1. How much investment can financial markets cope with?  A personal perspective  Financial correlations: Why are stocks correlated? [structure/exogenous] Why are correlations time dependent? [dynamics/endogenous]  Impact of investment strategies: portfolio theory a simple dynamical model dynamic instability of financial markets fitting real market data  Conclusions M. Marsili (ICTP) + G. Raffaelli (SISSA)

2. A personal perspective  External driving or to internal dynamics?  Interacting agents (Caldarelli et al, Lux Marchesi, …)  Minority games market ~ system close to phase transition (also in other models, e.g. Langevin, Lux, …)  ∞ susceptibility  response perturbation

3. Price taking behavior (the basis of all financial math!)  Traders (perturbation) are negligible (~1/N) with respect to the market  What if = ∞ The Market

4. Example: a minority game experiment  Find the best strategy on historical data of a Minority Game  (virtual) gain = 0.87  Rewind and inject the strategy in the game  The price process changes a lot  (real) gain = -0.0034!

5. The covariance matrix t = days

6.  Eigenvalue distribution random matrix theory and SVD (Laloux et al./Gopikrishnan et al. …)  Structure → economic sectors: Minimal Spanning Tree (Mantegna …) data clustering (Giada …) Facts: There is a non-trivial cluster structure

7. Facts: Economic networks (Battiston et al., Kogut, …)  Shareholding  Board of directors Does this has an effect on financial correlations?

8. Board of directors: yes Italian companies (with G. Caldarelli & co) Rank of c i,j with a link in the board of directors wrt all c i,j

9. What is in the covariance matrix? C i,j = B i,j + F i,j + i,j The economyFinance(white) noise

10. Dynamics of market mode

11. Key issue: feedback in the financial component  Behavioral: people buy when the market goes up (Airoldi ~ Cont-Bouchaud-Wyart)  Portfolio investment ……

12. A model:notations  vectors |v  =(v 1,…v n ),  v|=(v 1,…v n ) T  scalar product  v|w  = i v i w i  Matrices |w  v|={w i v j }

13. Preliminaries: portfolio theory  Problem: Invest |z  with fixed return =  r|z  = R and wealth =  1|z  = W so as to minimize risk  Solution:  No impact on market. But unique solution. All will invest this way!

14. A phenomenological model:  |x(t+1)  = |x(t)  + |b  + |(t)  +[+(t)]|z(t)  |b  = bare return |(t)  = bare noise E[|(t)  (t)|] = B bare correlation +(t) = portfolio investment rate E[(t) 2 ]=  Where  Average return and correlation matrix ( ~ 1/T average ) |r(t+1)  = (1-) |r(t)  + [|x(t)  -|x(t-1)  ] C(t+1) = (1-) C(t) + |x(t)  x(t)| |x(t)  =|x(t)  -|x(t-1)  -|r(t) 

15. Note:  Linear impact of investment  Impact through |z(t)  not |z(t)   Many agents |z k (t)  with (R k,  k, D k ) → one agent |z(t)  with (R, , D)  Only a single time scale 1/  A simple attempt to a self-consistent problem

16. Numerical simulations

17. “Mean field”:  →0  Self-consistent equations

18. Phase transition!  market mode parellel to |q  ( B=BI)  Critical point: W

19. What happens at the critical point?

20. Fitting real market data  Linear model + Gaussian hypothesis → compute likelihood (analytical)  Find the parameters which maximize the (log)likelihood

21. Where are real markets?

22. Conclusions  Feedback of portfolio strategies on correlations  There is a limit to how much investment can a market deal with before becoming unstable  Markets close to a phase transition  Large response (change in C) to small investment → “dynamic impact risk”

23. Thanks www.sissa.it/dataclustering/ www.ictp.trieste.it/~marsili/