How to explain the stock market volatility

1
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.

2
Volatility without bubbles

• Followers and market disturbances

3
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
  whosell 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.

4
History.

• Real Sphere.
• F, ? r.v zero mean.
• Period 0 fundamental0
• Period 1
• Period 2 shock gt F.
• Period 3
• Random shock gt F?
• Information
•  Positive feedback traders  know past pt-i .
• Passive Investors know F in period 2, ? in 3.
• Speculators receive (a signal of ) F at period 1
• Agents actions.
• Passive Invest. , Speculators.
• Mean-Var

5
A summary of history.
6
Equilibrium destabilisation
7
Equilibrium mechanics.

• In the absence of speculators (u 0)
• At period 1, no news gt p1 p0 0
• At period 2, Df0,
• Passive investors expect no trade from positive
  feedback, hence equilibrium is no trade and De
  0, p2 F
• Destabilisation (u gt 0)
• Perfect forecast of p2 by specul.
• -u(p1-p2)-(1-u)p1 0, gt p1 up2
• If p1 gt 0, Df2 gt 0 gt p2 gt F.
• Equilibrium.
• Period 2 equilibrium 0 ß p1 a (F - p2)
• With u1, p1 p2 p2 a F /(a - ß)
• Then p2 gt F
• Destabilisation by rational agents.

8
Herd behaviour..

• Herd behaviour and
• 1- Followers
• 2- Bubbles.

9
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
• R, go to A.
• N, go to A.
• V, go to B.
• 3 observes.
• Go to A, if 1 or 2, even if B
• 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.

10
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..

11
Herd behaviour and financial markets.

• A simple model with event uncertainty.
• V0, ½, 1
• A single type of informed tradersignal x
• x ½ , iif V ½
• P(x1/1) P(x0/0)pgt ½.
• If plt1, 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

12
Herd behaviour and financial markets.

• A simple model with event uncertainty.
• V0, ½, 1
• Informed tradersignal x
• x ½ , iif V ½
• P(x1/1) P(x0/0)pgt ½.
• But two types of informed traders H, L.
• p(H)1, p(L) gt 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)

13
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

14
Bubbles

• Without followers and/or herd behaviour

15
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.

16
The model.

• The Background.
• Vague
• Price p(t)exp(gt).
• The bubble.
• Price gtFV from t(0), (1-b(t-t(0)))p(t),
• b increasing.
• t(0) random, Poisson
• ?(t(0))1-exp(-?t(0))
• Bursts.
• for sure at t(0) ?,
• Bursts if cumulative selling pressure. ? lt1
• Definitive selling of one unit.

17
Information on the bubble.

• Sequential information of actors.
• Uniform density (1/?)
• Random awareness window
• t(0), t(0) ?.
• Facts and beliefs.
• After t(0),  bubble 
• After t(0)??,  order 1 bubble 
• After t(0)2?? ..

18
Asking help from a statistician.

• Beliefs, next.
• On the arrival of the bubble. (Bayes)
• ?(t(0)/ti)exp(??)-exp(?(ti-t(0))/exp(??)1.
• On the duration life of the bubble ?,/ it bursts
  at ?t(0),
• Note ? ?t(0)- ti
• duration depends on / ti
• ..
• ? (?/ti) exp(??)-exp(?(?-?)/exp(??)1.
•  Hasard rate 
• d??/(1- ?) ? /(1-exp(-?(?- ?)))
• Call it h

?
?
ti
t(0)
19
Individual decision to ride  the bubble ?

• Calculations.
• For fixed strategies of others
• Endogenous bursting t(0)T gt t(0)??.
• Min(T,?) bursting bubble.
• Loss / s, One unit. b(s-T)p(s) or b(t)p(t)
• Criteria compare
• h(?/ti)(b(t-T) and (g-r)., tti ?
• h instantaneous probability of crash
• or
• h(t/ti) and (g-r)/(b(t)).
• Hint h (?/ti)(? /(1-exp(-(?- ?))), ? T

20
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 ? (1/?)Log((g-r)/g-r-?b(?))
• Proof and conditions.
• ? /(1-exp(-?(t-?)))(g-r)/(b(t)).
• If for ? t- ??, lhs lt rhs ?
  /(1-exp(-???))lt(g-r)/(b(t)).
• tgt t- ??, the bubble does not burst.
• Comments
• Opinion dispersion intensity prevent bursting.

21
Equilibrium.

• Type 2 Equilibrium bubble bursts under attack.
• Each informed agent sells (if possible) after
  waiting ?
• ? b-1(g-r)(1-exp(-?(??))/?)-??
• Proof and conditions.
• If all have the trigger strategy ?,
• The bubble will burst at t(0) ???, (t(0) ?)
• ? ? ??? ?
• Equilibrium Condition ? ? ?
• ? /(1-exp(-?(??))(g-r)/ (b(???)).
• Comments.

22
Crash in information transmission.

• The multiplicity hypothesis..

23
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).
• 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.

d
High informed Demand
Non informed Demand
p
24
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
25
Getting multiple equilibria.

