Dressing of financial Dressing of financial correlations due to porfolio correlations due to porfolio

optimizationoptimizationand and

multi-assets minority gamesmulti-assets minority games

Palermo, Venerdi 18 giugno 2004

Unità di Trieste: G. Bianconi, G. Raffaelli e M. Marsili

Correlations between Correlations between different assetsdifferent assets

• Suppose we are in a market with assets and

observations

• The correlations are defined as

where è il return dell’asset i and is the volatility of asset i.

Ni ,2,1

Tt ,2,1

ji

jjii

ji

StSStStC

))()()((

)(,

iS i

Elements that shape the Elements that shape the correlationscorrelations

• Economy Correlations between different assets encode the real economic relations between stocks

• NoiseThere is noise in the data also due to finite T.The noise can explain much of the correlation matrix with the relavant exception of the highest eigenvalue which is .

• Financial marketAgents who optimize their porfolio generates correlation. For example if they investe in two assets in the same way they will create positive correlations .

)(max NO

But what is the role of portofolio But what is the role of portofolio optimization in dressing the optimization in dressing the

correlations between stocks ?correlations between stocks ?

General framework:General framework:

• Given the market, following the theory of Markowitz, agents invest in order to diversify their portfolio.

• Feedback: The new investements increase the correlations between stocks and the portfolio of the agents must be modified.

Analytic resultsAnalytic results

iiii tbtx )()()(

)'()'()( tttt

)'()()( , ttBtt jiji

Noise that comes from the economy

Bare return

Optimized portfolio

Number of agents that invest

Equation for the returns

0)( t

Correlation dressingCorrelation dressing

br

BrxrxEC ˆ))((ˆ

The mean return is

The observable correlations are dressed

If the matrix B of the bare correlations is diagonal the financial investement descrived by this model generates a correlation matrix with one different eigenvalue.

The maximal eigenvalueThe maximal eigenvalue

• Minimizing the risk taking fixed the return R and the normalization condition

• We obtain that the maximal eigenvalue of C in the simple case of =0

NRN

12

Rr 11

Other modelOther model

))(()( iiii tbtx

)'()()( , ttBtt jiji

Noise that comes from the economy

Bare returnIncrement of the optimized portfolio

Equation for the returns in continuous time

Optimization of the portfoliosOptimization of the portfolios

• The mean return and the correlations of the stocks are calculated on a time window T

• The portfolio i is choosen in order to minimize risk, taking fixed the return R, i.e. i minimize

))()'())(()'(('1

)( /)'(

1, trtxtrtxedt

TtC jjii

Tttt

ji

)11()(2

1 rRC

)1(1 rC

t

i

Ttt

i txedtT

tr1

/)'( )('1

)(

Parameters of the modelParameters of the model

1. Liquidity of the market2. Peole who invest with optimized portfolio

3. Time window T that is used to measure mean quantities of the market correlation matrix and returns.

4. We choose zero bare returns and unitary correlation matrix to start studying the model

Effect of Effect of on the portfolio on the portfolioand pricesand prices

As the number of agents who play optimizing the portfolio increases the changes in the portfolio decreases.

Instead the price fluctuations increse

with incresing

Minority GameMinority Game

• Minority Game is a very stylized model that describes how agents react to information present in the market and in which conditions the stylized facts present in the market do emerge.

• These stylized fact arise close to the phase transition between a predicitive phase and a unpredictable phase.

• The phase transition arise at a special value of the ratio between the number of infomation patterns P present in the market and the number of agents that play N.

• The order parameter is the Lyapunov function of the dynamics called predictability H such that H=0 for c and close to the phase transition.

NP /

2)( caH

Multi-assets Minority gameMulti-assets Minority game

We are generally interested in the generalization of the model when players can play in different assets so that the number of players in each assets is not constant but varies in time.

As a starting point we studied the two assets Minority Game when the information content of the two assets is not the same.

Agents Assets

2-assets Minority Game2-assets Minority Game

• There are N agents. Each one can play at each time in one of the two assets present in the market

• The information patterns for the two assets are different: there is an exogenous infomation for the two assets

• Given (1,-1) each agent has two strategies

with payoffs

and

with si taking the sign of the strategy with higher payoff.

P2,1

,ia

AatUtU iii ,,, )()1(

i

ii saA 2/)1(,

Phase diagram of the modelPhase diagram of the model

N

P AS

S

The parameters of the model are two

The Lyapunov function

2AH

•The model is soluble, we derive the phase diagram in figure with a critical line.•The attendance <s> has the opposite sign of +--

Comparison simulation and Comparison simulation and theorytheory

We show the behavior of the predictability H.The volatility and the attendance <s> on a cut of the phase diagram such that +---=0.4.

ConclusionConclusion

• We are currently working on models that mimic the impact of the financial market on the correlations between stocks. Financial market do in effect dress the correlation matrix and generates higher volatity.

• The second topic is the generalization of the Minority Game into a game with multi-assets. In the two-asset minority game we observed that players play in the asset with less information preferably.