VIEWS: 13 PAGES: 39 CATEGORY: Education POSTED ON: 12/5/2009 Public Domain
How to explain the stock market volatility ? Option 1 : – Stick to the REH. – Test sophisticated version : multiplicity, horizon – With possibly, some piece of « bounded rationality». Non standard rationality :neu Myopia, … Option 2 : – More radical individual bounded rationality. – Or at the collective level: rationality of expectations…. « Eductive test ». SREE… Évolutive learning. First option : a gallery of phenomena. – Erratic fluctuations of stock prices. – Herd behaviour… – Krachs and « multiplicity ». – Asymmetric information bubbles. – Irrational agents and bubbles. Volatility without bubbles Followers and market disturbances Soros and « Positive Feedback traders » Soros strategy ! – Bet not on fundamentals… – But on the future behaviour of the crowd… – « Insiders destabilize by driving the price up and up, selling out at the top to the outsiders who…sell out at the bottom » Kindleberger (1978) Rationalizing Soros strategy : the framework. ``Rational Speculator’’: – Private Info. on fundamental value – No market power, risk-averse… Mechanism : Rational investor destabilize the market – Positive feedback traders : buy when the price has increased recently. – Rational investors anticipate the overreaction of the market. History. Real Sphere. – Φ, θ r.v zero mean. – Period 0: fundamental=0 – Period 1: – Period 2: shock => + Φ. – Period 3: Random shock => Φ+θ Information – « Positive feedback traders »: know past pt-i . – Passive Investors: know Φ in period 2, θ in 3. – Speculators: receive (a signal of ) Φ at period 1 Agents actions. – Passive Invest. , Speculators. – Mean-Var A summary of history. Demands Positive Date Event Passive Speculators feedback .. Optimal 0 Reference 0 0 (0) Optimal : 1 Speculators learn 0 - p1 -(p1-p2) 2 Passive learn (p 1 - p 0 ) -(p2-) -(p2-) 3 Liquidation : + – – – Equilibrium : destabilisation Equilibrium mechanics. In the absence of speculators (u = 0) – At period 1, no news => p1 = p0 = 0 – At period 2, Df=0, – Passive investors : expect no trade from positive feedback, hence equilibrium is no trade and De = 0, p2 = Φ Destabilisation (u > 0) : – Perfect forecast of p2 by specul. – -u(p1-p2)-(1-u)p1 =0, => p1 = up2 – If p1 > 0, Df2 > 0 => p2 > Φ. Equilibrium. – Period 2 equilibrium : 0 = β p1 + α (Φ - p2) With u=1, p1 = p2 : p2* = α Φ /(α - β) Then p2* > Φ Destabilisation by rational agents. Herd behaviour.. Herd behaviour and 1- Followers… 2- Bubbles. Herd behaviour The logic : reminder.. – Two choices, A and B. – Two observations, R (A), V(B), N (No observation. – Sequential decisions : Agents, 1,2, … N have to decide sequentially. Cascade : – 1 observe R, go to A. – 2 observes 1. R, go to A. 2. N, go to A. 3. V, go to B…. – 3 observes…. 1. Go to A, if 1 or 2, even if B… 2. Go … – Then, … Characteristics. – Rational expectations equilibrium : it is rational to follow the crowd. – Fragile : everybody understands that a long queue at A carries little information. Herd behaviour and financial markets. A simple model : – (from Avery-Zemsky AER, (98)) Each informed agent receives a signal on the value of fundamental V. He buys, (sells) if his expected value greater,(smaller) than the price… The price of the asset equals its expected value given the history of trades (market maker…). Noise traders come into the sequence with probability 1-u and act randomly….. With standard signals, – no herd behaviour. The price reflects all previous public information… Wtih fixed price, back to previous story.. – The price converges towards the fundamental value.. Herd behaviour and financial markets. A simple model