Dressing of Financial Correlations: Portfolio Optimization and Multi-Assets Minority Games

1. Dressing of financial correlations due to porfolio optimizationand multi-assets minority games Palermo, Venerdi 18 giugno 2004 Unità di Trieste: G. Bianconi, G. Raffaelli e M. Marsili

2. Correlations between different 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.

3. Elements that shape the correlations • Economy Correlations between different assets encode the real economic relations between stocks • Noise There 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 market Agents who optimize their porfolio generates correlation. For example if they investe in two assets in the same way they will create positive correlations .

4. But what is the role of portofolio optimization in dressing the correlations between stocks ? 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.

5. Analytic results Equation for the returns Optimized portfolio Bare return Number of agents that invest Noise that comes from the economy

6. Correlation dressing 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.

7. The maximal eigenvalue • Minimizing the risk taking fixed the return R and the normalization condition • We obtain that the maximal eigenvalue L of C in the simple case of e=0

8. Other model Equation for the returns in continuoustime Bare return Increment of the optimized portfolio Noise that comes from the economy

9. Optimization of the portfolios • The mean return and the correlations of the stocks are calculated on a time window T • The portfolioqi is choosen in order to minimize risk, taking fixed the return R, i.e. qiminimize

10. Parameters of the model • Liquidity of the market 1/l • Peole who invest with optimized portfolio e • Time window T that is used to measure mean quantities of the market correlation matrix and returns. • We choose zero bare returns and unitary correlation matrix to start studying the model

11. Effect of e on the portfolioand prices As the number e of agents who play optimizing the portfolio increases the changes in the portfolio decreases. Instead the price fluctuations increse with incresing e.

12. Minority 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 a<ac and close to the phase transition.

13. Multi-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. Assets Agents

14. 2-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 (m1,m-1) each agent has two strategies with payoffs and with si taking the sign of the strategy with higher payoff.

15. Phase diagram of the model The parameters of the model are two AS The Lyapunov function S • The model is soluble, we derive the phase diagram in figure with a critical line. • The attendance <s> has the opposite sign of a+-a--

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

17. Conclusion • 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.