Multi-asset Minority Game Complex Markets Meeting Marseille,6-7 - PowerPoint PPT Presentation

Multi-asset Minority Game Complex Markets Meeting Marseille,6-7 October 2006 G.Bianconi (ICTP), A. De Martino (Roma), F. F. Ferreira (Sao Paolo) and M. Marsili(ICTP)

Canonical Minority Game The Minority Game is a synthetic market model devised to show how stylized facts emerge from speculators trading under minimal assumptions . The game introduce heterogeneity of speculators strategies and adapting behavior (learning of the agents) which generate cooperative phenomena.

Canonical Minority Game µ + = − U ( t 1 ) U ( t ) a A ( t ) γ γ γ i , i , i , φ =argmax γ U i, γ N s φ i = − 1 µ = ± Speculators a 1 − i , 1 µ = ± φ i = a 1 1 1 i , Market in the 1 µ = ∑ ∑ + γφ A ( t ) a ( 1 ) / 2 γ i i , state (or history) N γ = − = 1 , 1 i 1 , N µ=1 ,...P

Where is the price? Where is the price? µ µ = − = + a i 1 1. agent buy 1$ agent sell 1/p(t) a i 1 units + − 2. Demand= Supply= ( N A ) / 2 ( N A ) / 2 p ( t ) + N A + = 3. Market clearing p ( t 1 ) p ( t ) − N A 4. Thus + = + λ log( p ( t 1 )) log( p ( t )) A MM Physica A 299 2001

Observables + = + λ Return log( p ( t 1 )) log( p ( t )) A ( t ) 1 Expected return µ λ A given µ Volatility σ 2 = + − 2 (log( p ( t 1 )) log( p ( t ))) Predictability 1 2 = ∑ µ H A ( t ) P µ = 1 ,.., P

Phase transition • The model show a phase transition – between a predictable phase (antisymmetric) in which the average price of the asset can be forecast given the available information – to a unpredictable phase (symmetric) in which the average price cannot be predicted by the available information in the system • The symmetric phase correspond to a state in which there is ergodicity breaking and there are many steady states of the dynamics corresponding to an efficient market. These states are selected by different initial conditions.

Phase transition The parametr of the game is α =P/N For α<α c the system is in the unpredictable, or efficient phase which correspond also to the non-ergodic phase of the system (dependence on initial conditions) D. Challet, M. Marsili and R. Zecchina PRL (2000). Figure from A. De Martino and M. Marsili (2006)

But in this coordinate context where to stylized fact came from? In order to find stylized facts it is important to introduce two variations to the model 1. There are some producers in the market which inject information and always play with fixed strategies. 2. Speculators are allowed not to play : there is a new strategy (do not trade) for the speculators.

Gran-canonical Minority Game • In the Gran-Canonical Minority Game agent are allowed not to trade. • At each time there is an incentive to trade/not to trade ε . Each agent has one strategy played only is the corresponding payoff is positive. The update rule for the payoff function is given by µ + = − + ε U ( t 1 ) U ( t ) a A ( t ) i i i

Grand-Canonical Minority Game µ A + = − + ε U ( t 1 ) U ( t ) a i i i φ i =θ (U i ) Ns φ i = 0 Speculators φ i = 1 µ = ± a i 1 Np µ µ = ∑ φ + ∑ A ( t ) a a i i i = = + Producers i 1 , Ns i Ns 1 , N µ = ± a i 1 Market in the state (or history) µ=1,..., P

GC Minority Game-Observables ε ≠ For there is no 0 phase transition as ε ->0 the number of active players diverges For ε=0 there is a phase transition with the appearence of an unpredictable non-ergodic phase The parameter of the model are ns=Ns/P D. Challet and M. Marsili PRE (2003). and np=Np/P

Stylized facts in the Gran- Canonical Minority Game Critical line Critical line Critical region Critical region ε ns* ns=Ns/P 1. Finite size effects 2. And strong dynamical effects before reaching D. Challet and M. Marsili equilibrium PRE (2003).