with event uncertainty. – V=[0, ½, 1] – A single type of informed trader…signal x x= ½ , iif V= ½ P(x=1/1)= P(x=0/0)=p> ½. If p<1, then herd behaviour occurs with positive probability. – Increases with (½). – Typically : no herd behaviour at the outset for a number of periods – For a long period, the price remains close to ½ , while there is herd behaviour, buy or sell, possibly in the wrong direction… Herd behaviour and financial markets. A simple model with event uncertainty. – V=[0, ½, 1] – Informed trader…signal x x= ½ , iif V= ½ P(x=1/1)= P(x=0/0)=p> ½. – But two types of informed traders : H, L. p(H)=1, p(L) > ½. Proportion of H,L unknown : well W or poorly informed P market… A price bubble : – (½) close to 1, strong a priori for W. – p(L) close to ½ , no H trader in the poorly informed market – 50/50 in the W market. – The truth is (0,P) Herd behaviour and financial markets. Period 1 : (0-50) – Price almost fixed. – Herding (buy..). Period 2. – the market-maker believes that activity reflects good news (since the market is a priori well- informed) and that a poorly informed market generates herd behaviour that mimicks activity of a well-informed market. – price rises…. Period 3. – Fall in activity due to herding, – The market-maker learns that the market is poorly informed. – Hence drop in prices. Remarks on the limit of the model : no forward-looking behaviour… Bubbles… Without followers and/or herd behaviour About bubbles. The bubble problem : – Price greater than fundamental value. – Definition of FV – Ruled out in general equilibrium ? – Sunspots ? The internet bubble: – Reminder : March 2000, « Tulip mania » (1630), South Sea bubble (1720.Newton !) – « funds managers » between Charybde and Scylla. Irrational , to play or not. Error 1: JR, Tiger Hedge Fund : dissolved end 1999. Error 2 : SD, Quantum Fund : resignation 04-2000. The model. The Background. Vague : Price p(t)=exp(gt). The bubble. Price >FV from t(0), (1-b(t-t(0)))p(t), b increasing. t(0) random, Poisson (t(0))=1-exp(-λt(0)) Bursts…. for sure at t(0) + т, Bursts if cumulative selling pressure. к <1 Definitive selling of one unit. Information on the bubble. Sequential information of actors. Uniform density (1/η) Random awareness window t(0), t(0)+ η. Facts and beliefs. – After t(0), « bubble » – After t(0)+ηк, « order 1 bubble » – After t(0)+2ηк ….. Asking help from a statistician. Beliefs, next. – On the arrival of the bubble. (Bayes) (t(0)/ti)={exp(λη)-exp(λ(ti-t(0))}/{exp(λη)–1}. – On the duration life of the bubble ρ,/ it bursts at ζ+t(0), – Note ρ = ζ+t(0)- ti – duration depends on / ti – .. ρ ζ – … t(0) ti (ρ/ti)= {exp(λη)-exp(λ(ζ-ρ)}/{exp(λη)–1}. « Hasard rate » dρ/(1- ) = λ /(1-exp(-λ(ζ- ρ))) Call it h Individual decision to «ride » the bubble ? Calculations . – For fixed strategies of others : – Endogenous bursting : t(0)+T* > t(0)+ηк. – Min(T*,т) bursting bubble. – Loss / s, One unit. b(s-T*)p(s) or b(τ)p(τ) Criteria : compare – h(ρ/ti)(b(t-T*) and (g-r)., t=ti + ρ – h instantaneous probability of crash – or – h(t/ti) and (g-r)/(b(τ)). – Hint: h (ρ/ti)=(λ /(1-exp(-(ζ- ρ))), ζ = T* Equilibrium. Equilibrium with trigger strategies – « Trigger strategy »: witing time /ti . – Type 1 Equilibrium : bubble bursts exogenously. – Type 2 ……………………………………………endogenously. Type 1 Equilibrium. – Each informed agent sells possibly / a waiting period of – t’ = т –(1/λ)Log((g-r)/g-r-λb(т)) Proof and conditions. – λ /(1-exp(-λ(τ-ρ)))=(g-r)/(b(τ)). – If for ρ = τ- ηк, lhs < rhs : λ /(1-exp(-ληк))<(g-r)/(b(τ)). – t’> τ- ηк, the bubble does not burst. Comments – Opinion dispersion + intensity prevent bursting. Equilibrium. Type 2 Equilibrium : bubble