Consensus and Market Models:
Stylized facts in a Sznajd World
Lorenzo Sabatelli1,2 and Peter Richmond1
1
2
Department of Physics, Trinity College Dublin, Ireland
Hibernian Investment Managers, IFSC, Dublin, Ireland
E-mail: sabatell@tcd.ie
Collective behaviour in markets
• Trends
• Crashes
We also know that:
• Traders talk to each other (sometimes through third parties…)
• Traders have similar constraints ( market regulation)
• Traders avail of common sources of information (press,
databases, consultant reports, fundamental economic factors and models)
Consensus
Price changes depending on the difference between demand and supply
Demand and supply depend on
1.
number of agents keen on buying or selling
2.
amount of an asset individually traded by agents
Here we focus on
•
•
number of buyers and sellers
why they buy or sell
We model these features in terms of
Consensus dynamics
Sznajd Consensus Model:
‘United we stand, divided we fall’
Abraham Lincoln
Within human societies, it is generally easier to change someone‘s
opinion by acting within from within a group than by acting alone.
Place agents (spins) on 2-d lattice:
• Each site carries a spin S that may be up or
down (two possible opinions, yes or no)
• Two neighbouring parallel spins are able to
convince other neighbours of their opinion.
• If these two neighbours are not parallel or in step, they have no
influence on their neighbours.
Sznajd Model and synchronous updating
Allowing system to evolve via
–
–
random sequential updating
always leads to total orientation of spins (total consensus)
We choose a synchronous updating mechanism:
It admits possibility of contradictory information.
Effects of single interaction may last for a certain time
In Sznajd model, synchronous updating carries new feature
frustration
Frustration
• Parallel pairs of spins induce
neighbours to turn to their same state.
• However single spin may often belong,
simultaneously, to neighbourhood of more
than one couple of like-oriented spins
– up to 4, on 2D lattice
• If neighbouring pairs have different
orientations, state of individual spin does
not change -> Frustration
?
ΔP is absolute
difference of
opinions at t=0
ΔPc
Phase transitions
in a synchronously updated Sznajd Model
1  M c ~ 1/ L
N  N
M
N  N _
Variation on a log-log scale of 1-Mc with lattice size, L, in the absence of noise or zero
temperature. When the initial net magnetisation, M(0)>Mc the system evolves to complete
consensus (M(equilibrium)=1). When the initial net magnetisation, M(0)<Mc the system
never reaches complete consensus (M(equilibrium )<1).
Introducing random noise
(random change of individual opinions)
Non-monotonic Magnetization
Microscopic interpretation
Spins surrounded by four others of the same sign, so forming cross-shaped islands, form a
stable state. At zero temperature, they are prevented, at any time, from flipping by
frustration.
Random noise acts flipping one of the five spins, so allowing frustration to be removed.
What does the noise account for ?
• External factor influence
• Degree of ‘connection’ between
neighbouring agents
• Independence of thought
Basic stylized facts…
Minimum requirements for any market model
• Fat tails in the probability density distribution of returns,
volumes and volatility
• (Almost) no time-correlation between the directions of
price fluctuations
• Long lasting time correlation (power law decay) between
the absolute values of price fluctuations.
A Market Model
• Price formation as a Consensus Process
• Noise as a measure of the degree of
connection between agents and
independence of thought.
• Risk perception (measured through
volatility) may change the degree of
connection.
Sznajd-like price formation process
Given N unbiased agents
• p_initial=0.5, random configuration.
• before each transaction they interact for n
time steps (on average).
• Each of them may change opinion
(independently of the others) with
probability q.
Volatility shocks and q changes
• The updating of q may happen either through
deterministic or stochastic rules.
• In both cases q changes depend on the volatility changes
– E.g.: one may use following stochastic rule
q t   
Rt   Bt  1volatility
Bt  1volatility
 t 
 t   N 0,1
Rt   Pr ice _ changet  ~ Magnetizat ion t 
Bt   Bt  1  1    Rt 
Stylized facts in a Sznajd World
Example: for L=50 (2500 agents), p_initial=0.5, alpha=0.98, N_iterations=400
Lower tail
exponent=2.7
Upper tail
exponent=2.6
Lower cut-off
tail exponent=3.9
Upper cut-off
tail exponent=4.1
•(Almost) No-time-correlation between returns
•Significant time-correlation between absolute returns
Conclusions
• Opinion dynamics play role in price formation
• Sznajd consensus model may describe price
formation process
• Noise, as measure of independence of thought
and external influence, plays non trivial role in
determining consensus
• Noise changes driven by Volatility changes may
( partly) explain some relevant stylized facts
observed in financial time series
– fat tails in histograms of returns and volatility
– long memory and clustering of volatility)
Outlook :
What’s next ?
• More realistic topologies for more
realistic models
• Heterogeneous size of transactions
• Non-uniform time duration between
consecutive transactions
• Other stylized facts
References
1.
2.
3.
4.
5.
6.
7.
8.
Mantegna R.N. and Stanley H.E An Introduction to Econophysics, Cambridge
University Press 2000
Johnson N. F., Jefferies P., Huy P.M. Financial Market Complexity. Oxford
Finance 2003
Bouchaud J. P. and Potter M. Theory of Financial Risks : From Statistical
Physics to Risk Management. Cambridge University Press 2000.
Sznajd-Weron, K.; Sznajd, J. Opinion Evolution in Closed Community. Int. J.
Mod. Phys. C 11 1157-1166 (2000).
Stauffer, D. Monte Carlo Simulations of the Sznajd model, Journal of Artificial
Societies and Social Simulation 5, No.1 paper 4 (2002) (jasss.soc.surrey.ac.uk).
Stauffer, D. Frustration from Simultaneous Updating in Sznajd Consensus
Model. (cond-mat/0207598 Preprint for J. Math. Sociology)
Sabatelli, L.; Richmond, P. Phase transitions, memory and frustration in a
Sznajd-like model with synchronous updating. Int. J. Mod.Phys C 14 No. 9
(2003) (cond-mat/0305015).
Sabatelli, L.; Richmond, P. Non-monotonic spontaneous magnetization in a
Sznajd-like Consensus Model. To be published in Physica A (condmat/0309375)
Stylised facts
• Why is the pdf tail equal to 3?
– Is it a tunable parameter or more like a critical
exponent?
• What are minimum requirements needed to get
the familiar stylised facts?
• Are we looking at all the relevant stylised facts?
– Time?
• Shape, asymmetry of speculative peaks?
• Log periodic oscillations?
• Higher order correlations
Agents
• What form does heterogeneity take?
• How do people differ?
• What do they order at the restaurant?
– Is only time heterogeneity required for fat
tails and clustered volatility?
Networks
• How are we connected?
• How do connections and nodes evolve?
• Risk
– How do we evolve containment and
innoculation strategies?
– Diffuse rumour & heresy
– Social problems
• Eg Drug addiction, smoking, disease
Financial markets are only a part
of the subject
• Many other economic systems
• Impact on social and regulatory policy
• Electricity
• Telecommunications
• Transport
• Taxation
– Impact on wealth
• Real options,
• Weather derivatives, etc
• Real worlds (Ormorod)