Scheinkman, J. A. and Xiong, W. (2003): Overcon dence and Speculative Bubbles, JPE, vol. 111, no.6

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1 Scheinkman, J. A. and Xiong, W. (2003): Overcon dence and Speculative Bubbles, JPE, vol. 111, no.6 Presented by: Ildikó Magyari March 26, 2010 March 26, / 16

2 The main motivation of the paper (1): 1. The behavior of asset prices and trading volumes of assets during episodes of price bubbles: high price volatility, high prices Figure: Source: Ofek, E. and Richardson, M. (2001) March 26, / 16

3 The main motivation of the paper (2): 2. The behavior of trading volumes of assets during episodes of price bubbles: the coexistence of high prices and high trading volume Figure: Source : Ofek, E. and Richardson, M. (2001) March 26, / 16

4 How do they rationalize these in an analytical framework? They propose: a continuous- time, in nite horizon model of asset trading March 26, / 16

5 How do they rationalize these in an analytical framework? They propose: a continuous- time, in nite horizon model of asset trading assume overcon dence (the belief of agent that his information is more accurate than it is in reality) Overconfidence (psychological bias) Heterogeneous believes Disagreement among investors and trading Asset owner has the option to sell the asset to others with more optimistic beliefs Agents value this option and pay prices that exceed their own valuation of future dividend stream: Price = E [fundamental value] + E[option value] March 26, / 16

6 The model: basic setup and main assumptions 1. One risky asset characterized by a cumulative dividend process and fundamental variable: dd t = f t dt + σ D dzt D ) observable df t = λ f t f dt + σ f dzt f ) not observable - the asset is nite total supply equal to unity 2. Two categories of risk neutral agents, {A,B}: 2.1. they observe a vector of signals: dst A dst B = f t dt + σ s dzt A = f t dt + σ s dzt B =) agents use the observation of D and the vector of signals to infere the fundamental and the value of the asset ASSUMPTION: Zt A, Zt B, Zt D, Zt f are mutually independent processes March 26, / 16

7 The model: basic setup and main assumptions 2.2. each agent believes that the informativeness of his own signal is larger than the observed one =) heterogenous beliefs Agent A : his own signal : ds A t = f t dt + σ s φdz f t + σ s q1 φ 2 dz A t agent B 0 s signal : ds B t = f t dt + σ s dz B t Agent B : his own signal : dst B = f t dt + σ s φdzt f + σ s q1 φ 2 dzt B agent A 0 s signal : dst A = f t dt + σ s dzt A where φ : the correlation of the innovation in the signal with the innovation in the fundamental process () Overcon dence parameter =) overreaction to signal 3. Assumptions: each group is large, there is no short-selling, agents can borrow and lend at the same rate of interest r, each group has in nitely total wealth March 26, / 16

8 because of the presence of overcon dence ( φ 6= 0) agent A overreacts to surprises in s A March 26, / 16 Implications of the assumed signal structure for conditional beliefs Main characteristics of the stationary solution: variance of the stationary solution is the same for both categories of agents: decreases with φ =)precision that agents atribute to their own signal increases q [λ + (φσ f /σ s )] 2 + (1 φ 2 ) [(2σ 2 f γ /σ2 s ) + (σ 2 f /σ2 D )] [λ + (φσ f /σ s ) 1/σ 2 D + 2/σ2 s conditional mean of the beliefs

9 Implications of the assumed signal structure for conditional beliefs di erences in beliefs follow a mean-reverting di usion process: g A = bf B g B = bf A bf A bf B March 26, / 16

10 Implications of the assumed signal structure for conditional beliefs di erences in beliefs follow a mean-reverting di usion process: g A = bf B g B = bf A bf A bf B which evolve according to the following process: dg A = ρg A dt + σ g dw A g =) in the min d of agent A dg B = ρg B dt + σ g dw B g =) in the min d of agent B March 26, / 16

11 Implications of the assumed signal structure for conditional beliefs di erences in beliefs follow a mean-reverting di usion process: g A = bf B g B = bf A bf A bf B which evolve according to the following process: dg A = ρg A dt + σ g dw A g =) in the min d of agent A dg B = ρg B dt + σ g dw B g =) in the min d of agent B where: σ g = p 2φσ f (larger overcon dence (φ)) larger variation in the di erences q in opinions) ρ = [λ + (φσ f /σ s )] 2 + (1 φ 2 ) [(2σ 2 f /σ2 s ) + (σ 2 f /σ2 D )] (increasing φ =) slower mean reversion) March 26, / 16

12 The model: Implications for Asset Trading 1 Di erences in beliefs vary over time (e.g. more omptimistic investors will buy and bid up the price of the asset)=) induces trade March 26, / 16

13 The model: Implications for Asset Trading 1 Di erences in beliefs vary over time (e.g. more omptimistic investors will buy and bid up the price of the asset)=) induces trade 2 no short-selling, nite supply of assets, in nite number of prospective buyers =) bidder pays his reservation price March 26, / 16