bursts under attack. – Each informed agent sells (if possible) after waiting ρ* – ρ*= b-1{(g-r)(1-exp(-λ(ηк))/λ)}-ηк Proof and conditions. – If all have the trigger strategy ρ’, – The bubble will burst at t(0)+ ηк+ρ’, (t(0) + ζ) – ζ – ρ= ηк+ρ’ – ρ – Equilibrium Condition : ρ’= ρ= ρ* – λ /(1-exp(-λ(ηк))=(g-r)/ (b(ηк+ρ*)). Comments. Crash in information transmission. The multiplicity hypothesis.. A model with informed and non informed agents and noisy supply. The framework : – Asset value , H or B. – Proportion a informed. – Mean-variance pref. . – Noisy (noise traders). d High informed Equilibrium : Z, Beliefs Demand Z(p,I)=ad(I,p)+(1-a)d(NI, p)=e p(I,e) clears the market. – Beliefs NI bayesian – d(I,p) Dominant Str. Non informed – If p : Demand e =- Z(p,H) ou -Z(p,B) – Compute E(H/p) and E(s/p)= HE(H/p) +B(1-E(H/p)) – d(NI,p) = E(s/p) –p. p Equilibrium in the noisy model. Equilibrium : Z, Beliefs Z(p,I)=ad(I,p)+(1-a)d(NI, p)=e p(I,e) clears the market. – Beliefs NI bayesian – d(I,p) Dominant Str. – If p : e =- Z(p,H) ou -Z(p,B) – Compute E(H/p) and E(s/p)= HE(H/p) +B(1-E(H/p)) – d(NI,p) = E(s/p) –p. Properties – Total demand is decreasing. – But not necessarily NI demand. – Function / noise précision. Equilibrium is unique. p Z totale/si B Getting multiple equilibria. The 1987 crash according to Genotte-Leland. – Add informatics programs into the d picture: Automatic sale whenever the asset price decreases. – Again, Static framework… – Adding : A positively sloped curve…. Consequence : – Total deamand is no longer decreasing. – Multiplicity. – The crash : passage From a high equilibrium p to a low one. Or crash of expectational coordination ? « Eductive stability » of equilibria. A reminder of Desgranges The setting : – Information transmission à la Grossman-Stiglitz. – Each small agent receive a piece of noisy information. – Noise traders. – Aggregate equilibrium excess demand reflects the average information of the society…. generates individual and aggregate… The analysis. – There exists a unique equilibrium. – But not necessarily strongly rational. – Contradiction between the confidence in the market transmission and the amount of information transmitted. A model with informed and non informed agents and noisy supply. The framework : – Asset value , H or B. – Proportion a informed. – Mean-variance pref. . – Noisy (noise traders). d Equilibrium : Z, Beliefs High informed Demand Z(p,I)=ad(I,p)+(1-a)d(NI, p)=e p(I,e) clears the market. – d(I,p) Dominant Str. Delta – Beliefs NI bayesian – If p : e = - Z(p,H) ou -Z(p,B) Non informed Demand – Hence : d(NI,p) = E(s/p) –p. The key parameters : guess.. – Delta, the « amount » of information.. – The variance of the noise… – The proportion of informed p traders… Eductive coordination in the noisy model. A first answer : – The equilibrium is eductively stable iif ( normal noise): – (1-)2<42 – With , prop.informed, ,gap, 2 variance of noise. – Product of an amplification effect and a sensitivity effect. – Comments. A second answer. – The equilibrium is eductively stable iif – Aggregate equilibrium demand is enough decreasing. – With few informed agents, a Necessary condition is that non informed demand is decreasing. – Comments. Crash in expectational coordination. Completing the market creates susnpot equilibria. Bowman-Faust (1997) The model. – 3 periods, 2 agents, log. Utility.. – one firm…equity is exchanged.. – 0 decision on firm’s investment, reaped at 2. – 1 one of the agent, randomy picked, desires immediate consumption, (zero value), the other one postponement… The equilibrium with one asset – Is not P.O : not zero consumption in the bad event.. – No sunspot. An