t −1
d
R a + 1 + (a − 1)mt −1
mt = tanh
x t −1
β dxt −2 − Rxt −1 βC ( Rx − x ) Rx + − t − 1 t t − 1 2a1 σ 2 a 2
(1.20)
where a = a2 /a1 and mt = n1,t − n2,t . The lemma below (Lemma 3.1 in C & H and proved in their appendix) is used to establish the existence and the stability of equilibrium:
Lemma. Let meq = tanh −
tanh
βC 2
, m∗ = 1 −
2aR , d+R(a−1)
and x∗ be the positive solution (if it exists) of
β d−R βC ∗ 2 ( R − 1 ) R + ( x ) − = m∗ . 2σ 2 a1 a 2
(1.21)
• For 0 < d < R, E1 = (0, meq ) is the unique globally steady state of Eq. (1.18). • For R < d < (a + 1)R there are two possibilities: ◦ If m∗ < meq , then E1 is the unique globally steady state of Eq. (1.18). ◦ If m∗ < meq , then (1.18) has three steady states E1 , E2 , E3 ; E1 is unstable. • For d > (a + 1)R, Eq. (1.18) has three steady states E1 , E2 = (x∗ , m∗ ), E3 ; E1 is unstable. The lemma indicates that when trend chasers extrapolate only weakly (0 < d < R), the fundamental steady state E1 is globally stable no matter what risk aversion investors have. When d > R, the stability of the fundamental steady state E1 depends on the ratio a, which measures the relative risk aversion. When trend chasers extrapolate strongly, (d > (a + 1) R), the fundamental steady state E1 becomes unstable and bifurcates into two additional nonzero steady states E2 , and E3 . In the case where R < d < (a + 1) r the fundamental steady state is stable when a is large, i.e., when trend chasers become more risk averse than fundamentalists. (Statements made about trend chasers and fundamentalists hold for contrarians and fundamentalists simply by reversing the inequality signs.) And when a = 1, the results of B & H [2] are recovered. 2. Heat baths Before we develop the heat bath model, we will present a brief introduction to mean field models. Discussing these types of models will deepen the reader’s understanding of the heat bath model as well as make it easier in Section 3 to compare agent-based and heat bath models. In statistical physics systems of many interacting particles – such as the agents described in Section 1 – are analytically intractable because they tend not to have or do not have closed-form solutions. To achieve a closed-form solution, physicists use a single particle and estimate the influence of all the other particles on it as an average influence. The resulting model is called a mean field model because it replaces many individual entities with their average. Ising models are a canonical example of mean fields. In an Ising model there are discrete variables (spins) that can be in one of two states, more often than not ±1. The spins are arranged in a lattice, and each spin interacts at most with its nearest neighbors. Mean field theory makes possible the solution of the Ising model analytically by replacing all of its interactions with an average or effective interaction. This reduces the multi-body (the collection of individual agents) problem to a onebody problem. The relative ease of solving mean field problems means that insights into the behavior of the system can be obtained at a reasonable cost, especially when compared to the computational cost of agent-based models. A mean field model can be characterized as centralized because variables require global information. Because they require global information, mean field models provide valuable information from a global perspective, e.g., the volatility clustering often seen in financial return series. In contrast, the natural tendency of the agent-based models is to define agent behavior in terms of observables accessible to the individual agent, thereby favoring a local versus global viewpoint. The evolution of system-level observables does emerge from both C & H’s and B & H’s work, but they do not use these observables to drive their models’ dynamics. Therefore, the B & H and C & H models do not generate global stylized facts such as volatility clustering.2
