Empirical symptoms of catastrophic bifurcation transitions on financial markets:

A phenomenological approach

Marzena Kozłowska, Tomasz Gubiec, Ryszard Kutner, Tomasz Werner

Faculty of Physics, University of Warsawand

Zbigniew Struzik University of Tokyo

&RIKEN Brain Science Institute, Japan

VII Sympozjum Fizyki w Ekonomii I Naukach SpołecznychUMCS Lublin, 14-17 May 2014

Abrupt climate change

Catastrophic (critical) bifurcation transition ina real life.

M. Scheffer et al., Nature 491 (2009), 53-59

In medicine: epileptic seizure and asthma attack.

In geophysics: earth quake, volcano eruption, abrupt shift in ocean circulation or in climate.

In ecosystem degradation: changes in states of coral reefs, colaps of vegetation in semi-arid ecosystem.

In physics: first order phase transitrions.

Index WIG of the Warsaw Stock Exchange (GPW)2004-02-06 – 2007-07-06 - 2009-05-18.

Here, we consider the lhs bubble

X ( y )=( X (0 ) +A1 ) E α (− ( y / τ )α )− A 1cos (ωy ) cos ( Δω ) , y=t c−t

uogólniony eksponens Mittag−Lefflera : E β (−( y / τ )α )=∑ (− ( y / τ )α )n

Γ (1+βn ), β=α

DAX: 2003-09-04 – 2007-07-13 - 2009-07-01.Here, we consider the lhs bubble

Beginning of new single-family house-buildings in US

1990-01 - 2009-03

Detrended signal and stochastic dynamics

x t+1−x t =f ( x t ;P )+ηt

x t : time−dependent daily signalP : control parameter

η t : δ−correlated (0, σ2 ) rand . var .

f ( xt ;P )=?

Detrended signal≡ x t

1 [ td ]≡1991−04−16 the beginning of Warsaw Stock Exchange

The noise of signal WIG2004.02.06 - 2007.07.06

Noise≡ x t+1− x t

Sale of new single-family houses in US 1990-01 - 2009-03

WIG: variance (monthly sample average).

Definition of threshold by spike

F lickering phenom enon

Shrinking in term ittencies

WIG: accumulative variance.

Bimodal situation

B im odalsystem

Detrended successive signals for WIG.Two time scales:

(i) daily t (fast) one and (ii) monthly (slow) one.Too large data dispersion for nonlinear analysis

x t =AR (1 ) x t−1+b+η t−1 , AR (1 )=1+λ, b=− λx1''∗

ACF (1 )=Cov (x t , xt±1 )

Var ( xt )=1+λ=AR (1 ) AR (1 ) , ACF (1 ) , λ : slowly varying

λ is our key quantity

Inverted triangles : λ≈0 . 0Circles λ≈0 . 4

Application of catastrophe theory to financial markets.Hypothesis: catastrophic bifurcation transitionComplete approach requires nonlinear analysis

Application of catastrophe theory to financial markets.Catastrophic (critical) slowing down

λ is negative below thresholdλ reaches zero at thresholdλ after threshold is negative

Blue : − λ=1−AR (1 )Red : −λ= 1− ACF (1 )

− λ sm ile :

−λ1− λ1 ''

− λ1 ''

− λ1 ''

−λ1

−λ1

Catastrophic bifurcation transition:force f and potential U

Plot of recovery rate

WIG: (Sub)catastrophic bifurcation transition

ALERT: invstors should leave such a stock market

DAX: (Sub)catastrophic bifurcation transition

ALERT: invstors should leave such a stock market

x1''∗

x1∗

General conclusion: schematic scenario

of stock market evolution generatedby bifurcation transitions