Sentiment and speculation in a market with heterogeneous beliefs Ian Martin Dimitris Papadimitriou September, 2020 Martin & Papadimitriou Sentiment and speculation September, 2020 1 / 42
Introduction Agents disagree about the probabilities of good/bad news Optimists go long; pessimists go short If the market rallies, optimists get rich; if the market sells off, pessimists get rich So prices embed ex post winners’ beliefs This sentiment effect boosts volatility, and hence risk premia Sentiment induces speculation : agents trade at prices that they think are not warranted by fundamentals, in anticipation of adjusting their positions in future Martin & Papadimitriou Sentiment and speculation September, 2020 2 / 42
Related literature An incomplete and somewhat arbitrary list Heterogeneous beliefs ◮ Keynes (1936, Chapter 12); Harrison and Kreps (1978); Scheinkman and Xiong (2003); Geanakoplos (2010); Simsek (2013); Basak (2005); Banerjee and Kremer (2010); Atmaz and Basak (2018); Zapatero (1998); Jouini and Napp (2007); Bhamra and Uppal (2014); Kogan, Ross, Wang, and Westerfield (2006); Buraschi and Jiltsov (2006); Sandroni (2000); Boroviˇ cka (2020); Blume and Easley (2006); Cvitani´ c, Jouini, Malamud, and Napp (2011); Chen, Joslin, and Tran (2012); . . . Heterogeneous risk aversion ◮ Dumas (1989); Chan and Kogan (2002); Longstaff and Wang (2012); . . . Martin & Papadimitriou Sentiment and speculation September, 2020 3 / 42
Setup Agents indexed by h ∈ ( 0 , 1 ) are endowed with one unit of a risky asset The asset evolves from t = 0 , . . . , T on a binomial tree with exogenous terminal payoffs The interest rate is normalized to zero Agent h thinks the probability of an up-move is h Agents have log utility over terminal wealth No learning (today; see the paper for results with learning) Martin & Papadimitriou Sentiment and speculation September, 2020 4 / 42
α = 9, β = 5 3.0 α = β = 5 2.5 α = 2, β = 4 2.0 1.5 α = β = 1 1.0 0.5 1.0 h 0.0 0.2 0.4 0.6 0.8 The mass of agents with belief h follows a beta distribution, pdf f ( h ) ∝ h α − 1 ( 1 − h ) β − 1 where α, β > 0 Martin & Papadimitriou Sentiment and speculation September, 2020 5 / 42
Equilibrium (1): individual optimization Solve backwards: the price of the risky asset is p d or p u next period Agent h has wealth w h at the current node If he chooses to hold x h units of the asset, then wealth next period is w h − x h p + x h p u in up state and w h − x h p + x h p d in the down state So portfolio problem is max x h h log [ w h − x h p + x h p u ] + ( 1 − h ) log [ w h − x h p + x h p d ] First order condition: � � h − 1 − h x h = w h p − p d p u − p Martin & Papadimitriou Sentiment and speculation September, 2020 6 / 42
Helpful to rewrite the FOC in terms of the risk-neutral probability of an up-move, p ∗ , which is defined via p = p ∗ p u + ( 1 − p ∗ ) p d : p ∗ = p − p d p u − p d We then have � � − 1 − h h w h h − p ∗ x h = w h = p − p d p u − p p u − p d p ∗ ( 1 − p ∗ ) Martin & Papadimitriou Sentiment and speculation September, 2020 7 / 42
If agent h behaves optimally, wealth next period is h w h + x h ( p u − p ) = w h p ∗ in the up-state, and 1 − h w h − x h ( p − p d ) = w h 1 − p ∗ in the down-state In either case, all agents’ returns are linear in their beliefs, h This is true at every node Martin & Papadimitriou Sentiment and speculation September, 2020 8 / 42
The wealth distribution Everyone starts with equal wealth at time 0 After m up and n down steps, person h ’s wealth is λ path h m ( 1 − h ) n Aggregate wealth equals p , so � 1 λ path h m ( 1 − h ) n f ( h ) dh = p 0 and hence B ( α, β ) λ path = B ( α + m , β + n ) p Martin & Papadimitriou Sentiment and speculation September, 2020 9 / 42
wealth share 2.0 1.5 · d 1.0 du duu 0.5 1.0 h 0.2 0.4 0.6 0.8 After m up and n down steps, agent h ’s share of aggregate wealth is w h B ( α, β ) B ( α + m , β + n ) h m ( 1 − h ) n p = The richest agent is h = m / ( m + n ) , who looks right in hindsight Martin & Papadimitriou Sentiment and speculation September, 2020 10 / 42
