A kinetic equation modelling irrationality and herding of agents - - PowerPoint PPT Presentation

A kinetic equation modelling irrationality and herding of agents

Bertram D¨ uring1 Ansgar J¨ ungel2 Lara Trussardi2

1 University of Sussex, United Kingdom 2 Vienna University of Technology, Austria

Lyon - July 7, 2015

www.itn-strike.eu B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 1 / 16

Index

1

Introduction

2

Main mathematical results

3

Numerical results

4

Outlook

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 2 / 16

Herding

Herd behavior: a large number

• f people acting in the same

way at the same time Stock market: greed in frenzied buying (named bubbles) and fear in selling (named crash)

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 3 / 16

Herding

Herd behavior: a large number

• f people acting in the same

way at the same time Stock market: greed in frenzied buying (named bubbles) and fear in selling (named crash)

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 3 / 16

Irrationality and aim

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 4 / 16

Irrationality and aim

Goal

To describe the evolution of the distribution of the value of a given product (w ∈ R+) in a large market by means of microscopic interactions among individuals in a society

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 4 / 16

Model

Background literature:

1 G. Toscani, Kinetic models of opinion formation (2006) 2 M. Levy, H. Levy, S. Solomon, Microscopic simulation of Financial

Market (2000)

3 M. Delitala, T. Lorenzi, A mathematical model for value estimation

with public information and herding (2014) The model is based on binary interactions. It describes two aspects of the opinion formation:

◮ interaction with the public information (rational investor) ◮ effect of herding and imitation phenomena (irrational investor)

We also have a drift term: process which modifies the rationality of the agents (x ∈ R).

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 5 / 16

Public information - microscopic view

Fixed background W which represents the fair asset value.

Interaction rule

w∗ = w − αP(|w − W |)(w − W ) + ηD(|w2|) w − → •

I −

→ w∗ w∗: asset value after exchanging information with the background W α ∈ (0, 1) mesures the attitude of agents in the market to change their mind P(·) describes the local relevance of the compromise D(·) describes the local diffusion for a given value η is a random variable with mean zero and variance σ2

I

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 6 / 16

Herding - microscopic view

Interaction rule

w∗ = w − βγ(w, v)(w − v) + η1D(|w|) v∗ = v − βγ(w, v)(v − w) + η2D(|w|)

• v

w v∗ w∗ The function γ describes a socio-economic scenario where the agents are over-confident in the product. β ∈ (0, 1/2) mesures the attitude of agents in the market to change their mind D(·) describes the local diffusion for a given value η1, η2 is a random variable with mean zero and variance σ2

H

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 7 / 16

Boltzmann like collision operator

Let f = f (x, w, t) : R × R+ × R+ → R: number of individuals with rationality x and asset value w at time t.

1

Collision I (public information): w ∗ = w − αP(|w − W |)(w − W ) + ηD(|w 2|) QI(f , f ), φ =

• f (x, w, t)[φ(w ∗) − φ(w)]M(W )dxdwdW

2

Collision H (herding):

• w ∗ = w − βγ(w, v)(w − v) + η1D(|w|)

v ∗ = v − βγ(w, v)(v − w) + η2D(|v|)

QH(f , f ), φ =

• f (x, w, t)f (y, v, t)[φ(w∗) − φ(w)]dxdwdv

Both collision operators can be seen as a balance between a gain and a loss of agents with asset value w.

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 8 / 16

Boltzmann equation

Evolution law of the unknown f = f (x, w, t):

Boltzmann equation

∂ ∂t f (x, w, t) + ∂ ∂x

• Φ(x, w)f (x, w, t)
• =

1 τI(x)QI(f , f ) + 1 τH(x)QH(f , f ) where Φ describes how the drift changes with time. Let R > 0 constant which represents the range within which bubbles and crashes do not occur; Φ(x, w) = −δκ, |w − W | ≤ R κ, |w − W | > R κ > 0, δ > 0

Aim

Analysis of moments Diffusion limit Numerical experiments

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 9 / 16

Properties

Recall the Boltzmann equation and the collision kernels:

∂ ∂t f (x, w, t) + ∂ ∂x [Φ(x, w)f (x, w, t)] = 1 τI QI(f , f ) + 1 τH QH(f , f )

QI(f , f ), φ =

• f (x, w, t)[φ(w∗) − φ(w)]M(W )dxdwdW

QH(f , f ), φ =

• f (x, w, t)f (y, v, t)[φ(w∗) − φ(w)]dxdwdv

1 The mass is conserved: f (·, ·, t)L1(Ω) =

• f in
• L1(Ω) for a.e. t ≥ 0.

