Inefficient Markets VI Damien Challet [email protected] October 22, 2014 Damien Challet Inefficient Markets VI Previous episodes Adaptive Complex Market Hypothesis Price returns: interesting noise? Price returns: not chaos Simple agent-based models fundamentalists trend followers not playing feedback from gains to strategy use Interaction Phase transitions: DANGER Damien Challet Inefficient Markets VI Fractals Mandelbrot (1924-2010) fractal object: invariant under change of scale (zoom, de-zoom) (romanesco) Damien Challet Inefficient Markets VI Fractals Mandelbrot set Damien Challet Inefficient Markets VI Critical point: fractal At critical point: fractal spatial properties ≡ self-similar system ≡ scale invariance ≡ system same at all scales Damien Challet Inefficient Markets VI Scale invariance and power-laws Measure of something over self-similar system g(x) Change of scale by factor a: g(ax) = h(a)g(x) Hypothesis: g(1) = 1 =⇒ g(a) = h(a) g(xy) = g(x)g(y) Only solution: g(x) = x−α Damien Challet Inefficient Markets VI Benford’s law Astronomer Newcomb (1881) Logarithm tables: first pages are the most dirty Measures and proposes Pdigit (d) Damien Challet Inefficient Markets VI Benford’s law Astronomer Newcomb (1881) Logarithm tables: first pages are the most dirty Proposes Pdigit (d) Physicist Benford (1936) law valid in variety of data sources numbers in newspapers length of rivers taxation forms accounting books NOT phone books Hill (1995) Random numbers from a random set of probability distributions → Benford’s law Damien Challet Inefficient Markets VI Benford’s law First digit law Pdigit (d is the first digit) 1 Pdigit (d) = log10 1 + d Pdigit (d) independent from unit (meters, feet, etc) P(x) ∝ 1x , xmax R d+1 Pdigit (d) = P(x) R xdmax dxP(x) 1 Damien Challet 1 = log10 1 + d Inefficient Markets VI Complex networks: how much complex? Benchmark (zero model): Erdös-Renyi (ER) random graphs k: number of links, P(k) ∝ exp(−k/p) (no hubs) Complex networks k: number of links, P(k) ∝ k−α (hubs) Clustering Assortativity Damien Challet Inefficient Markets VI Complex networks: clustering open triple closed triple The friend of my friend is also my friend clustering coefficient CC = 3x number of triangles number of links Small world: CC CCER = c/N → 0 Damien Challet Inefficient Markets VI Complex networks: assortativity Assortative: hubs link to hubs Intuitively: E(knearest neighbours |k) = ∑k0 k0 P(k0 |k) E(knearest neighbours |k) grows with k in assortative networks Damien Challet Inefficient Markets VI Processes on networks Transfer Roads Airplane routes Electric wiring Information Internet wires Emails Money Contagion sentiment risk disease panic default Damien Challet Inefficient Markets VI Network: systemic risk E.g.: links between businesses Caldarelli (2007) Contagion sentiment risk disease panic default Cascading defaults Damien Challet Inefficient Markets VI Epidemic propagation on heterogeneous networks Vespgnagni et al (1999): SIS/SIR Site i susceptible → infected → susceptible (SIS) Site i susceptible → infected → removed (SIR) If power-law link distribution, disease propagate much more easily Optimal immunisation strategy: hubs Application (computer) virus rumors financial defaults Damien Challet Inefficient Markets VI Systemic crisis of banking systems (TRENDY) Overnight network of lending Bank i lends ci→j to bank j Directed network Assume bank k defaults Default propagation? If many banks affected , systemic risk What about many banks defaulting at the same time for the same reason (cf. Greece)? Damien Challet