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Agent-based Versus Macroscopic Modeling ofCompetition and Business Processes in Economics
and Finance
Vygintas Gontis, Aleksejus Kononovicius and Bronislovas Kaulakys
Institute of Theoretical Physics and Astronomy, Vilnius University, Lithuania,vygintas@gontis.eu, www.gontis.eu
Galway, July 12, 2012
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 1 / 23
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Outline
1 Power-law statistics arising from the nonlinear stochastic differential
equations
2 The nonlinear stochastic differential equations as a background of
financial fluctuations
3 Microscopic versus macroscopic modeling
4 Kirmans herding model as a statistical background of microscopicdescription
5 Introduction of macroscopic feedback on microscopic behavior
6 Bass product diffusion as a one direction Kirman process
7 Kirmans herding as a background of financial fluctuations
8 Herding model with three groups of agents
9 Summary
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 2 / 23
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The class of non-linear SDE with power law statistics
dx = ( 2 )x21
dt + xdWs
P(x) x, S(f) 1
f, = 1
3
2 2
P(x) x exp xmin
x m
xxmax
m
dx =
2 +
m2
xmin
x
m
xxmax
m
x21 + xdWs
Publications
Kaulakys, B.; Gontis, V. & Alaburda, M. (2005), Phys. Rev.E, 71,051105.
Kaulakys, B.; Ruseckas, J.; Gontis, V. & Alaburda, M. (2006),
Physica A, 365, p. 217-221.
Ruseckas, J. & Kaulakys B. (2010), Phys.Rev.E, 81, 031105.
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 3 / 23
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Power-law statistics arising from the nonlinear
stochastic differential equations
A simple case of nonlinear SDE
0 100 000 200 000 300 000 400 000 500 000
t1
2
5
10
20
50
x
dx = x3/2dW
10-18
10-16
10-14
10-12
10-10
10-8
10-6
10-4
10-2
100
102
100
101
102
103
104
105
P(x)
x
10-3
10-2
10-1
100
101
102
103
10-2
10-1
100
101
102
103
S(f)
f
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 4 / 23
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Power-law statistics arising from the nonlinear
stochastic differential equations (continued)
More power-law statistics
dx = (
2)x21dt + xdW , P(x) x , S(f) 1/f
= 13
2
2 , S T2
, P(S) S1.3
, P(T) T1.5
, P() 1.5
Avelanches or bursts Size versus duration S T2V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 5 / 23
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The stochastic model with a q-Gaussian PDF and
power spectrum S(f) 1/f
SDE with q-Gaussian PDF and power spectrum S(f) 1/f
dx =
2
(1 + x2)1xdt + (1 + x2)
2 dW
x
x
x0 , P(x) 1
1+x22
, = 1
3
2
2 , =
3
2 , = 3.
10-10
10-8
10-6
10-4
10-2
100
-1000 -500 0 500 1000
P(x)
x
10-3
10-2
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
100
101
102
S(f)
f
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 6 / 23
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Statistics of burst duration T generated by nonlinear
SDE
V. Gontis, et al, ACS, 15(1) (2012), pp. 1250071 (13).
p()hx
(T) T3/2, T 2h2x
j2,1
,
p()
hx (T)
exp
j2,1
T
2h2x
T , T2h2xj2,1
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 7 / 23
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The stochastic model of return in financial markets
reproducing q-Gaussian PDF and power spectrum
The background nonlinear SDE for financial markets
dx =
2 (
x
xmax)2
(1 + x2)1
((1 + x2)12 + 1)2
xdts +(1 + x2)
2
(1 + x2)12 + 1
dWs
We solve SDE introducing the variable steps of integration
hk = 2
(
x2k + 1 + 1)
2
(x2k + 1)
1,
xk+1 = xk + 2
2
x
xmax
2xk +
x2k + 1k
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 8 / 23
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Stochastic model of return versus empirical data of
NYSE and Vilnius stock exchanges
Comparison of empirical statistics of absolute returns, = 600s,traded on the NYSE (black curves) and VSE (light gray curves) with
model (gray curves). Model parameters are as follows: = 5;2t = 1/3 10
6s1; 0 = 3.6; = 0.017; = 2.5; r0 = 0.4;xmax = 1000.
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
10-2
10-1
100
101
102
P(r)
r
(c)
PDF of normalized absolute return.
102
103
104
105
10-7
10-6
10-5
10-4
10-3
10-2
S(f)
f
(d)
Power spectral density of absolute
return.V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 9 / 23
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Microscopic versus macroscopic modeling
Microscopic and Macroscopic correspondence is a major
challenge in the research of complex systems:
Usually only macroscopic behavior can be extracted from the
empirical data,
The ambiguity of microscopic interactions lies in the veryfoundation of social systems
One has consider the possible feedback of macroscopic state on
microscopic behavior
This conditions the significance of intelligence free (probabilistic)agent systems with well defined macroscopic behavior
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 10 / 23
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Herding behavior - Kirmans stochastic ant colony
model
One step probabilities
p(X X + 1) = (N X) (1 + hX) t = +N2t,p(X X 1) = X(2 + h(N X)) t =
N2t ,can be rewritten for continuous x = X/N as
+(x) = (1 x) 1N + hx ,
(x) = x2
N + h(1 x)
,where X is a number of agents exploiting chosen trading strategy, N is
a total number of agents in the system. Here the large number of
agents N is assumed to ensure the continuity of variable x.
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 11 / 23
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Master equation for the probability density function
P(x, t) of continuous variable x
tP(x, t) = N2
(E 1)[(x)P(x, t)]+ (E1 1)[+(x)P(x, t)]
.