14 The model: Implications for Asset Trading 1 Di erences in beliefs vary over time (e.g. more omptimistic investors will buy and bid up the price of the asset)=) induces trade 2 no short-selling, nite supply of assets, in nite number of prospective buyers =) bidder pays his reservation price 3 this price = the agent s fundamental valuation + the expected gain from selling the asset at some point in the future at the demand price of agents in the other group: 8 i, j 2 fa, Bg 2 t+τ Z pt i = supet i 4 c3 e r (s t) dd s + e r τ p j 5 t+τ τ0 t March 26, / 16

15 The model: Implications for Asset Trading 1 Di erences in beliefs vary over time (e.g. more omptimistic investors will buy and bid up the price of the asset)=) induces trade 2 no short-selling, nite supply of assets, in nite number of prospective buyers =) bidder pays his reservation price 3 this price = the agent s fundamental valuation + the expected gain from selling the asset at some point in the future at the demand price of agents in the other group: 8 i, j 2 fa, Bg 2 t+τ Z pt i = supet i 4 c3 e r (s t) dd s + e r τ p j 5 t+τ τ0 t 4 they conjecture the demand price of the current owner = current owner s fundamental valuation + the value of the resale option pt i = p i f bi t, gt i = f f b r + i t f r + λ + q g t i March 26, / 16

16 The model: Implications for the Equilibrium the equilibrium option value is the solution to: q gt i g = sup Et i i t+τ τ0 r + λ + q g j t+τ c e τr - for 8 c0, this implies a unique trashold value k : the minimum amount of di erence in opinions that generates a trade March 26, / 16

17 The model: Implications for the Equilibrium the equilibrium option value is the solution to: q gt i g = sup Et i i t+τ τ0 r + λ + q g j t+τ c e τr - for 8 c0, this implies a unique trashold value k : the minimum amount of di erence in opinions that generates a trade the optimal policy: exercise immediately if g i > k ; otherwise wait until the rst time: g i k =)trade occurs, the owner group switches g j k and the whole process restarts from k March 26, / 16

18 The model: Implications for the Equilibrium the equilibrium option value is the solution to: q gt i g = sup Et i i t+τ τ0 r + λ + q g j t+τ c e τr - for 8 c0, this implies a unique trashold value k : the minimum amount of di erence in opinions that generates a trade the optimal policy: exercise immediately if g i > k ; otherwise wait until the rst time: g i k =)trade occurs, the owner group switches g j k and the whole process restarts from k q g i = pt i f r + f b t i f r +λ =) when a trade occurs this di erence is q ( k ) b : is called a bubble March 26, / 16

19 The model: Implications for the Equilibrium the equilibrium option value is the solution to: q gt i g = sup Et i i t+τ τ0 r + λ + q g j t+τ c e τr - for 8 c0, this implies a unique trashold value k : the minimum amount of di erence in opinions that generates a trade the optimal policy: exercise immediately if g i > k ; otherwise wait until the rst time: g i k =)trade occurs, the owner group switches g j k and the whole process restarts from k q g i = pt i f r + f b t i f r +λ =) when a trade occurs this di erence is q ( k ) b : is called a bubble duration between trades: depends on how fast g i reaches the value k, τ(x, k) = inffs : g i t+s > kg, plus E [τ(x, k)] = 0 if c = 0 March 26, / 16

20 The model: Implications for the Equilibrium the presence of q g i induces an extra source of volatility of prices March 26, / 16

21 The model: Implications for the Equilibrium the presence of q g i induces an extra source of volatility of prices Remember: price conjecture: pt i = p i f bi t, gt i = f f b r + i t f r + λ + q g t i March 26, / 16

22 The model: Implications for the Equilibrium the presence of q g i induces an extra source of volatility of prices Remember: price conjecture: pt i = p i f bi t, gt i = f f b r + i t f r + λ + q g t i total price volatility = the volatility of the fundamental value in the asset owner s mind+ the volatility of the option component : η (x) = p 2φσf r + λ h 0 (x) h 0 (k ) + h 0 ( k ), 8 x < k which increases in φ March 26, / 16

23 Characteristics of the equilibrium under small trading cost (c -> 0) Magnitude of the extra volatility component: when c = 0, η (x) " as φ " and σ f ", η (x) # as r " March 26, / 16

24 Characteristics of the equilibrium under small trading cost (c -> 0) Magnitude of the extra volatility component: when c = 0, η (x) " as φ " and σ f ", η (x) # as r " Magnitude of the bubble: i. when c = 0 ) k = 0 =) whenever di erences in beliefs ( g i ) 6= 0 trade occurs March 26, / 16

25 Characteristics of the equilibrium under small trading cost (c -> 0) Magnitude of the extra volatility component: when c = 0, η (x) " as φ " and σ f ", η (x) # as r " Magnitude of the bubble: i. when c = 0 ) k = 0 =) whenever di erences in beliefs ( g i ) 6= 0 trade occurs. ii. b increases with σ g and ρ, for a given di erence in beliefs < k March 26, / 16