2 In Section 3 the author will discuss how agent-based models can reproduce volatility clusters.
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The heat bath model of Krawiecki et al. [4] has I = 1, 2, . . . , N agents (spins) with orientations σi (t ) = ±1, which correspond to the decision to sell (−1) or to buy (+1) a share of stock at discrete time steps t. The orientation of agent i at time t depends on the local field: Ii ( t ) =
N 1
N j =1
Aij (t )σj (t ) + hi (t )
(2.1)
where Aij are time-dependent interaction strengths among agents and hi (t ) is an external field reflecting the effect of the environment (e.g., access to external information, which can differ between agents3 ). The interaction strengths and external fields change randomly in time. In Eq. (2.2): Aij (t ) = Aξ (t ) + aηij (t ) hi (t ) = hςi (t )
(2.2)
where ξ (t ), ηij′ (t ) and ζi (t ) are random variables with no correlation in either space or time and are uniformly distributed along the interval (−1, 1). A is a measure of the randomly varying average interaction strength between agents, a is a measure of the random deviations of the individual interaction strengths from the average, and h is a measure of the random influence of the environment. The strengths Aij (t ) can assume both positive and negative values corresponding to a tendency to imitate or avoid the orientation of the players in the market. Krawiecki et al. demonstrate that the noises ξ (t ) and ζi (t ) are necessary to get real market results such as volatility clustering. The orientation of all the agents are updated synchronously according to the discrete choice probability σi (t + 1) = 1 with probability p or −1 with probability 1 − p. The uncertainty in the investment decision (where p = 1/{1 + exp[−2Ii (t )]}) is analogous to heat bath dynamics. In terms of modeling the time series and distribution of financial returns, Krawiecki et al. make the (common) assumption that the relative change in a security’s price is proportional to the difference between the number of buying and selling decisions and use the following formula where the security’s returns are the forward changes in the logarithms of prices P (t ) over a time scale ∆t : G∆t (t ) = ln P (t ) − ln P (t − ∆t ). N By introducing an average orientation x(t ) = N −1 τ =0 x(t − τ ) of the agents, they get dP /dt ∝ xt. Discretizing time in dP /dt ∝ xt yields G∆t (t ) ∝
∆ t −1
x(t − τ )
(2.3)
τ
where G∆t (t ) is the return of the security. G∆t (t ) allows us to interpret Eqs. (2.1) and (2.2) in the following manner: since the average interaction strength Aξ (t ) is common to all connections, it is a measure of the average reaction of agents to price changes, influencing their decisions via the term Aξ (t )x(t ) in the mean field Ii (t ). The term aηij (t ) describes the fluctuating interaction network and hζ (t ) the fluctuating environment. The local field in Eq. (2.1) describes how agent i anticipates price changes by averaging the information of the opinions of his interaction partners and the external information accessible to him. A mean field approximation exists for the Krawiecki et al. model, where x(t ) can be reduced to a 1-d map. This mean field model allows Krawiecki et al. to generate time series that resemble the stylized facts of empirical price returns, such as large bursts of volatility and their corresponding volatility clustering, as well as power laws. The mean field approximation sets are A ̸= 0, a = 0, and ςi (t ) = ς (t ) for all i. The dynamics of x(t ) become: x(t + 1) = tanh[Aξ (t ) + hς (t )] ≈ Aξ (t )x(t ) + hς (t )
(2.4)
where the approximate equality holds for ⌊Aξ (t )x(t ) + hζ (t )⌋ ≪ 1. Eq. (2.4) is a generic model for on–off intermittency if h = 0 and for attractor bubbling if h > 0. The approximate linear form of Eq. (2.4) belongs to a more general class of stochastic processes with multiplicative noise. Although the Krawiecki et al.-generated time series results given Eq. (2.4) could be either on–off intermittency or attractor bubbling, there is a qualitative difference between these phenomena, write Krawiecki et al. In the case of on–off intermittency the bursts are caused solely by the multiplicative noise ξ (t ) when its amplitude A exceeds the blowout bifurcation4 threshold. In the case of attractor bubbling the multiplicative noise amplifies only to macroscopic sizes the small deviations from the fixed point caused by the additive noise. Krawiecki et al. suggest – by visual inspection – the attractor bubbling scheme bears the best resemblance to the empirical stylized facts. This is because it has frequent large bursts of volatility and volatility clustering, separated by long-lasting laminar phases when the map of Eq. (2.3) is governed mainly by the additive noise hζ (t ).