Inefficient Markets VI Binary interactions and global sentiment: framework 2-by-2 interaction Opinion of agent i: si ∈ {−1, 0, 1} Agent j influences agent i: link Jij 6= 0 J: network of influences Effective influence: Jij sj Total influence on agent i: ∑j Jij sj Total influence of agent i: si ∑j Jji Damien Challet Inefficient Markets VI Interactions: framework Convention: Jij > 0 ⇐⇒ i likes to agree with j Frustration if Jij si sj < 0 Total frustration H = − ∑ Jij si sj ij Central principle Dynamics of si should tend to minimise H Damien Challet Inefficient Markets VI Market sentiment dynamics: example Michard-Bouchaud (2005) N agents si ∈ {−1, 1} Opinion Personal a priori towards si = +1 φi heterogeneity Global pressure F(t) Social pressure ∑j Jij sj (t) Frustration of i −si [F + φi + ∑j Jij sj (t)] Damien Challet Inefficient Markets VI Opinion dynamics Michard-Bouchaud (2005) Dynamics ! si (t + 1) = sign F(t) + φi + ∑ Jij sj (t) j Total frustration H = −NF ∑ si − ∑ φi si − ∑ Jij si sj i i ij a.k.a random-field ({φi }) Ising ({si }) model (RFIM) Damien Challet Inefficient Markets VI Opinion shifts: absence of social influence Jij = 0: ( +1 si = sign (F + φi ) = −1 φi > −F φi < −F Global opinion O(t) = if R(x) = Rx −∞ P(φ )dφ O0 = − 1 si (t) N∑ i and N 1 R(−F) + (1 − R(−F)) = 1 − 2R(−F) | {z } | {z } fraction of - fraction of + Damien Challet Inefficient Markets VI Opinion shifts: social influence Jij = J/N: complete graph, no distance, mean-field model ( +1 φi > −F − JO si (t + 1) = sign (F(t) + φi + JO(t)) = −1 φi<−F−JO O = 1 − 2R(−F − JO) Small imitation J 1: R(−F − JO) ' R(−F) − P(−F)OJ O' Damien Challet O0 1 − 2P(−F)J Inefficient Markets VI Abrupt opinion shifts Response function: reaction to small perturbation dO dF maximum at maximum of P(φ ) P(φ ) ∼ N(φ0 , σ ): max P(φ ) ∝ 1/σ J small enough: no divergence J = Jc ∝ 1/ max P(φ ) ∝ σ : divergence of Damien Challet dO dF Inefficient Markets VI Abrupt opinion shifts J small enough: no divergence J = Jc ∝ σ : divergence of dO dF J > Jc : O = 1 − 2R(−F − JO) has three solutions, hysteresis Damien Challet Inefficient Markets VI Opinion shifts Near critical point J . Jc 1 dO = G dF Jc − J F − Fc (J) (Jc − J)3/2 G : known exactly, indep from P(φ ) G (0) = cst < ∞ G (x → ∞) ∝ x−2/3 Damien Challet Inefficient Markets VI Test for presence of social interaction Near critical point J . Jc dO 1 = G dF Jc − J max dO dF 1 Jc −J dO dF large F − Fc (J) (Jc − J)3/2 =h∝ near its max, in window w ∝ (Jc − J)3/2 h ∝ w−2/3 Without interaction: dO dF = 2P(−F): e.g. Gaussian h ∝ w−1 Damien Challet Inefficient Markets VI Example: birth rates Damien Challet Inefficient Markets VI Example: cell phone Damien Challet Inefficient Markets VI Example: hand clapping Damien Challet Inefficient Markets VI Validation I Damien Challet Inefficient Markets VI Validation II Damien Challet Inefficient Markets VI RFIM and finance Iori (1999): RFIM + thresholds si (t) ∈ {−1, 0, +1} signal Yi (t) = ∑ Jij sj (t) + ε(t) + νi (t) j where ε(t) ∼ N(0, σε ), νi (t) ∼ N(0, σν ) thresholds θi ∼ N(0, σθ ) +1 si (t + 1) = −1 0 Yi (t) > θi (t) Yi (t) < −θi (t) otherwise Random ε(t) sweeps over critical point Damien Challet Inefficient Markets VI RFIM and finance Iori (1999): RFIM + thresholds P(ξ ) unbiased: max at ξ = 0 Random ε(t) sweeps over critical point if J large enough Damien Challet Inefficient Markets VI Summary Phase transitions: scale-invariance extreme sensitivity opinion dynamics Binary models: si = ±1 Interaction on networks Damien Challet Inefficient Markets VI
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