Local Interactions in a Market with Heterogeneous Expectations

Transcription

1 1 / 17 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

2 2 / 17 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

3 3 / 17 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

4 4 / 17 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

5 5 / 17 BASELINE FRAMEWORK: ASSET PRICING 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 : { max U i,t+1 (W i,t+1 ) max E z i,t z t 1 h [W t+1 ] a } i,t 2 V [W t+1] Agent i selects the trading type h i H to form expectations on p t. Market Clearing (zero supply of risky asset): n h Et 1 h [p t+1 + y t+1 ] Rp t t aσ 2 = 0 p t = 1 n h t Et 1 h [p t+1 ] + ȳ h H }{{} R h H individual mean-variance demand

6 6 / 17 Given the fundamental price p, define x t = p t p : x t = 1 n h t Et 1 h [x t+1 ] R h H Trading Types: h H {f, c} E f t 1 [x t+1] = 0 E c t 1 [x t+1 ] = g x t 1 Dynamics n c t n t }{{} proportion of c-type x t = g R n tx t 1 n f = 1 n t }{{} proportion of f-type e βπc t 1 n t = t = e βπf t 1 + e βπt 1 c = zt 1 h (x t Rx t 1 ) c h π h t

7 7 / 17 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 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

8 8 / 17 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 θ t = k kn k,t 1 P (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

9 9 / 17 SWITCHING 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: g k (θ t ) = k ( ) k F t (k, a) θt a (1 θ t ) k a a a=0 Probability for a chartist with k links to switch to f: q k (θ t ) = k ( ) k R t (k, a) θt a (1 θ t ) k a a a=0

10 10 / 17 Different selection mechanisms: 1. Brock and Hommes (1998). F t (k, a) = 1 R t (k, a) = t, a, k 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

11 11 / 17 DYNAMICS x t = g R [ 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 ). θ t = k k P (k) n k,t 1 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

12 12 / 17 GENERALIZATION OF PANCHENKO ET AL With Panchenko et al. diffusion protocol, the LoM is: ( ) ( n k,t = n k,t 1 θt k + (n k,t 1 1 θt k + (1 n k,t 1 ) 1 (1 θ t ) k)) t In equilibrium, the system is described by: x = g R x [ k>0 P (k) x ((1 θ) k ] 1) x ((1 θ) k θ k ) (1 θ k ) θ = k>0 k P (k) [ x ((1 θ) k ] 1) k x ((1 θ) k θ k ) (1 θ k )

13 13 / 17 FUNDAMENTAL STEADY STATES: 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: ( 0 k 2 ) > 1 and 1 ( 0 k 2 ) > k 0 k }{{} avg. deg of neighbor ( ) 3. For β < β 1 = ln k 2 /c k s.s. E2 exists: Regular Random Scale-Free k 2 = k 2 k 2 = k + k 2 k 2 = β 1 regular < β 1 random < β 1 scale free

14 Fixed Point Analysis vs Simulations: ( k = 3, β = 1, g < 1 + r) Simulations match well approximation of analytical mappings. 14 / 17

15 AVERAGE 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

16 16 / 17 SIMULATIONS: k = 2, β = 1, g > 1 + r Economy is sensitive to network typology for realistic range of k

17 17 / 17 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