3 This can be the presence of asymmetric information or uncertainty about future dividend yields. 4 A blowout bifurcation takes place when the Lyapunov exponent in the transversal direction of the invariant subspace changes from negative to positive.
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Krawiecki et al.’s mean field results, the laminar phases with x ∼ 0, correspond to the disordered phase with no preferred orientation (the system is near equilibrium). x ∝ O(1) bursts occur and these correspond to the ordered phase with a majority of agents sharing one orientation, such as when herding occurs and financial arbitrage becomes possible. To summarize the Krawiecki et al. heat bath model, the mean field model quantitatively reflects the multi-agent dynamics. The necessary conditions for the applicability of the mean field model are (1) random fluctuations in time of the average interaction strength between agents, corresponding to their average reaction to price changes; (2) an uncertainty in decision making analogous to heat bath dynamics; and (3) in the thermodynamic limit small additive noise, simulating the effect of the external environment. When these conditions are used to simulate financial prices, stylized facts such as volatility clustering and the power-law distribution of returns appear. The volatility clustering, power-law distributions, and other phenomena are shown to be the result of attractor bubbling. 3. Mean field and agent-based modeling A difference between agent-based modeling and mean field models is that agent-based modeling directs the modeler’s attention to the individual entities in the domain, while mean field models focus on variables. These variables may be extensive (e.g., the total population or agent density) or intensive (e.g., parameters of individual agents). Intensive variables may reflect individual agents or (as averages) across the system as a whole. The Krawiecki et al. mean field model works with averages, while the C & H model focuses on the activity of individuals in the system. The heat bath model can be characterized as centralized, since its variables require global information. It provides valuable information from a global perspective, e.g., volatility clustering, but misses the information contained in agent-based entities. The Krawiecki et al. model neglects the interaction between spins. To quote from Krawiecki et al., ‘‘The details of the interactions among pairs of agents, and the details of the variation of the mean interaction strength seem unimportant for the qualitative result’’. The heat bath model replaces the average of the interactions with the interaction of the averages. This replacement corresponds to the assumption that the spins are statistically independent. But, in real markets spins are not independent, and it is this lack of independence that makes modeling financial data interesting. It is what can give agent-based models an upper hand over mean field models. There are ways mean field models can work with agents as entities, and there are ways agent-based models can generate the global features of markets. For mean field models increasing the dimensionality of the mean field allows it to (reasonably) reproduce the granularity of agent-based models. When the dimensions of the mean field model are extended, however, a significant problem arises. The mean field model, be it in a planar or nonplanar5 form, is NP-complete.6 As [5] writes, ‘‘New results show that the computational barrier lies not so much in the extra dimension as in the nonplanarity7 of the essential underlying graph,8 which explains why physicists have been stymied even in certain two-dimensional generalizations of the Ising model. Although it does not completely put the kibosh on the search for exact solutions (for one thing, the P-versus-NP question is still famously open), recent work sheds new light on the likely limitations of techniques that, because of their success in the plane, had theorists chasing wild geese into the third dimension’’. Onsager did derive a closed-form solution of the 2-d model when there is no external field, but there is no closed-form solution when the external field does exist in the 2-d planar model or when