Local Interactions in a Market with Heterogeneous Expectations - - PowerPoint PPT Presentation

Local Interactions in a Market with Heterogeneous Expectations

Mikhail Anufriev1 Andrea Giovannetti2 Valentyn Panchenko3

1,2University of Technology Sydney 3UNSW Sydney, Australia

Computing in Economics and Finance 2018 Milano

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THE SCENE

• Many non-specialist investors allocate money into pension funds
• Funds can be classified into a number of types, e.g.,

◮ value (fundamental) ◮ momentum (chartists)

• Investors are able to switch funds

◮ Every period investors receive performance reports for the past period ◮ Investors talk to their network of friends and may switch to a different fund if it performed better

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LITERATURE

Behavioral Asset Pricing

• Brock and Hommes (JEDC, 1998)

◮ model with switching based on past performance

• Panchenko, Gerasymchuk and Pavlov (JEDC, 2013)

◮ local interaction in BH asset pricing model ◮ no switching if all neighbors are of the same type ◮ analytic solution for random graph, assuming homogeneous degree ◮ simulations for small world model

Spread of behaviors on networks:

• Lopez-Pintado (2008, GEB)

◮ related to S-I-S model on network (e.g., Vespignani, 2001) ◮ two types of agents: Susceptible agent becomes Infected if the number of Infected neighbours crosses a threshold

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CONTRIBUTION

• 1. Network is characterized by degree distribution P(k)
• 2. We can study the market dynamics for general classes of

networks:

◮ regular random networks (same degree k) ◮ Poisson networks ◮ scale free (power-law) networks

• 3. Analytical results using mean field approximation,

finer approximation than in Panchenko et al. (2013)

• 4. Our approach can handle broad classes of diffusion mechanisms

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BASELINE FRAMEWORK: ASSET PRICING MODEL

Brock and Hommes, 1998, JEDC

• Large population of traders N = {1, 2, ..., N} trading combination
• f risk-free bond with return R and risky asset z with stochastic

dividend yt

• t = 1, 2, ...
• Agents. Myopic optimizers with CARA preferences in wealth Wi,t:

max

zi,t Ui,t+1(Wi,t+1)

⇔ max

zi,t

• Eh

t−1 [Wt+1] − a

2V [Wt+1]

• Agent i selects the trading type hi ∈ H to form expectations on pt.
• Market Clearing (zero supply of risky asset):
• h∈H

nh

t

Eh

t−1 [pt+1 + yt+1] − Rpt

aσ2

• individual mean-variance demand

= 0 ⇒ pt = 1 R

• h∈H

nh

t Eh t−1 [pt+1] + ¯

y

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• Given the fundamental price p∗, define xt = pt − p∗:

xt = 1 R

• h∈H

nh

t Eh t−1 [xt+1]

• Trading Types: h ∈ H ≡ {f, c}

Ef

t−1 [xt+1] = 0

Ec

t−1 [xt+1] = g · xt−1

nc

t ≡ nt proportion of c-type

nf = 1 − nt

• proportion of f-type
• Dynamics

         xt = g R · ntxt−1 nt = ∆t = eβπc

t−1

eβπf

t−1 + eβπc t−1

πh

t

= zh

t−1(xt − Rxt−1) − ch

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INFORMATION NETWORK

• As in Panchenko et al. (2013) performance of types h is only

locally observable:

◮ for switching you need information from agents of other types ◮ agent i gathers information from ki other agents - neighbours ◮ P(k) can capture cognitive overload, inattention

Timeline

• 1. Agents survey their neighborhood
• 2. Agents select their type hi,t (based on past performance)
• 3. Demand for risky asset is generated
• 4. Price pt determined via a Walrasian market clearing
• 5. Agents portfolios are updated, dividend realizes.
• 6. Agents observe performance of their strategies

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MEAN-FIELD APPROXIMATION

• Nodes are homogeneous conditional on their own degree k
• Neighbors types are independent from each other

A random link in the network points to a chartist with probability θt =

• k

knk,t−1 · P(k)/k, where nk,t−1 fraction of chartists for agents with degree k, and k =

k k · P(k) average degree of the network

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SWITCHING

Let a counts the number of chartists in neighborhood ki,t.