Equilibrium (2): market clearing From the FOC, � � B ( α, β ) h − 1 − h B ( α + m , β + n ) h m ( 1 − h ) n p x h = p − p d p u − p � �� � w h In aggregate the agents hold one unit of the asset: � 1 x h f ( h ) dh = 1 0 The equilibrium price is therefore p = ( m + α ) p d p u + ( n + β ) p u p d ( m + α ) p d + ( n + β ) p u Martin & Papadimitriou Sentiment and speculation September, 2020 11 / 42
In terms of risk-neutral probability, at time t = m + n , H m , t p d p ∗ = H m , t p d + ( 1 − H m , t ) p u where � 1 m + α hw h f ( h ) H m , t = t + α + β = dh p 0 is wealth-weighted average belief For comparison, in a homogeneous economy with up-prob H , Hp d p ∗ = Hp d + ( 1 − H ) p u Martin & Papadimitriou Sentiment and speculation September, 2020 12 / 42
all cash representative agent h = 0 h = p * h = H m , t h = 1 shorts balanced levered optimists h − p ∗ Share of wealth agent h invests in the risky asset is H m , t − p ∗ Representative agent—“Mr. Market”—with h = H m , t invests fully in the risky asset The agent with h = p ∗ invests fully in the bond Martin & Papadimitriou Sentiment and speculation September, 2020 13 / 42
Agents disagree on the risk premium agent h ’s perceived risk premium = ( h − p ∗ )( H m , t − p ∗ ) p ∗ ( 1 − p ∗ ) But they agree on objectively measurable quantities, such as risk-neutral variance = ( H m , t − p ∗ ) 2 p ∗ ( 1 − p ∗ ) Notice that H m , t − p ∗ = agent h ’s risk premium h − p ∗ risky share of agent h = risk-neutral variance In particular, the risk premium perceived by the representative agent equals risk-neutral variance Martin & Papadimitriou Sentiment and speculation September, 2020 14 / 42
A pricing formula Result If the risky asset has terminal payoffs p m , T then its initial price is 1 p 0 = T � c m p m , T m = 0 where � T � B ( α + m , β + T − m ) c m = m B ( α, β ) Result (Signing the effect of heterogeneity on prices) If 1 / p m , T is convex (concave) in m, the price p 0 falls (rises) as heterogeneity increases Martin & Papadimitriou Sentiment and speculation September, 2020 15 / 42
Example 1: geometric payoffs, uniform belief distn. p = 2.25 p = 1.69 p = 1.50 H 1,1 = 0.67 p * = 0.50 p = 0.96 p = 1.00 p = 1.13 H 0,0 = 0.50 p * = 0.29 p = 0.68 p = 0.75 H 0,1 = 0.33 p * = 0.20 p = 0.56 p : price. p : price in homogeneous economy. H m , t : identity of rep agent. p ∗ : risk-neutral prob (cutoff between longs and shorts). Martin & Papadimitriou Sentiment and speculation September, 2020 16 / 42
Sharpe ratio 3 2 · 1 d u 1.0 h 0.2 0.4 0.6 0.8 - 1 - 2 Mr. Market perceives a higher Sharpe ratio in “up” than “down” This is the opposite of what any individual thinks Martin & Papadimitriou Sentiment and speculation September, 2020 17 / 42
Example 2: A risky bond T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism: ◮ h = 0 . 50 thinks default prob is less than 10 − 15 ◮ h = 0 . 25 thinks default prob is less than 10 − 6 ◮ h = 0 . 10 thinks default prob is less than 0.006% ◮ h = 0 . 05 thinks default prob is less than 8% ◮ h = 0 . 01 thinks default prob is more than 60% Initially, h = 0 . 50 is the representative agent What price does the bond trade at? Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42
Example 2: A risky bond T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism: ◮ h = 0 . 50 thinks default prob is less than 10 − 15 ◮ h = 0 . 25 thinks default prob is less than 10 − 6 ◮ h = 0 . 10 thinks default prob is less than 0.006% ◮ h = 0 . 05 thinks default prob is less than 8% ◮ h = 0 . 01 thinks default prob is more than 60% Initially, h = 0 . 50 is the representative agent What price does the bond trade at? at $95.63 Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42
Example 2: A risky bond T = 50 periods to go. Uniform beliefs Bond defaults (recover 30) in the bottom state. Else pays 100 In order of increasing pessimism: ◮ h = 0 . 50 thinks default prob is less than 10 − 15 ◮ h = 0 . 25 thinks default prob is less than 10 − 6 ◮ h = 0 . 10 thinks default prob is less than 0.006% ◮ h = 0 . 05 thinks default prob is less than 8% ◮ h = 0 . 01 thinks default prob is more than 60% Initially, h = 0 . 50 is the representative agent What price does the bond trade at? at $95.63 Who would go short, at this price? Martin & Papadimitriou Sentiment and speculation September, 2020 18 / 42
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