26 Characteristics of the equilibrium under small trading cost (c -> 0) Magnitude of the extra volatility component: when c = 0, η (x) " as φ " and σ f ", η (x) # as r " Magnitude of the bubble: i. when c = 0 ) k = 0 =) whenever di erences in beliefs ( g i ) 6= 0 trade occurs. ii. b increases with σ g and ρ, for a given di erence in beliefs < k. Remember: variance and mean of the di erences in beliefs σ g = p 2φσ f March 26, / 16

27 Characteristics of the equilibrium under small trading cost (c -> 0) Magnitude of the extra volatility component: when c = 0, η (x) " as φ " and σ f ", η (x) # as r " Magnitude of the bubble: i. when c = 0 ) k = 0 =) whenever di erences in beliefs ( g i ) 6= 0 trade occurs. ii. b increases with σ g and ρ, for a given di erence in beliefs < k. Remember: variance and mean of the di erences in beliefs σ g = p 2φσ f. q ρ = [λ + (φσ f /σ s )] 2 + (1 φ 2 ) [(2σ 2 f /σ2 s ) + (σ 2 f /σ2 D )] March 26, / 16

28 Characteristics of the equilibrium under small trading cost (c -> 0) Magnitude of the extra volatility component: when c = 0, η (x) " as φ " and σ f ", η (x) # as r " Magnitude of the bubble: i. when c = 0 ) k = 0 =) whenever di erences in beliefs ( g i ) 6= 0 trade occurs. ii. b increases with σ g and ρ, for a given di erence in beliefs < k. Remember: variance and mean of the di erences in beliefs σ g = p 2φσ f. q ρ = [λ + (φσ f /σ s )] 2 + (1 φ 2 ) [(2σ 2 f /σ2 s ) + (σ 2 f /σ2 D )]. i. volatility of the fundamental value: σ f " =) σ g "=)resale option more valuable=) b "=) q (x) " March 26, / 16

29 Characteristics of the equilibrium under small trading cost (c -> 0) Magnitude of the extra volatility component: when c = 0, η (x) " as φ " and σ f ", η (x) # as r " Magnitude of the bubble: i. when c = 0 ) k = 0 =) whenever di erences in beliefs ( g i ) 6= 0 trade occurs. ii. b increases with σ g and ρ, for a given di erence in beliefs < k. Remember: variance and mean of the di erences in beliefs σ g = p 2φσ f. q ρ = [λ + (φσ f /σ s )] 2 + (1 φ 2 ) [(2σ 2 f /σ2 s ) + (σ 2 f /σ2 D )]. i. volatility of the fundamental value: σ f " =) σ g "=)resale option more valuable=) b "=) q (x) ". ii.informativeness of the signal and dividend ow: i s = σ f /σ s " or i D = σ f /σ D "=) ρ "=) b "=) q (x) " March 26, / 16

30 Characteristics of the equilibrium under small trading cost (c -> 0) Magnitude of the extra volatility component: when c = 0, η (x) " as φ " and σ f ", η (x) # as r " Magnitude of the bubble: i. when c = 0 ) k = 0 =) whenever di erences in beliefs ( g i ) 6= 0 trade occurs. ii. b increases with σ g and ρ, for a given di erence in beliefs < k. Remember: variance and mean of the di erences in beliefs σ g = p 2φσ f. q ρ = [λ + (φσ f /σ s )] 2 + (1 φ 2 ) [(2σ 2 f /σ2 s ) + (σ 2 f /σ2 D )]. i. volatility of the fundamental value: σ f " =) σ g "=)resale option more valuable=) b "=) q (x) ". ii.informativeness of the signal and dividend ow: i s = σ f /σ s " or i D = σ f /σ D "=) ρ "=) b "=) q (x) ". iii.degree of overcon dence:φ "=) σ g " and ρ # =)two o setting e ects? =) Numerical simulation March 26, / 16

31 Characteristics of the equilibrium under small trading cost (c-> 0) The impact of φ ": =) b ", η (k ) ", k " on τ two o setting e ects: (1) k " (this change is second order) =) τ " (2) σ g "=) τ V E [τ(k, k )] # March 26, / 16

32 Characteristics of the equilibrium under increasing trading cost March 26, / 16

33 Summary of the main ndings 1 asset trading model in which agents observe a vector of signals and information concerning future dividend stream March 26, / 16

34 Summary of the main ndings 1 asset trading model in which agents observe a vector of signals and information concerning future dividend stream 2 they cannot observe the fundamental value but based on the availalble information they form beliefs about it March 26, / 16

35 Summary of the main ndings 1 asset trading model in which agents observe a vector of signals and information concerning future dividend stream 2 they cannot observe the fundamental value but based on the availalble information they form beliefs about it 3 Overconfidence (Psychological bias) Heterogeneous beliefs Disagreement among investors Price = E [fundamental value] + E[option value] Increasing Overconfidence higher prices, higher price volatility, increasing trading volume March 26, / 16

36 Thank you for your attention! March 26, / 16