• 1. Ft(k, a): probability of f → c given (k, a)
• 2. Rt(k, a): probability of c → f given (k, a)

(Possibly Rt(k, a) = Ft(k, a)) Probability for a fundamentalist with k links to switch to c: ˜ gk(θt) =

k

• a=0

Ft(k, a) k a

• θa

t (1 − θt)k−a

Probability for a chartist with k links to switch to f: ˜ qk(θt) =

k

• a=0

Rt(k, a) k a

• θa

t (1 − θt)k−a

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• Different selection mechanisms:
• 1. Brock and Hommes (1998).

Ft(k, a) = 1 − Rt(k, a) = ∆t, ∀a, k ≥ 0.

• 2. Panchenko et al (2013).

Ft(k, a) = 1 − Rt(k, a) =    for a = 0 ∆t for 0 < a < k 1 for a = k

• 3. Generalization - smooth transition from 0 to 1 depending on a

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DYNAMICS

                     xt = g R [

k P(k)nk,t] · xt−1

. . . nk,t = nk,t−1 − nk,t−1˜ qk(θt) + (1 − nk,t−1)˜ gk(θt) . . . θt =

• k

k · P(k) k nk,t−1

• The economy is described by a system of (k + 2) equations
• Each nk,t tracks the evolution of c-type traders endowed with k

links

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GENERALIZATION OF PANCHENKO ET AL. 2013

• With Panchenko et al. diffusion protocol, the LoM is:

nk,t = nk,t−1θk

t +

• nk,t−1
• 1 − θk

t

• + (1 − nk,t−1)
• 1 − (1 − θt)k

·∆t

• In equilibrium, the system is described by:

       x = g Rx

k>0 P(k)

• ∆x((1 − θ)k − 1)

∆x((1 − θ)k − θk) − (1 − θk)

• θ

=

• k>0 kP(k)

k

• ∆x((1 − θ)k − 1)

∆x((1 − θ)k − θk) − (1 − θk)

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FUNDAMENTAL STEADY STATES: g < (1 + r)

Consider E = (x, θ) the stationary states of the system.

• 1. Fundamental s.s. E0 = (0, 0), E1 = (0, 1) always exist.
• 2. Fundamental s.s. E2 = (0, ¯

θ) exists iff neighboring traders have

• n average at least one neighbor of either type:

∆0 1 − ∆0 · k2 k

• avg. deg of neighbor

> 1 and 1 − ∆0 ∆0 · k2 k

• > 1
• 3. For β < β1 = ln
• k2

k

c s.s. E2 exists: Regular Random Scale-Free k2 = k2 k2 = k + k2 k2 = ∞ β1

regular

< β1

random

< β1

scale−free

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Fixed Point Analysis vs Simulations: (k = 3, β = 1, g < 1 + r)

• Simulations match well approximation of analytical mappings.

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AVERAGE PRICE DEVIATION |x| FOR k = 2, g > 1 + r

• Ordering of primary and secondary bifurcations seems to depend
• n the network type:

Regular < Scale-Free < Random.

• Amplitude of |x|: Regular > Scale-Free > Random.

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SIMULATIONS: k = 2, β = 1, g > 1 + r

• Economy is sensitive to network typology for realistic range of k

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CONCLUSIONS

• Analytically tractable model for random networks with P(k)

◮ importance of neighborhood size

• Major features generated by the BH model are preserved under

various communication structures

• Importance of the network structures

◮ depending on P(k) faster bifurcations - less stability ◮ short period between primary and secondary bifurcations

• Future work - other network features, e.g. clustering

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