Insider Trading and Multidimensional Private Information

Transcription

1 Insider Trading and Multidimensional Private Information Tomasz Sadzik, UCLA, Chris Woolnough, NYU March 2014 Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

2 Overview Why asset prices diverge from their fundamental values? Inflated prices (or mispricing in general) are an important and a well-established empirical fact. But they are hard to explain in a rational framework, given arbitrage arguments. We provide an answer: A purely Bayesian model of inflated prices, with no behavioral elements, or agency problems. Our model is just a hair breadth away from the standard model with well-behaved prices. It is the standard model that is nongeneric! Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

3 Overview In our model there is one informed trader who trades over time. He knows the fundamental value as well as the the exogenous demand shock. (multidimensional private information). In this model, with just information about fundamentals prices are stable (Kyle 85). But his equilibrium strategy is to inflate the price, no matter how small the exogenous demand shock. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

4 Contributions - detailed Contributions: Analytically solved model, up to an ODE, with multidimensional uncertainty. Contrast with binary second dimension. Novel "Indeirect arbitrage argument": Insider in the end reveals all the info, also about payoff irrelevant exogenous demand shock. "Familiar yet new" argument for why insider exacerbates demand shocks. Here it is part of the optimal equilibrium obfuscation strategy. Prices are inflated even if the demand shock is arbitrarily small. So: "Stable prices" are a fragile outcome, Price is strategically inflated by the Insider. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

5 Literature Review Dynamic Asset Pricing with 1-dimensional private information: Glosten Milgrom (1985), Kyle (1985), Back (1992),... Arbitrage arguments and information revelation:..., Ostrovsky (2009). Multidimensional uncertainty (quality of information) and mispricing: Keynes (1936), Romer (1993), Easley O Hara (1987), Avery and Zemsky (1998), Green and Banerjee (2013). Multidimensional uncertainty and contrarian behavior: Allen Gale (1992), Foster Viswanathan (1994), Fishman Hagerty (1995). Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

6 The Model At t = 0: Single Informed trader observes value of the asset v N ( 0, s 2 v ), and mean demand from liquidity traders m N ( 0, s 2 m), v m. At t =, 2,..., 1 : (I)nformed trader submits order u t, (L)iquidity trader submits order ε t N (m, ), (M)arket maker observes the total order u t + ε t and sets the price p t = E[v u + ε,..., u t + ε t ]. At time t = 1 I gets u t [v p t ] Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

7 Special case 1: Static model Suppose there is only one round of trading. Linear solution: price = λ (u + ε), where λ = profit = u (v λ (u + ε)), FOC implies the equilibrium strategy Cov (u+ε,v ) Var (u+ε), solve for λ. u = 1 2λ v 1 2 m, Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

8 Special case 2: Kyle 85 Suppose now I has information only about v (or s 2 m = 0), but the trading is dynamic. Linear solution: u t = β t [v p t ], β t > 0, E [p t v] = vt. v 0.5 v 0.02 Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

9 Warm-up Equilibrium here is I s dynamic trading strategy that is optimal given that M knows this strategy. M learns from observed order flow about v and m using Kalman formula: d (E t [v]) = λ t (u t + ε t E t [u t + ε t ]) = λ t (u t + y t ) dt + λ t db t, d (E t [m]) = φ t (u t + ε t E [u t + ε t ]) = φ t (u t + y t ) dt + φ t db t, where λ t = Cov (u t + ε t,v t ), φ t = Cov (u t + ε t,m t ). Throughout we will use the following state variables: x t = v p t, y t = m E t [m]. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

10 Preliminary Result 1 Proposition. For = 0 there is a unique linear Markov equilibrium. Insider uses a linear strategy of the form: u t = β t x t + δ t y t. The value function is quadratic: E [payoff I t, x t, y t ] = axt 2 + b t x t y t + cyt 2 + d t. The law of motion of the state variables are: dx t = λ t (u t + y t ) dt λ t db t, dy t = φ t (u t + y t ) dt φ t db t. All the parameters a, b t, c, d t, β t, δ t, λ t, φ t are solved upto an ODE. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

11 Preliminary Result 1 How do we pin down the parameters? Insider is indifferent at any point of time. Note that his flow payoffs are u t x t, linear in his policy. This relates the learning parameters λ t and φ t and the value function. The learning parameters must be "justified" by Bayes formula. This relates the strategy of the Insider (β t and δ t ) and the learning parameters λ t and φ t, All the information must be revealed by the end (see next lemma). This relates the learning parameters λ t and φ t to the exogenous Var (x 0 ), Var (y 0 ) and Cov (x 0, y 0 ). Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

12 Preliminary Result 2 Lemma. Let [ x 2 Σ t = E t x t y t x t y t yt 2 Then Σ 1 = 0, i.e. all the information is revealed by the end of trading (p 1 = v and E 1 [m] = m with probability 1). ]. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