the model is 3-d or higher. For the 2-d planar problem it has been shown that arriving at a solution is tantamount to solving the NP-complete problem of finding the largest set of vertices in a planar, degree-3 graph with no two vertices in the set connected by an edge. So, while dimensions can be added to a mean field, the computational time could rise significantly. NP-completeness, however, does not mean things are completely hopeless. The complexity result Cipra discusses bars algorithms only from solving all instances of the problem in polynomial time. To add granularity to the heat-bath model the first dimension could be fundamentalists and the second technical traders. Each would be operating in an environment where an external field is present, and that field would be the same for each investor. Fundamentalists’ orientation, at least to start with, will always generate a near-equilibrium state, i.e., for them any change in price will always correspond to x ∼ 0, reflecting their belief that prices will always return to their fundamental value. For technical traders the average reaction to price changes will vary. If x(t ) ∝ O(h), technical traders will have no preferred orientation. If x(t ) ∝ O(1), technical traders will exhibit an ordered state, with a majority of agents sharing one orientation, such as in the case of herding. This ordered state will conflict with the equilibrium state of the fundamentalists
5 In a planar model interactions occur only with nearest neighbors. In a nonplanar model interactions occur (or bonds are formed) between nearest and next nearest neighbors. 6 A problem is NP-complete, non-polynomial complete, if it is in the set of NP problems where any solution to the problem can be verified in polynomial time. The most notable characteristic of NP-complete problems is that no fast solution is known for them. The time required to solve an NP-complete problem using any currently known algorithm can increase quickly as the size of the problem grows. For example, the time to solve a moderately large version of an NP-Complete problem would require years. 7 This criterion includes 2-d models with next nearest neighbor interaction in addition to the nearest neighbor kind. 8 This graph can be characterized as one where vertices represent particles or atoms in a crystal and edges represent the bonds between adjacent atoms. An external field can be represented graphically as an extra vertex with edges to all the vertices.
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and will generate the same results as Eq. (1.19) because the orientation of the technical traders and potentially of the fundamentalists will vary. For the 2-d mean field model to generate results similar to Eq. (1.19) a nonplanar version of the mean field model with an external field is needed. In a planar model no bonds are formed between nearest and next nearest neighbors. But, as was written in Section 1, if β = +∞, in each period all the investors switch to the strategy with the highest fitness in the previous period. If β = 0, the mass of investors distributes itself evenly across all the available strategies. For 0 < β < ∞ a fraction of each group of investors switches to the strategy with the highest fitness in the previous period. This interaction between investor types could lead to bonds being formed between nearest and next nearest neighbors in the 2-d model. Approximate solutions for a 2-d model in the presence of an external field have been developed by Weiss [6], and the reader is referred to his work for details. Another way of untangling the problems related to multi-dimensional mean field models is knot theory. Interested readers are referred to Kauffman [7] for an introduction and Nelson [8] to read about recent advances. To show how an agent-based model can generate global properties of financial markets, we will briefly review the work of Gaunersdorf and Hommes [9]. Their solution involves going from the linear equations of C & H to nonlinear equations, in particular a nonlinear equation that is known to produce any given autocorrelations. Their nonlinear difference equation to generate autocorrelations is:
(xt , yt ) = (a1 xt −1 + · · · + aL xt −L + yt −1 , 1 − 2y2t −1 ).
