How much investment can financial markets cope with?

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 cresponse = c perturbation

Price taking behavior(the basis of all financial math!) • Traders (perturbation) are negligible (~1/N) with respect to the market • What if c=∞ 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 = days

Facts: There is a non-trivial cluster structure • Eigenvalue distributionrandom matrix theoryand SVD(Laloux et al./Gopikrishnan et al. …) • Structure → economic sectors:Minimal Spanning Tree(Mantegna …)data clustering(Giada …)

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? The economy Finance (white) noise Ci,j = Bi,j + Fi,j +Wi,j

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 • …

A model:notations • vectors |v=(v1,…vn), v|=(v1,…vn)T • scalar productv|w =Si viwi • Matrices|wv|={wivj}

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

A phenomenological model: • |x(t+1) = |x(t) + |b + |h(t)+[e+x(t)]|z(t)|b = bare return|h(t) = bare noise E[|h(t) h(t)|] = B bare correlatione+x(t) = portfolio investment rate E[x(t)2]=D • Where • Average return and correlation matrix (m ~ 1/Taverage) |r(t+1) = (1-m) |r(t) + m[|x(t)-|x(t-1)]C(t+1) = (1-m) C(t) + m|dx(t)dx(t)| |dx(t)=|x(t)-|x(t-1)-|r(t)

Note: • Linear impact of investment • Impact through |z(t) not |dz(t) • Many agents |zk(t) with (Rk, ek, Dk) → one agent |z(t) with (R, e, D) • Only a single time scale 1/m • A simple attempt to a self-consistent problem

Numerical simulations

“Mean field”: m→0 • Self-consistent equations

Phase transition! • market mode parellel to |q (B=BI) • Critical point: l 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/

How much investment can financial markets cope with?