How much investment can financial
markets cope with?
M. Marsili (ICTP) + G. Raffaelli (SISSA)
 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
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 c
response = c perturbation
Price taking behavior
(the basis of all financial math!)
 Traders (perturbation) are negligible
(~1/N) with respect to the market
The Market
 What if c=∞
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
Ci , k =
 x (t )  x x (t )  x 
 x (t )  x   x (t )  x 
t
i
i
k
k
2
t
i
i
2
t
k
k
xi (t ) = log  pi (t ) pi (t  1)
1 T
xi =  xi (t )
T t =1
t = days
Facts:
There is a non-trivial cluster structure
 Eigenvalue distribution
random matrix theory
and 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: yes
Italian companies (with G. Caldarelli & co)
Rank of ci,j with
a link in the board
of 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
 …

ˆ
ˆ
ˆ
ˆ
ˆ
C = BF C W
A model:notations
 vectors
|v=(v1,…vn),
 scalar product
v|w =Si viwi
 Matrices
|wv|={wivj}
v|=(v1,…vn)T
Preliminaries: portfolio theory
 Problem: Invest |z with fixed
return = r|z = R and
wealth = 1|z = W
so as to minimize risk
 Solution:
1

z = arg min  z C z    z r  R    z 1  W 
z , , 2


 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 correlation
e+x(t) = portfolio investment rate E[x(t)2]=D
 Where
1

z (t ) = arg min  z C(t ) z    z r (t )  R    z 1  W 
z , , 2


 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

ˆ
Cˆ = Bˆ  dCˆ Cˆ  W
Numerical simulations
“Mean field”: m→0
 Self-consistent equations
r = b e z
Phase transition!
 market mode parellel to |q (B=BI)
4.5
 Critical point:
4

3.5
3
2.5
0.2
0.4
0.6
0.8
W
1
1.2
1.4
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/