Beauty Contests and Fat Tails in Financial Markets Makoto Nirei 1 , - PowerPoint PPT Presentation

Beauty Contests and Fat Tails in Financial Markets Makoto Nirei 1 , 2 Koichiro Takaoka 3 Tsutomu Watanabe 4 1 Policy Research Institute, Ministry of Finance 2 Institute of Innovation Research, Hitotsubashi University 3 Faculty of Commerce and Management, Hitotsubashi University 4 Faculty of Economics, University of Tokyo SWET Hokkaido University August 8, 2015 1 / 43

Empirical literature on fat tails in finance ◮ Stock returns follow a fat-tailed distribution ◮ Evident in the high-frequency domain (Mandelbrot 1963; Fama 1963) ◮ The tail regularity could span historical crashes (Jansen and de Vries, REStat 1991; Longin, JB 1996) ◮ Leptokurtic (4th moment greater than the normal) ◮ Trading volumes also show a fat tail (Gopikrishnan, Plerou, Gabaix, and Stanley 2000) ◮ “It takes volume to move prices” 2 / 43

Fat tails of stock returns S&P 500 index, 1 minute interval, 6 years coverage. Source: Mantegna and Stanley, 2000, Cambridge 3 / 43

Source: Mantegna and Stanley 4 / 43

Source: Bouchaud and Potters, 2000, Cambridge 5 / 43

Tail distributions ◮ Gaussian φ ( x ) ∝ e − ( x − µ ) 2 / 2 σ 2 ◮ Parabola in a semi-log plot ◮ Exponential tail Pr( X > x ) ∝ e − λ x ◮ Linear in a semi-log plot ◮ Power law tail Pr( X > x ) ∝ x − α ◮ Linear in a log-log plot ◮ Does not have a finite variance if α < 2 ◮ ...nor a finite mean if α ≤ 1 (e.g. Cauchy) 6 / 43

Tail matters ◮ Fat tail affects risks ◮ volatility ◮ option price ◮ value at risk ◮ Power-law tail suggests the same mechanism for price fluctuations, small and large ◮ fractal, self-similar, scale-free ◮ crash ◮ high frequency data 7 / 43

Plan of the paper ◮ Develop a simultaneous-move rational-herding model of securities traders with private signal ◮ Derive a distribution of equilibrium aggregate actions ◮ Match with an empirical fat-tailed distributions of stock trading volumes and returns ◮ Provide an economic reason why the fat tail has to occur 8 / 43

Signal ◮ Two states of the economy: H (High) and L (Low) ◮ True state is H . ◮ Common prior belief Pr( H ) = Pr( L ) = 1 / 2 ◮ Each informed trader receives private signal X δ, i i.i.d. across i , which follows cdf F s δ in state s = H , L with common support Σ where sup Σ = ¯ Σ < ∞ . Also f s δ ( x ) > 0 for any x ∈ Σ. ◮ Likelihood ratio ℓ δ = f L δ / f H is strictly decreasing, and satisfies δ max x ∈ Σ | ℓ δ ( x ) − 1 | < δ ◮ Define the following likelihoods Pr( x δ, i < x | H ) = F L Pr( x δ, i < x | L ) δ ( x ) λ δ ( x ) ≡ F H δ ( x ) Pr( x δ, i ≥ x | H ) = 1 − F L Pr( x δ, i ≥ x | L ) δ ( x ) Λ δ ( x ) ≡ 1 − F H δ ( x ) ◮ λ δ ( x ) > ℓ δ ( x ) > Λ δ ( x ) > 0; λ ′ δ ( x ) < 0, Λ ′ δ ( x ) < 0 9 / 43

Market microstructure ◮ An asset that is worth 1 in H and 0 in L ◮ n informed traders decide to buy ( d n , i = 1) or not ( d n , i = 0). ◮ Each informed trader submits demand function d n , i ( p ). ◮ Trading volume is denoted by m n = � n i =1 d n , i . ◮ Aggregate demand function D ( p ) = � n i =1 d n , i ( p ) / n ◮ Uninformed traders submit supply function S ( p ) ◮ S (0 . 5) = 0, S ′ > 0, S (¯ Σ) = ¯ p < 1 ◮ Auctioneer clears the market D ( p ∗ n ) = S ( p ∗ n ) ◮ supp P ∗ n = [0 . 5 , ¯ p ] 10 / 43

Rational Expectations Equilibrium For each realization of information profile ( x δ, i ) n i =1 , a rational expectations equilibrium consists of price p ∗ n , trading volume m ∗ n , demand functions d n , i ( p ), and posterior belief r n , i such that ◮ for any p , d n , i ( p ) maximizes i ’s expected payoff evaluated at r n , i = r ( p n , x δ, i ) ◮ r n , i is consistent with p n and x δ, i for any i ◮ the auctioneer delivers the orders d ∗ n , i = d n , i ( p ∗ n ) and clears n = � n the market, S ( p ∗ n ) = m ∗ n / n , where m ∗ i =1 d ∗ n , i 11 / 43

Informed trader’s optimal behavior ◮ Trader i maximizes expected payoff: r n , i − p n if buying and 0 otherwise. ◮ p n ( m ) denotes the price level such that S ( p n ) = m / n ◮ i ’s optimal threshold policy: � 1 if x δ, i ≥ ¯ x ( m ) d n , i ( p n ( m )) = (1) 0 otherwise where ¯ x the threshold level of private signal at which i is indifferent between buying and not. 12 / 43