With the Taylor expansion of operators E and E1 (up to the second
term) we arrive at the approximation of the Master equation
tP(x, t) = Nx[{+(x)(x)}P(x, t)]+
1
22x[{
+(x)+(x)}P(x, t)].
Introducing custom functions
A(x) = N{+(x) (x)} = 1(1 x) 2x,D(x) = +(x) + (x) = 2hx(1 x) + 1N (1 x) +
2N x
we get Fokker-Planck equation
tP(x, t) = x[A(x)P(x, t)] +1
22x[D(x)P(x, t)].
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 12 / 23
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SDE of population dynamics in Kirmans model
Herding dynamics is well approximated by the SDE:
dx = A(x)dt + D(x)dW = [1(1 x) 2x] dt +
2hx(1 x)dW.
Two
qualitatively different regimes and the intermediate phase observed in
the symmetric Kirmans model (a) and two asymmetric regimes (b).
P(x) x1
h1(1 x)
2h1
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 13 / 23
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Introduction of macroscopic feedback on microscopic
behavior - variable time scale
J. Ruseckas, B. Kaulakys, V. Gontis, EPL (2011), 96, 60007.A. Kononovicius, V. Gontis, PhysA (2012), 391, 1309.
We introduce interevent time (x) into the transition probabilities:
+(x) =1 x
(x)1
N
+ hx , (x) = x(x)
2N
+ h(1 x) .The same derivation produces the SDE for x
dx =1(1 x) 2x
(x)dt +2hx(1 x)
(x)dW.
(x)
x
1 x
.
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 14 / 23
B d diff i di i Ki
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Bass product diffusion as a one direction Kirman
process
dX(t)
dt= [N X(t)][p+
q
NX(t)], X(0) = 0,
where X(t) - the number of product users at time t; N is a number ofpotential users, p is the coefficient of innovation, q is the coefficient of
imitation. It is a case when a new user adopts the product with
probability
p(X X + 1) = (N X)
p+ qNX
, p(X X 1) = 0.The functions defining the macroscopic system description in the limit
N are as follows
A(x) = N+(x) = (1x) (p+ qx) , D(x) = +(x) =(1 x)
N(p+ qx) .
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 15 / 23
C i f i d i i B
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Comparison of macroscopic and microscopic Bass
diffusion descriptions
(a) N = 1000, t = 0.1; (b) N = 1000, t = 1; (c) N = 10000,t = 0.1; (d) N = 10000, t = 1.
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 16 / 23
D fi iti f i d t
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Definition of price and returns
Walrassian scenario based on excess
demand D formed by fundamentalistsNf and chartists Nc.
Df(t) = Nf(t) lnPf(t)P(t)
Dc(t) = r0Nc(t)(t), here Nc(t)(t) isa difference between chartist sellers
and chartist buyers, r0 is scaling term.
P(t) = Pf(t) exp
r0
Nc(t)
Nf(t)(t)
,
r(t) = r0 x(t)
1 x(t) (t)
x(t )
1 x(t ) (t )
,
r(t) = r0y(t) [(t) (t )] = r0x(t)
1 x(t)(t).
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 17 / 23
St h ti diff ti l ti f th d l ti
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Stochastic differential equation for the modulating
return - volatility
Continuous variable y(t) = x(t)
1
x(t)stands for the modulating absolute
return or volatility in the model. One can by Ito transform of variable
arrive at the SDE for y
dy = 1
+ y2h 2
(y) (1 + y)dts + 2hy
(y)(1 + y)dWs.
With the assumption (y) = y in the limit y >> 1 we can consideronly terms of the highest power in the SDE. This produces a simplified
nonlinear SDE for y of general class
dy = (
2)y21 + ydWs, P(y) y
, S(f) 1
f
with = 3+2 ; = + 1 +2h and = 1 +
2/h+21+ .
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 18 / 23
Variety of reproducible and values
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Variety of reproducible and values
Wide spectra of obtainable and values. Model parameters were set
as follows: = 1,
1
h = 0.1,
2
h = 0.1 (red plus), 0.5 (green cross), 1(blue stars), 1.5 (magenta open squares), 2 (cyan filled squares) and 3(orange open circles). Black curves correspond to the limiting cases:
(a) 1 = 2 and 2 = 5, (b) 1 = 0.5, 2 = 2
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Herding model with three groups of agents
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Herding model with three groups of agents
Nf - fundamentalists, Nopt - chartists
optimists, Npes - chartists pessimists,
(t) = xopt(t) xpes(t).
dnf = (1 2nf)c,fdt +
2(1 nf)nfdW1,
d = 2hc,cdt +
2hc,c(1 2)dW2.
(nf, ) = 11 +
1nf
nf .
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 20 / 23
Two time scales of financial fluctuations numerical
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Two time scales of financial fluctuations - numerical
evidence
Power spectral density of return
with trading activity scenario = 1Power spectral density of return
with trading activity scenario = 2
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 21 / 23
Summary
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Summary
We generalized macroscopic treatment of microscopic herding
model proposed by A. Kirman:
This reveals evident relation between Bass new product diffusion
model and one directional Kirmans herding,
Gives a microscopic background for the stochastic modeling of
financial variables by the class of nonlinear stochastic differentialequations,
Developed double stochastic model of return in financial markets
reproduces PDF, Power spectral density and Burst duration
empirically defined in NYSE and Vilnius Stock Exchanges,Further we consider financial market with three groups of
heterogenous agents and two time scales of their interaction.
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 22 / 23
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Thank you!
V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 23 / 23
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