generalized minority games with adaptive trend-followers and contrarians a. tedeschi, a. de martino,...
TRANSCRIPT
Generalized minority games with adaptive trend-followers and
contrarians A. Tedeschi,
A. De Martino, I. Giardina, M.Marsili
• Interaction of different types of agents in market
• N agents formulate a binary bid: (buy/sell)
• The quantity is the excess demand
• When is large/small the risk perceived by the
agents is high/low and they act as fundamentalists/trend-followers.
• If each agent is rewarded with a good choice is
Some initial considerations
1ia
i
iaN
tA1
)(
)(tA
AFap ii 3AAAF
• Contrarians/trend-followers are described by minority/majority game players (rewarded when acting in the minority/majority group)
• Our model allows to switch from one group to the other
• Trend-following behavior dominates when price movements are small, whereas traders turn to a contrarian conduct when the market is chaotic
Introduction
N
igia
NtA
1
~1
0igp
• Each time t, N agents receive an information Pt ,...,1
• Based on the information, agents formulate a binary bid (buy/sell)
• Each agent has S strategies mapping information into actions 1,1iga
• Each strategy of every agent has an initial valuation updated according to
tAFatptp igigig1
• The excess demand is where )(maxarg~ tpg igg
The Model
Our Model
• In minority game
• In majority game
• In our model 3)( AAAF
AAF
AAF )(
-3
-2
-1
0
1
2
3
-4 -3 -2 -1 0 1 2 3 4
The ε parameter
• ε is a tool to interpolate between two market regimes: agents change their conduct at some threshold value A* depending on ε
• This threshold value A* can be verified in real markets from order book data by reconstructing
where O=order and dR= price increment
• We neglect the time dependency of ε (being on much larger time scales than ours)
dROdRPi |)sgn(
The Observables
• Study of the steady state for of the valuation as a function of α=P/N
• The volatility (risk)
N
22 A
• The predictability (profit opportunities)2
| AH
• The fraction of frozen agents ϕ
• The one-step correlation 2
)1()(
tAtA
D
Numerical simulations: volatility
• Small ε: pure majority game behavior
• Increasing ε: smooth change to minority game regime
• ε going to infinity: minimum at phase transition for standard min game
Numerical simulations: predictability
• Increasing ε: H <1 at small α as in min game, H→1 for large α as in maj game
• No unpredictable regime with H=0 is detected at low α, even in the limit ε going to infinity
Numerical simulations: frozen agents
• For large α, one finds a treshold separating maj-like regime with all agents frozen from min-like regime where Φ=0
• For large ε, Φ has a min game charachteristic shape
• In the low α, large ε phase, agents are more likely to be frozen than in a pure min game
Theoretical estimate for the large α regime
• We can give a theoretical estimate (that fits with simulations at large α) of the crossover from min to maj regime.
• The ε crossover value can be computed considering that at large α agents strategies are uncorrelated and A(t) can be approximated with a gaussian variable.
• With these assumptions we analytically estimate the crossover value at ε=1/3 for α>>1 (in a consistent manner from both maj and min sides). Numerically we find ε≈0.37.
Numerical simulations: correlation
• For small ε, D is positive, so the market dynamics is dominated by trend-followers
• The contrarian phase becomes larger and larger as ε grows and, for ε>>1, the market is dominated by contrarians
Numerical simulations: probability distribution
• For α=0.05, the distribution of A(t) shows heavy tails. The distribution peak moves as 1/√ε: the system is self-organized around the value of A such that F(A)=0
• For α=2 and A not too large with a weak dependence on ε42)(log bAAAP
Numerical simulations: Single Realization
• Time series of the excess demand A(t): spikes in A(t) occur in coordination with the transmission of a particular infomation pattern
• Time series of price : we observe formation of sustained trends and bubbles
tl
lAtR )()(
Conclusions
• In our model, market-like phenomenology (heavy tails, trends and bubbles) emerges when the competiton between trend-followers and contrarians is stronger
• Further developments for real market models: grand-canonical extensions, real market history and time-dependent ε coupled to the system performance