(3.1)
The variable yt follows a chaotic process with zero mean and zero autocorrelations at all lags. Since yt is generated independently of all past values of xt , the series xt and yt are uncorrelated. The variable xt thus follows a linear AR(L) process driven by a chaotic series with zero autocorrelations at all lags and thus has the desired autocorrelation structure of volatility clustering. To quote Gaunersdorf and Hommes: In this paper we present a simple nonlinear structural model where prices are driven by a combination of exogenous random news about fundamentals and evolutionary forces underlying the trading process itself. Volatility clustering becomes an endogenous variable caused by the interaction between heterogeneous traders, fundamentalists and technical analysts, having different trading strategies and expectations about future prices and dividends [· · ·]. The heterogeneous market is characterized by irregular switching between phases of low volatility when price changes are small and phases of high volatility where small price changes due to random news are reinforced and become large due to trend following trading rules. Volatility clustering is thus driven by heterogeneity and conditional evolutionary learning. Due to the heterogeneity in expectations, our model is a nonlinear dynamical system [· · ·]. Nonlinear dynamic models can generate a wide variety of irregular patterns [· · ·] a nonlinear chaotic model buffeted by dynamic noise with almost no autocorrelation in returns but at the same time persistence in absolute and squared returns provides a structural explanation of the unpredictability of returns and at the same time preserving the structure of the second moment of the return series [volatility]. As Gaunersdorf and Hommes write, their model has technical traders responding to the global phenomenon of price changes, i.e., they have technical traders respond to the market’s own price history. By doing so, if particular price patterns occur, then successive price reactions are triggered that eventually build into an increase or decrease in volatility. Technical traders in Gaunersdorf and Hommes watch and learn from the pricing process – a global phenomenon – which is a pricing process they help create. The action of technical traders then can be compared to the multiplicative noise that amplifies the additive noise of the random news events. As noted by Krawiecki et al. (2000) and Nakao [10], in the presence of small but positive additive noise the basic process, in this case financial prices, will be amplified and attain large values by a positive value of the multiplicative noise. Both authors identify these large values as being nothing else but bursts. By incorporating global phenomena and a nonlinear reaction to the same, Gaunersdorf and Hommes generate volatility clusters with high persistence in absolute and squared returns—the same stylized global phenomenon generated by the heat bath model. The final topic in this section is a look at the bifurcations and the related bifurcation dynamics each model generates: The on–off intermittency examined in Krawiecki et al. (2000) is one that occurs when the system loses stability as a result of a blowout bifurcation. On–off intermittency exhibits dynamics that are confined to the phase space of the original basin of attraction and are characterized by a small positive Lyapunov exponent as well as by all the properties briefly described in Krawiecki et al. (2000). For attractor bubbling to arise a riddled basin is needed. A riddled basin is one where the domain of attraction is a fractal domain that can be observed on one side of the blowout bifurcation, where the transverse Lyapunov exponent is negative (versus positive for on–off intermittency). A riddled basin occurs through a riddled bifurcation.9 And after a supercritical riddling bifurcation occurs, attractor bubbling and local riddling arise.
9 In a riddling bifurcation: (1) the bifurcation is subcritical, (2) the attractor will start to have a riddled basin of attraction and (3) the existence of (1) and (2) will mean that small noisy perturbations will eventually drive the system away from its invariant subspace.
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Krawiecki et al. (2000) conclude that attractor bubbling is the more likely dynamic for the observed power laws and volatility bursts. However, the full dynamics of the heat path process such as phase diagrams and Lyapunov exponents are not available in Krawiecki et al., so we think a case can be made that the bursts and power laws of the heat bath model can be described via noisy on–off intermittency. We say this because if weak additive noise is present in a system, Platt et al. [11] observe that intermittency, which before their work had been observed only in a narrow supercritical region of the blowout bifurcation, can also be present in a wider region, including the subcritical side of the blowout bifurcation. This wider form of intermittency is called noisy on–off intermittency. Platt et al. propose a random multiplicative model (as used in the heat bath model) that incorporates the effect of the additive noise by introducing a lower bound to the standard on–off intermittency model. Nakao [10] demonstrates that noisy on–off intermittency generates power laws and bursts. Noisy on–off intermittency, then, is a possible second candidate that can account for the power laws and bursts