Local Interactions in a Market with Heterogeneous Expectations Mikhail Anufriev 1 Andrea Giovannetti 2 Valentyn Panchenko 3 1 , 2 University of Technology Sydney 3 UNSW Sydney, Australia Computing in Economics and Finance 2018 Milano 1 / 17
T HE S CENE ◦ 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 2 / 17
L ITERATURE 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 3 / 17
C ONTRIBUTION 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 4 / 17
B ASELINE FRAMEWORK : A SSET P RICING MODEL Brock and Hommes, 1998, JEDC ◦ Large population of traders N = { 1 , 2 , ..., N } trading combination of risk-free bond with return R and risky asset z with stochastic dividend y t ◦ t = 1 , 2 , ... ◦ Agents. Myopic optimizers with CARA preferences in wealth W i,t : � t − 1 [ W t +1 ] − a � E h ⇔ max z i,t U i,t +1 ( W i,t +1 ) max 2 V [ W t +1 ] z i,t ◦ Agent i selects the trading type h i ∈ H to form expectations on p t . ◦ Market Clearing (zero supply of risky asset): E h t − 1 [ p t +1 + y t +1 ] − Rp t = 0 ⇒ p t = 1 � � n h n h t E h t − 1 [ p t +1 ] + ¯ y t aσ 2 R h ∈H h ∈H � �� � individual mean-variance demand 5 / 17
◦ Given the fundamental price p ∗ , define x t = p t − p ∗ : x t = 1 � n h t E h t − 1 [ x t +1 ] R h ∈H ◦ Trading Types : h ∈ H ≡ { f, c } E f E c t − 1 [ x t +1 ] = 0 t − 1 [ x t +1 ] = g · x t − 1 n f = 1 − n t n c t ≡ n t � �� � � �� � proportion of f-type proportion of c-type ◦ Dynamics = g x t R · n t x t − 1 e βπ c t − 1 n t = ∆ t = e βπ f t − 1 + e βπ c t − 1 π h = z h t − 1 ( x t − Rx t − 1 ) − c h t 6 / 17
I NFORMATION N ETWORK ◦ 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 k i other agents - neighbours ◮ P ( k ) can capture cognitive overload, inattention Timeline 1. Agents survey their neighborhood 2. Agents select their type h i,t (based on past performance) 3. Demand for risky asset is generated 4. Price p t determined via a Walrasian market clearing 5. Agents portfolios are updated, dividend realizes. 6. Agents observe performance of their strategies 7 / 17
M EAN - 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 � θ t = kn k,t − 1 · P ( k ) / � k � , k where n k,t − 1 fraction of chartists for agents with degree k , and � k � = � k k · P ( k ) average degree of the network 8 / 17
S WITCHING Let a counts the number of chartists in neighborhood k i,t . 1. F t ( k, a ) : probability of f → c given ( k, a ) 2. R t ( k, a ) : probability of c → f given ( k, a ) (Possibly R t ( k, a ) � = F t ( k, a ) ) Probability for a fundamentalist with k links to switch to c : k � k � � t (1 − θ t ) k − a θ a g k ( θ t ) = ˜ F t ( k, a ) a a =0 Probability for a chartist with k links to switch to f : k � k � � t (1 − θ t ) k − a θ a q k ( θ t ) = ˜ R t ( k, a ) a a =0 9 / 17
◦ Different selection mechanisms: 1. Brock and Hommes (1998). F t ( k, a ) = 1 − R t ( k, a ) = ∆ t , ∀ a, k ≥ 0 . 2. Panchenko et al (2013). 0 for a = 0 F t ( k, a ) = 1 − R t ( k, a ) = ∆ t for 0 < a < k 1 for a = k 3. Generalization - smooth transition from 0 to 1 depending on a 10 / 17
D YNAMICS g R [ � x t = k P ( k ) n k,t ] · x t − 1 . . . n k,t = n k,t − 1 − n k,t − 1 ˜ q k ( θ t ) + (1 − n k,t − 1 )˜ g k ( θ t ) . . . k · P ( k ) � θ t = n k,t − 1 k � k � ◦ The economy is described by a system of ( k + 2) equations ◦ Each n k,t tracks the evolution of c -type traders endowed with k links 11 / 17
G ENERALIZATION OF P ANCHENKO ET AL . 2013 ◦ With Panchenko et al. diffusion protocol, the LoM is: � � � � 1 − (1 − θ t ) k �� n k,t = n k,t − 1 θ k 1 − θ k t + n k,t − 1 + (1 − n k,t − 1 ) · ∆ t t ◦ In equilibrium, the system is described by: ∆ x ((1 − θ ) k − 1) � � g Rx � x = k> 0 P ( k ) ∆ x ((1 − θ ) k − θ k ) − (1 − θ k ) ∆ x ((1 − θ ) k − 1) � � k> 0 kP ( k ) � θ = ∆ x ((1 − θ ) k − θ k ) − (1 − θ k ) � k � 12 / 17
F UNDAMENTAL S TEADY S TATES : g < (1 + r ) Consider E = ( x, θ ) the stationary states of the system. 1. Fundamental s.s. E 0 = (0 , 0) , E 1 = (0 , 1) always exist. 2. Fundamental s.s. E 2 = (0 , ¯ θ ) exists iff neighboring traders have on average at least one neighbor of either type: � � k 2 � � � � k 2 � � ∆ 0 1 − ∆ 0 · > 1 and · > 1 1 − ∆ 0 � k � � k � ∆ 0 � �� � avg. deg of neighbor � � � 3. For β < β 1 = ln � k 2 � c s.s. E 2 exists: � k � Regular Random Scale-Free � k 2 � = � k � 2 � k 2 � = � k � + � k � 2 � k 2 � = ∞ β 1 β 1 β 1 < < regular random scale − free 13 / 17
Fixed Point Analysis vs Simulations: ( � k � = 3 , β = 1 , g < 1 + r ) ◦ Simulations match well approximation of analytical mappings. 14 / 17
A VERAGE PRICE DEVIATION | x | FOR � k � = 2 , g > 1 + r ◦ Ordering of primary and secondary bifurcations seems to depend on the network type: Regular < Scale-Free < Random. ◦ Amplitude of | x | : Regular > Scale-Free > Random. 15 / 17
S IMULATIONS : � k � = 2 , β = 1 , g > 1 + r ◦ Economy is sensitive to network typology for realistic range of � k � 16 / 17
C ONCLUSIONS ◦ 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 17 / 17
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