13 Preliminary Result 2 Proof: That all information about v is revealed is intuitive, from the arbitrage argument : If v = E t [v] then there are some unrealized gains from trade for I. Formally: Suppose that E [ x1 2 ] = σ 2 > 0. I is indifferent between original strategy and speeding up trade so that E [ x1 ε] 2 = σ 2. But now he can make additional profits between time 1 ε and 1. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

14 Preliminary Result 2 Proof: The same arbitrage argument does not work for m, as there are no direct gains from trade on m. However, if v = p t but m = E t [m] then v would deterministically move away from E t [v]. This creates indirect gains from trade. Formally: Suppose that E [ x1 2 ] [ ] = 0 but E y 2 1 = σ 2 > 0. I is indifferent between original strategy and "speeding up trade" so that E [ x1 ε] 2 [ ] = 0 and E y 2 1 ε = σ 2. But since d (v p t ) = dx t = λ t (u t + y t ) dt λ t db t, the price will move now deterministically away from v. Therefore I can now make additional profits between time 1 ε and 1. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

15 Preliminary Result 3 Recall that u t = β t (v p t ) + δ t (m E t [m]). In the static model we have u = β (v 0) 1 (m 0), 2 i.e., δ < 0 (Insider trades against, dampens the demand shock). Lemma. For every t in [0, 1] δ t > 0 (I amplifies the shock). Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

16 Preliminary Result 3 Heuristically: why the difference with the static model? First, simply, you can postpone trading on "biased market belief about exogenous shock" until later. Both in the static and dynamic model, different signs of biases are Insider s "sweet spot" (e.g., asset underpriced and supply shock). Only in the dynamic model Insider can steer market s beliefs in desired direction. To get negatively correlated biases, set β, δ > 0. Together: In dynamic model the Insider can first steer the biases in the right direction by setting δ > 0, and only then start making money. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

17 Preliminary Result 3 Proof idea: δ 0 < 0 can be shown to be inconsistent with Insider s indifference: As we said above, opposite signs of x t and y t is Insider s sweet spot. With δ 0 << 0 order flow moves x 0 and y 0 in opposite directions. But now, think about buying when x 0 = y 0 = 0 It pushes y and x up (good), while driving them apart (also good). δ 0 ( 1, 0) is also inconsistent with Insider s indifference. market puts too much weight on the influence of the exogenous shock, which can be exploited by the Insider. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

18 Main Result Proposition. Fix s 2 v = 1. Suppose v = 0 and m = 3s m. Then: s m 1 s m 0.1 s m 0.01 s m Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

19 "Proof" 1/2 Fix (x 0, y 0 ) = (0, 3s m ), condition all expectations on this event. Lets show a weaker result: Why max t E[p t ] >> 0 even as s m 0? Given (x 0, y 0 ) = (0, 3s m ) how will the expected price evolve? Given positive demand shock amplified by the Insider, initially the price climbs up (M thinks demand might be due to v high). As price goes up, I starts selling the asset, dampening the positive order flow. Price will stop rising when the expected order flow is zero, or β t E[0 p t ] + (δ t + 1) E[y t ] = 0 To show: δ t +1 β t 1 s m as s m 0 (I GREATLY amplifies the shock). Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

20 "Proof" 2/2 Suppose instead that δ t +1 β t 1. In this case the order flow is driven mostly by x t and hardly by y t. But then it is impossible for the M to learn about y t. Generally: to successfully learn about two variables from one signal, the signal must covary with each variable to the same order of magnitude. Formally: Kalman formula (projection theorem) says that: [ ] [ ] [ ] [ λt Cov (ut + ε = t, x t ) x 2 = E t x t y t φ t Cov (u t + ε t, y t ) x t y t yt 2 β t δ t + 1 In the covariance matrix, the right column is much smaller than the left collumn. So unless δ t >> β t, we have a system of two equations with basically one unknown. ]. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

21 Extensions The main message remains the same for several extensions of the model: Allow I s information about shock or fundamentals to be incomplete, so that he learns as well. Make I risk averse (CARA). We believe the logic behind the hypersensitive prices is robust to parametric assumptions. The intuition does not rely on normality of noise, which helps get analytic solutions. The intuition does not seem to rely on the particular kind of addition private info (shock). We start working on the private information about own payoff function. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22

22 Conclusions Analytically solved model with multidimensional uncertainty. Novel "Indirect arbitrage argument". New argument for why Insider exacerbates demand shock. Prices are inflated even if the demand shock is arbitrarily small. So: "Stable prices" are a fragile outcome, Price is strategically inflated by the Insider. Bottomline: Inflating prices is the optimal dynamic strategy of the Insider with multidimensional private information, who privately observes small yet unexpected demand shock. Tomasz Sadzik, UCLA, Chris Woolnough, NYU Insider ( ) Trading and Multidimensional Private Information March / 22