Phase Transition between a Phase Transition between a predictable phase and an predictable phase and an ε=0 unpredictable phase at ε=0 unpredictable phase at 0 10 At the phase transition At the phase transition 1. H goes to zero H goes to zero 1. − 2 H/N < φ> 10 2. The number of active The number of active 2. speculators < φ > H/N U 0 = − 3000 speculators H/N U 0 =3000 depends on their prior depends on their prior <φ> U 0 = − 3000 − 4 10 <φ> U 0 =3000 beliefs beliefs 3. The volatility also if The volatility also if 3. affected by prior affected by prior − 6 10 0 1 2 beliefs. beliefs. 10 10 10 n s

In the critical regions there are In the critical regions there are non trivial dynamical effects non trivial dynamical effects 1 − − ∆ ∑ = 2 ( t t ' ) / v ( t ) A ( t ' ) e ∆ t ' 20 ε+τ=0.01 2 1.5 v(t) n s =10 v(t) 10 1 0.5 0 0 0 1000 2000 3000 4000 0 500 1000 1500 2000 20 20 ε+τ=0.001 15 n s =40 v(t) v(t) 10 10 5 0 0 1000 2000 3000 4000 0 100 0 500 1000 1500 2000 20 n s =160 ε+τ=0.0001 v(t) 50 v(t) 10 0 0 1000 2000 3000 4000 t/P 0 0 500 1000 1500 2000 t/P ∆ ≈ − * v ( t ) ns ns ≈ − ε * T P ( ns ns ) / eq

Volatility and finite size effects in Volatility and finite size effects in the critical region the critical region • The system in this phase has significant finite size 0.10 effects in the volatility L=3000 L=6000 • There is a minimum in the volatility as a function of ε L=12000 0.05 σ 2 theory • in the infinite size limit converges to the 0.00 theoretical results. ε − 2 10 − 4 10 − 6 10

What role have speculators in a multi-asset market ? Data Clustering Given a certain structure of the correlations in the market which is the role of speculators which do not take into account risk M.Marsili Quantitative Finance (2002) considerations? Minimal Spanning Trees Bonanno et. al. PRE (2002)

Minority Game with multiple assets • How do speculators distribute their investments on different assets with different amount of information? • How do incentive of trading affect their behavior? • How does risk and correlations affect the composition of their portfolio?

2- -asset Canonical Minority Game asset Canonical Minority Game 2 µ + = − U ( t 1 ) U ( t ) a A ( t ) γ γ γ γ i , i , i , φ i =argmax γ U i, γ N φ i = − µ 1 = ± a 1 − Speculators i , 1 φ i = µ 1 = ± a 1 1 i , Different information in the two assets. • Asset γ =1 in the state (or history) µ =1,...P 1 • Asset γ= -1 in state 1 µ = ∑ + γφ A ( t ) a ( 1 ) / 2 (or history) µ =1,...P 2 γ i i , s N = i 1 , N

Parameters and observables of the model P Each market has a different information, γ { } α = γ ∈ − 1 , 1 γ thus the parameters of the game are N 1 2 ∑ ∑ = = µ H H H A γ γ γ γ The Lyapunov function of the game is P γ µ γ s 1 There is one volatility for each asset σ = ∑ σ σ = 2 2 2 2 A γ γ γ N γ 1 = ∑ φ m There is one “magnetization” of the speculators i N i

Correlations The correlation between the two asset is given by − φ + φ 1 1 1 µ µ µ µ µ µ j = ∑ ≈ ≈ i A A a a a a a a 0 + − + − + − + − + − + − + − i , j , i , j , i , j , N 2 2 i , j And it vanish since the strategies of the speculators in the two assets are uncorrelated This is an effect of the behavior of the trades aimed at detecting excess demand in the market with no consideration of the correlations between the assets Even if the strategies of each agent in the two asset were correlated ≈ µ µ µ µ A A 0 ≠ a a a a + − + − + − + − + − i , i , i , i ,

Exploitation of the available information • The dynamics tries to exploit maximally the information present in the system. This translates in the minimization of a Lyapunov function, • the predictability H 1 2 = ∑ ∑ µ H A γ P γ µ γ Speculators tend to make the market as efficient as possible minimizing the excess demand trying to get to a unpredictable phase in which <A µ >=0 for every value of µ =1,…P γ .

Static approach Describes the equilibrium state of the Canonical Minority-Game as the one minimizing H 2 ⎛ ⎞ 1 µ = ∑ ∑ ∑ + γ ⎜ ⎟ H a γ ( 1 m ) γ i = φ i , ⎝ ⎠ defined on the variables NP m γ µ γ i i i γ In order to find H, the system is replicated and mediated over the different realization of the strategies. Thus we introduce a replicated partition function on the variables } { } { − β µ a = ∏ H ( m , a ) n a ∫∫ Z dm e γ i i , i ( 0 , 1 ) γ i , a ,