How to explain the stock market volatility by FJ84JKFD

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									               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<42
     – 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.

								
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