option : – Completes the market : PO with the option.. – Creates sunspots, in fact multiplicity… Partial equilibrium model of inventories Guesnerie-Rochet (1993). The model : – A manna of crop at each period w(t), t=1,2 – Part on the market, the other goes to inventories. – Inventory Cost : Cx2 /2, – P(t)=k{d(t) - w(t) (+/–) S(t)}, S(t), quantity on the market. Profitability of inventories. – Mean-variance utility : E() – (1/2)b(Var ) – ΔP(t)=k{Δ d(t) -2X- Δw(t)}, Var(w(2)) = v2 – x() = k(X_ -2X(e))/{bk2v2+C} – Inventory mass N – X= kN(X_ -2X(e))/{bk2v2+C} Equilibrium inventories : – X* = X_/{2+C/kN + bkv2/N} – Plausible….. The Equilibrium. The profitability of inventories – x() = k(X_ -2X(e))/{bk2v2+C} – Mass of inventory holders N -2kN/{bk2v2+C} – X= kN(X_ -2X(e))/{bk2v2+C} Equilibrium inventories : – X* = – X_/{2+C/kN + bkv2/N} – Plausible….. – … X* X The « eductive » process :the inventory variant An « eductive » story : – Expectations X(e), – Realisations : -2kN/{bk2v2+C} – -2kNX(e))/{bk2v2+C} Results : – N<{bk2v2+C}/2k – Bad More traders Less risk averse Less uncertainty Less costly.. X Inventories with futures markets : equilibrium. M mass of « speculators » : intervene on the market of futures, price p(f), one unit of wheat to morrow. Hedging behaviour from primary traders : – N[p(f)-p(1)]/C = (N+M)[p(2)-p(f)]/bk2v2. – Intuition : uncertainty cost born by N+M agents. Computation of equilibrium – N[p(f)-p(1)]/C = (N+M)[p(2)-p(1)+p(1)-p(f)]/bk2v2. – X[1+((N+M)/N)(C/ bk2v2. )]=(N+M)(….-2kX)/bk2v2 – Previously X = X*/{2+C/kN + bkv2/N}. Now : X = X*/{2+C/kN + bkv2/(N+M)} – X* random… – The ex ante variance of prices has decreased… Inventories with futures markets : the « eductive analysis » M mass of « speculators » : intervene on the market of futures, price p(f), one unit of wheat to morrow. Hedging behaviour from primary traders : – N[p(f)-p(1)]/C = (N+M)[p(2)-p(f)]/bk2v2. Now : X = X*/{2+C/kN + bkv2/(N+M)} – Intuition : uncertainty cost born by N+M agents. – The variance of prices has decreased… Eductive stability : – More complex : timing futures, inventory decisions, period 1 market – C/kN + bkv2/(N+M) >2 – Intuition. N(M), N decreasing function of M. M>0 is bad. Excess confidence. A cognitive bias – Well established by psychologists ? A model: investors. – A r.v v , mean. 0, 2 signals t(1)=v+e(1), t(2)=v+e(2), – e(1),e(2) zero mean, precision(e(i))=ρ , – 2 categories of investors A and B – A, (resp.B) overestimates precision t(1),cρ,(resp.t(2),cρ), c>1 A model: the firm. – Long term value w = u+v+e’, u mean u >0 – A signal s, on u, centered on u, precision ρ(s). – u,v,e’, s, normals precision denoted ρ(.) – Absence of cognitive bias : E(w/s,t)= – u + [ρ(s)/(ρ(u)+ ρ(s))][s - u] + i[ρ/(ρ(v)+2ρ)][t(i)] – +i[1/(ρ*+2)][t(i)], avec ρ*= ρ(v)/(ρ). Excess confidence. A model: the firm. – Long term value w = u+v+e’, u mean u >0 – A signal s, on u, centered on u, precision ρ(s). – u,v,e’, s, normals precision denoted ρ(.) Cognitive bias. – With bias : E(w/s,t)= – u + [ρ(s)/(ρ(u)+ ρ(s))][s- u] + – for A : [cρ/(ρ(v)+ρ(1+c)][t(1)] + [ρ/(ρ(v)+ρ(1+c)][t(2)] – Difference between a priori belief of A and B : – [(c-1)/(ρ*+1+c)][t(1)-t(2)] – Exchange of stocks after observation of t (with or without s) Si t2 > t1 B too optimistic /w : B buys stocks A at his own valuation Si t1 > t2 then A too optimistic, but A has to buy from B. Ex ante Value of the firm : – V(0) = u+ [(c-1)/(ρ*+1+c)][ [ρ*(c+1)/2c](standard deviation (v)) – Hint : Increase of standard deviation is valuable for initial owners…. D(p(t),p(t+1)=1 p(t) c(t+1) A(d+p(t+1))/p(t) p’ p’’ p* p(t+1) A-p(t) A c(t) •Effet de revenu domine effet de substitution.