observed in the heat bath model. In C & H volatility bursts occur in the presence of additive noise, modeled in their case by adding an IID noise to the existing constant dividend process. Volatility bursts occur in the range of the Hopf bifurcation, which occurs when a ∈ (1.1, 1.2), i.e., the risk aversion of the counterparties is approximately equal and the system is roughly stable.10 Noisy on–off intermittency can occur near the Hopf bifurcation as Hammer et al. [12] demonstrate, and the small positive Lyapunov exponents in the 1.1–1.2 interval (as appears in C & H) appears to confirm noisy on–off intermittency as a possible solution. As C & H write, ‘‘· · · with noise, the solution fluctuates in a quite irregular fashion and exhibits bursts of volatility. This simulation makes the point that non-linear models of the financial market dynamics (even when stable) ‘process’ external noise is far more complicated than is possible with linear models’’. The appeal to nonlinear models is confirmed in the work of Gaunersdorf and Hommes. 4. Conclusions We describe a mean field model as one that relies on system-level averages or agent variables. We examine the Krawiecki et al. heat bath model and show that it can easily reproduce such financial stylized facts as volatility clustering and power laws. And, while we propose another phenomenon to describe the results of Krawiecki et al. – noisy on–off intermittency versus attractor bubbling – this is an issue of interpretation and not one that affects the basic results of the Krawiecki et al. model. As noted above, mean field models have limited accuracy but they provide concise insight into the behavior of a system that is often obscured by discrete models. They are small, fast models that can quickly produce plausible market-pricing data for long simulation times, they more than likely are sufficient for many financial purposes, and they yield important analytical insights. We also discuss ways that mean field models can approach the granularity of agent-based models using graph theory. While the results of such a model would be approximate, they are approximate because the whole space cannot be searched in a reasonable amount of time, not because the answers in the completed search are approximate. Agent-based models, in particular those of B & H and C & H, which emphasize heterogeneous agent models, are attempts to derive the necessary and sufficient conditions that drive financial markets as well as to identify the factors that drive those markets. The models of C & H and B & H highlight the importance of differing attitudes of risk on securities prices. They also show that the dynamics of asset prices are also affected by the differing risk attitudes of different kinds of investors and the impact of different learning schemes. However, with the C & H and B & H results, because of their high dimensionality it is difficult to distill the salient features of the models into a simpler model based on easily understood principles. An attempt is made in this paper to distill part of the C & H model: the interaction of noise and the associated nonlinear dynamics. It is the author’s hypothesis that there is a closer relation relationship between mean field models and agent based models than some expect. As shown here, mean field models can potentially generate the necessary high level features one is looking for. The results of the mean field model can then be given some level of granularity – the sufficient conditions and factors – via agent-based models. In the author’s view, both models need to be presented in any complete attempt to understand markets or other phenomena where agents are at work and where averaging across agents is expected to produce high level insights. References [1] W. Brock, C. Hommes, A rational route to randomness, Econometrica 65 (5) (1997) 1059–1095. [2] W. Brock, C. Hommes, Heterogenous beliefs and routes to chaos in a simple asset pricing model, Journal of Economic Dynamics and Control (1998) 1235–1274. [3] C. Chiarella, X. He, Heterogeneous beliefs, risk and learning in a simple asset pricing model, Computational Economics (2002) 95–132. [4] A. Krawiecki, J.A. Holyst, D. Helbind, Volatility clustering and scaling of financial time series due to attractor bubbling, Physical Review Letters (2002). [5] B.A. Cipra, The Ising model is np-complete, SIAM News 33 (6) (2000).
10 a, the ratio of fundamental risk aversion to trend followers’ risk aversion, is modeled by C & H more often than not in the range of 0.2–1.5.
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[6] Y. Weiss, Comparing the mean field method and belief propagation for approximate inference in MRF’s, in: D. Saad, M. Opper (Eds.), Advanced Mean Field Theory: Theory and Practice, MIT Press, 2001. [7] L.H. Kauffman, Knots and Physics, World Scientific, 2001. [8] S. Nelson, The combinatorial revolution in knot theory, Notices of the American Mathematical Society (2012). [9] A. Gaunersdorf, C. Hommes, A Nonlinear Structural Model for Volatility Clustering, SSRN, 2000. [10] H. Nakao, Asymptotic power law of moments in a random multiplicative process with additive noise, Physical Review E (1998). [11] N. Platt, S.M. Hammel, J.F. Heagy, Effects of additive noise on on–off intermittency, Physical Review Letters (1994). [12] P.W. Hammer, N. Platt, S.M. Hammel, J.F. Heagy, B.D. Lee, Experimental observation of on–off intermittency, Physical Review Letters (1994).