• The 1987 crash according to Genotte-Leland.
• Add informatics programs into the 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
• to a low one.

d
p
26
Or crash of expectational coordination ?

•  Eductive stability  of equilibria.

27
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.

28
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).
• Equilibrium Z, Beliefs
• Z(p,I)ad(I,p)(1-a)d(NI, p)e
• p(I,e) clears the market.
• d(I,p) Dominant Str.
• Beliefs NI bayesian
• If p e - Z(p,H) ou -Z(p,B)
• 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 traders

Delta
29
Eductive coordination in the noisy model.

• A first answer
• The equilibrium is eductively stable iif ( normal
  noise)
• ?(1-?)?2lt4?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.

30
Crash in expectational coordination.

• Completing the market creates susnpot equilibria.

31
Bowman-Faust (1997)

• The model.
• 3 periods, 2 agents, log. Utility..
• one firmequity is exchanged..
• 0 decision on firms 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

32
Partial equilibrium model of inventories
Guesnerie-Rochet (1993).

• The model
• A manna of crop at each period w(t), t1,2
• Part on the market, the other goes to
  inventories.
• Inventory Cost Cx2 /2,
• P(t)kd(t) - w(t) (/) S(t), S(t), quantity on
  the market.
• Profitability of inventories.
• Mean-variance utility E(?) (1/2)b(Var ?)
• ?P(t)k? d(t) -2X- ?w(t), Var(w(2)) v2
• x() k(X_ -2X(e))/bk2v2C
• Inventory mass N
• X kN(X_ -2X(e))/bk2v2C
• Equilibrium inventories
• X X_/2C/kN bkv2/N
• Plausible..

33
The Equilibrium.

• The profitability of inventories
• x() k(X_ -2X(e))/bk2v2C
• Mass of inventory holders N
• X kN(X_ -2X(e))/bk2v2C
• Equilibrium inventories
• X
• X_/2C/kN bkv2/N
• Plausible..

X
34
The  eductive  process the inventory variant

• An  eductive  story
• Expectations X(e),
• Realisations
• -2kNX(e))/bk2v2C
• Results
• Nltbk2v2C/2k
• Bad
• More traders
• Less risk averse
• Less uncertainty
• Less costly..

35
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
• Np(f)-p(1)/C (NM)p(2)-p(f)/bk2v2.
• Intuition uncertainty cost born by NM agents.
• Computation of equilibrium
• Np(f)-p(1)/C (NM)p(2)-p(1)p(1)-p(f)/bk2v2.
  
• X1((NM)/N)(C/ bk2v2. )(NM)(.-2kX)/bk2v2
• Previously X X/2C/kN bkv2/N.
• Now X X/2C/kN bkv2/(NM)
• X random
• The ex ante variance of prices has decreased

36
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
• Np(f)-p(1)/C (NM)p(2)-p(f)/bk2v2.
• Now X X/2C/kN bkv2/(NM)
• Intuition uncertainty cost born by NM agents.
• The variance of prices has decreased
• Eductive stability
• More complex timing futures, inventory
  decisions, period 1 market
• C/kN bkv2/(NM) gt2
• Intuition. N(M), N decreasing function of M. Mgt0
  is bad.

37
Excess confidence.

• A cognitive bias
• Well established by psychologists ?
• A model investors.
• A r.v v , mean. 0, 2 signals t(1)ve(1),
  t(2)ve(2),
• e(1),e(2) zero mean, precision(e(i))? ,
• 2 categories of investors A and B
• A, (resp.B) overestimates precision
  t(1),c?,(resp.t(2),c?), cgt1
• A model the firm.
• Long term value w uve, u mean u gt0
• A signal s, on u, centered on u, precision ?(s).
  
• u,v,e, s, normals precision denoted ?(.)
• Absence of cognitive bias E(w/s,t)
• u ?(s)/(?(u) ?(s))s - u
  ?i?/(?(v)2?)t(i)
• ?i1/(?2)t(i), avec ? ?(v)/(?).

38
Excess confidence.

• A model the firm.
• Long term value w uve, u mean u gt0
• A signal s, on u, centered on u, precision ?(s).
  
• u,v,e, s, normals precision denoted ?(.)
• Cognitive bias.
• With bias E(w/s,t)
• u ?(s)/(?(u) ?(s))s- u
• for A c?/(?(v)?(1c)t(1)
  ?/(?(v)?(1c)t(2)
• Difference between a priori belief of A and B
• (c-1)/(?1c)t(1)-t(2)
• Exchange of stocks after observation of t (with
  or without s)
• Si t2 gt t1 B too optimistic /w B buys stocks A
  at his own valuation
• Si t1 gt t2 then A too optimistic, but A has to
  buy from B.
• Ex ante Value of the firm
• V(0) u (c-1)/(?1c)? ?(c1)/2c?(standar
  d deviation (v))
• Hint Increase of standard deviation is valuable
  for initial owners.

39
D(p(t),p(t1)1
c(t1)
A(dp(t1))/p(t)
p
p
p
c(t)
A
A-p(t)

• Effet de revenu domine
• effet de substitution.