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Gontis Galway

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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,[email protected], 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

    V. Gontis, A. Kononovicius & B. Kaulakys () Economics and Finance Galway, July 12, 2012 19 / 23

    Herding model with three groups of agents

    http://find/
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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

    http://find/
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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

    http://find/

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