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 theorya simple dynamical modeldynamic instability of financial marketsfitting real market data

Conclusions

M. Marsili (ICTP) + G. Raffaelli (SISSA)

A personal perspective

External driving or to internal dynamics?

Interacting agents(Caldarelli et al, Lux Marchesi, …)

Minority gamesmarket ~ system close to phase transition(also in other models, e.g. Langevin, Lux, …)

∞ susceptibility

response perturbation

Price taking behavior(the basis of all financial math!)

Traders (perturbation) are negligible (~1/N) with respect to the market

What if =∞

The Market

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!

The covariance matrix

T

tii

iii

t kkt ii

t kkiiki

txT

x

tptptx

xtxxtx

xtxxtxC

1

22,

)(1

)1()(log)(

)()(

)()(

t = days

Eigenvalue distributionrandom matrix theoryand 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

Facts: Economic networks(Battiston et al., Kogut, …)

Shareholding Board of directors

Does this has an effect on financial correlations?

Board of directors: yesItalian companies (with G. Caldarelli & co)

Rank of ci,j witha link in the boardof directors wrt all ci,j

What is in the covariance matrix?

Ci,j = Bi,j + Fi,j +i,j

The economy Finance (white) noise

Dynamics of market mode

Key issue: feedback in the financial component

Behavioral: people buy when the market goes up (Airoldi ~ Cont-Bouchaud-Wyart)

Portfolio investment …

ˆˆˆˆˆ CFBC

A model:notations

vectors |v=(v1,…vn), v|=(v1,…vn)T

scalar product v|w =i viwi

Matrices |wv|={wivj}

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!

WzRrzzzz

z1

2

1minarg

,,

C

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/Taverage)

|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)

WzRtrzztztz

z1)()(

2

1minarg)(

,,

C

Note:

Linear impact of investment Impact through |z(t) not |z(t) Many agents |zk(t) with (Rk, k, Dk)

→ one agent |z(t) with (R, , D) Only a single time scale 1/ A simple attempt to a self-consistent problem

ˆˆˆˆˆ CCBC

Numerical simulations

“Mean field”: →0

Self-consistent equations

zbr

Phase transition!

market mode parellel to |q (B=BI)

Critical point:

0.2 0.4 0.6 0.8 1 1.2 1.4

2.5

3

3.5

4

4.5

W

What happens at the critical point?

Fitting real market data Linear model +

Gaussian hypothesis→ compute likelihood (analytical)

Find the parameters which maximize the (log)likelihood

Where are real markets?

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”

Thanks

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