Threshold rule and revealed information Given the threshold rule, the information revealed by “buy” and “not-buy” actions are λ δ (¯ x ) and Λ δ (¯ x ). When p n ( m ) realizes, the information revealed to a buying trader is: x ( m )) m − 1 x ( m )) n − m Λ δ (¯ λ δ (¯ (2) The threshold is determined by: 1 x ( m )) n − m Λ δ (¯ x ( m )) m − 1 ℓ δ (¯ p n ( m ) − 1 = λ δ (¯ x ( m )) (3) 13 / 43

Upward sloping aggregate demand function ◮ Lemma 1: For sufficiently large n , ¯ x ( m ) is strictly decreasing in m and D ( p n ( m )) is non-decreasing in m . ◮ Proof: dm = − log (Λ δ ( x ) /λ δ ( x )) − { S ′ ( p n ( m )) p n ( m )(1 − p n ( m )) n } − 1 � d ¯ x n � � ( n − m ) λ ′ δ ( x ) /λ δ ( x ) + ( m − 1)Λ ′ δ ( x ) / Λ δ ( x ) + ℓ ′ δ ( x ) /ℓ δ ( x ) � x = Use λ ′ δ < 0, Λ ′ δ < 0, ℓ ′ δ < 0, and λ δ ( x ) > Λ δ ( x ). ◮ A higher price indicates that there are more traders who receive high signals → strategic complementarity 14 / 43

Existence of equilibrium ◮ Proposition 1: For sufficiently large n , there exists an equilibrium outcome n ) for each realization of ( x δ, i ) n ( p ∗ n , m ∗ i =1 . ◮ Proof: ◮ Construct a reaction function m ′ = Γ( m ) ≡ D ( p n ( m )) n : the number of traders with x δ, i ≥ ¯ x ( m ). ◮ Γ is non-decreasing, and thus Tarski’s fixed point theorem applies. ◮ Multiple equilibria may exist. We focus on the minimum equilibrium outcome m ∗ n . 15 / 43

Minimum outcome m ∗ n as a first passage time ◮ A counting process Y o ( x ) ≡ � n i =1 I X δ, i ≥ x , where X δ, i follows density f H δ ◮ M ∗ n is equivalent to the first passage time m such that Y o (¯ x n ( m )) = m . ◮ Change of variable t = ¯ x − 1 n ( x ) − 1. ( t corresponds to m − 1 for t = 0 , 1 , . . . , n − 1.) Then, t follows ˜ f δ, n ( t ) ≡ f H x ′ δ (¯ x n ( t + 1)) | ¯ n ( t + 1) | . ◮ Transform Y o ( x ) to Y ( t ), satisfying Y o ( x = ¯ x n ( m )) = Y ( t = m − 1). ◮ M ∗ n is the first passage time for Y ( t ) = t . 16 / 43

Y ( t ) follows a Poisson process asymptotically as n → ∞ ◮ The number of traders who switch to buy during ( t , t + dt ) follows a binomial distribution with population n − Y ( t ) and probability q δ, n ( t ) dt ≡ ˜ f δ, n ( t ) dt / F H δ (¯ x n ( t + 1)) ◮ Y (0) follows a binomial distribution with population n and probability q o δ, n ≡ 1 − F H δ (¯ x n (1)). ◮ Lemma 2: As n → ∞ , Y ( t ) asymptotically follows a Poisson process with intensity: log ℓ δ (¯ x n ( t + 1)) lim ℓ δ (¯ x n ( t + 1)) − 1 n →∞ 17 / 43

Change-of-time for the first passage time distribution ◮ τ φ δ, n ( · ) denotes the first passage time of the Poisson process Y ( t ) with intensity function φ δ, n ( t ) reaching t . 18 / 43

Change-of-time for the first passage time distribution ◮ τ φ δ, n ( · ) denotes the first passage time of the Poisson process Y ( t ) with intensity function φ δ, n ( t ) reaching t . ◮ Suppose that Y (0) = c . Then, τ φ δ, n ( · ) is the first passage time of Y ( t ) − Y (0) starting 0 and reaching t − c . 18 / 43

Change-of-time for the first passage time distribution ◮ τ φ δ, n ( · ) denotes the first passage time of the Poisson process Y ( t ) with intensity function φ δ, n ( t ) reaching t . ◮ Suppose that Y (0) = c . Then, τ φ δ, n ( · ) is the first passage time of Y ( t ) − Y (0) starting 0 and reaching t − c . ◮ N ( t ) denotes the Poisson process with intensity 1. τ 1 denotes the first passage time of N ( t ) reaching t . 18 / 43

Change-of-time for the first passage time distribution ◮ τ φ δ, n ( · ) denotes the first passage time of the Poisson process Y ( t ) with intensity function φ δ, n ( t ) reaching t . ◮ Suppose that Y (0) = c . Then, τ φ δ, n ( · ) is the first passage time of Y ( t ) − Y (0) starting 0 and reaching t − c . ◮ N ( t ) denotes the Poisson process with intensity 1. τ 1 denotes the first passage time of N ( t ) reaching t . � t ◮ Change of time: Y ( t ) is transformed to N ( 0 φ δ, n ( s ) ds ) 18 / 43

Change-of-time for the first passage time distribution ◮ τ φ δ, n ( · ) denotes the first passage time of the Poisson process Y ( t ) with intensity function φ δ, n ( t ) reaching t . ◮ Suppose that Y (0) = c . Then, τ φ δ, n ( · ) is the first passage time of Y ( t ) − Y (0) starting 0 and reaching t − c . ◮ N ( t ) denotes the Poisson process with intensity 1. τ 1 denotes the first passage time of N ( t ) reaching t . � t ◮ Change of time: Y ( t ) is transformed to N ( 0 φ δ, n ( s ) ds ) ◮ �� t � � � τ φ δ, n ( · ) ≡ inf t ≥ 0 | N φ δ, n ( s ) ds ≤ t − c 0 where inf ≡ ∞ 18 / 43