2 The first moment converges toward W :

• wf dxdw → W .

We compute it: ∂t

• wf dxdw +
• Φw∂xf dxdw
• =0

= 1 τI QI, w

=0

+ 1 τH QH, w

• =−αwf +W

3 The second moment converges toward 0:

• w2f dxdw → 0.

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 10 / 16

Fokker-Planck limit system

Rescale: τ = αt, y = αx = ⇒ g(y, w, τ) = f (x, w, t) ∂ ∂t f (x, w, t)+Φ(x, w) ∂ ∂x f (x, w, t) = 1 α 1 τI Q(α)

I

(f , f )+ 1 α 1 τH Q(α)

H (f , f )

Compute the limit α → 0, σ → 0 such that λ = σ2/α

Fokker-Planck equation

∂g ∂t + ∂ ∂x [Φg] = (K[g]g)w + (H[g]g)w + (D(w)g)ww K(w, τ) =

• R+

γ(v, w)(w − v)g(v)dv, D(w) > 0 H(w, τ) =

• R+

(w − W )P(|w − W |)M(W )dW

Difficulties

This equation is: non-linear, non-local, degenerate.

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 11 / 16

Existence theorem

Theorem

For x ∈ R, w ∈ R+ let consider the problem gt + Φgx = (K[g]g)w + (H[g]g)w + (D(w)g)ww (1) with g(x, w, 0) = g0(x, w) and g|w=0 = 0. Then there exist a weak solution g ∈ L2(0, T; L2(R × R+)) to (1). Idea of the proof:

1 reduction on bounded domain QT = Ω × Ω′ × (0, T) where

Ω × Ω′ ⊂ R × R+

2 approximate elliptic problem: for τ > 0 time discretisation and

addition of the term εgxx

3 Leray-Schauder fixed point theorem 4 estimates for (τ, ε) → 0 5 diagonal argument on the domain: Ω × Ω′ R × R+ B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 12 / 16

Numerical model

We implement the Boltzmann equation for f (x, w, t) (2D-model). We need to reduce the model to a bounded domain: for the rationality x ∈ [−1, 1] and for the asset value w ∈ [0, 1]. The scheme is divided into:

◮ drift: flux-limiters method (Lax-Wendroff scheme and up-wind scheme) ◮ collision with the public source & herding collision: slightly modified

Bird method

Goal

1 To check the analytical results regarding the asymptotic analysis 2 To understand the role of the parameters in the formation of bubbles

and crashes

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 13 / 16

Moments

Let fix W = 0.3. The mass: ρ =

• R
• R+

f indxdw is conserved

200 400 600 800 1000 1200 1400 1600 1800 2000 0.2 0.4 0.6 0.8 1

time

First moment w “converges” to W = 0.3

200 400 600 800 1000 1200 1400 1600 1800 2000 0.02 0.04 0.06 0.08 0.1

time

Second moment w “converges” to 0

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 14 / 16

Role of α

w∗ = w − αP(|w − W |)(w − W ) + ηD(|w2|)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 35 40 45

alpha % bubble

beta=0.5 beta=0.05 beta=0.005 0.2 0.4 0.6 0.8 2 4 6 8 10 12 14 16 18

alpha % crash

beta=0.5 beta=0.05 beta=0.005

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 15 / 16

Role of α

w∗ = w − αP(|w − W |)(w − W ) + ηD(|w2|)

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

time

α = 0.05

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

time

α = 0.5

B.D¨ uring, A.J¨ ungel, L.Trussardi Kinetic equation for irrationality and herding Lyon - July 7, 2015 15 / 16

Outlook

Summary:

◮ importance of the reliability of public information ◮ herding promotes occurence of bubbles and crashes: may lead to

strong trends with low volatility of asset prices, but eventually also to abrupt corrections.

Furthes studies

◮ Fokker-Planck simulation ◮ Investigate all the parameters and try to understand better their role

(counter action for herding)

Thanks for your attention

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