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Model 3 (Strategic informed trader)Kyle (Econometrica 1985)

The economyA group of three agents trades a risky asset for a risk-less asset.• One insider trader;• Liquidity (or noise) traders;• Risk neutral dealer.Liquidation value of the asset is v : N( w, ²).

Insider trader is a risk neutral speculators with some private information regarding the liquidation value of the risky asset v.

Noise traders are agents that come to the market for reason other than speculation; the aggregate market order submitted by noise traders is a random variable m: N(0, ²m) .

Dealers are risk neutral speculators that provide liquidity to the market:

The random variables m and v are independently distributed.

Information structureThe insider trader knows the realization of v but he does not know the realization of m.Dealers do not know the realization of v nor the realization of m.

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Trading technology

1) Insider trader and liquidity traders chose the quantity they want to trade. There is no restriction on the volume of trade.

• x is the quantity traded by the insider trader (Market order)• m is the quantity traded by the noise traders (Market orders)

2) Dealers observe the aggregate net order flow q = ( x + m ) and set a price p to clear the market (set of limit orders).

Market participant payoff functions

Insider trader’s payoff: I = x(v – p )

Dealer’s payoff: D = (x+m)( p – v)

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Equilibrium conditionsEquilibrium conditions

1) Competition among dealers leads them to execute the aggregate trade at a price p* such that the dealers’ expected profit is zero:

E[D ]= E[(x+m)( p – v )| x+m] = 0

p* = p( x + m ) =E[v| x + m ] (4)

Remark: E[v| x + m ] depends on the insider trader’s strategy.

2) The insider trader’s strategy depends on the dealers pricing rule p(x+m)The insider trader chooses the quantity x* that satisfies:

x* = x(v) = argmaxx E[x(v – p(x + m))] (5)

Remark: x( v ) depends on the dealer pricing rule p(.).

Definition: An equilibrium of this economy is a pricing function p( q ) and a insider trader’s strategy x(v) that satisfy simultaneously conditions (4) and (5).

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Result 5: There is a linear equilibrium where

x( v ) = (v – w)and

p( x + m ) = w + (x + m )

whereand

Insider trader equilibrium payoff ex ante is

E[D] = m /2

Remarks:• The informed trader optimal quantity depends on the variance of the

noise traders order flow.• The informed trader expected payoff increases with the variance of the

noise traders order flow and the variance of v.• The equilibrium price is not efficient.• The deepness of the market increases with the ratio m /.

2

2

m 2

2

2

1

m

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Proof:Recall that: If y is N(my, ²y) and z be N(mz, ²z) and let Cov(y,z) = yz

Then

E[y|z = z] = my + (z – mz ) yz/²z

Step (1)What is the dealer price if he expects the insider trader to use the strategy x = (v –

w)?We know that

p( x + m ) =E[v| x + m ]

we have that x + m is N(0, ()² + ²m)) and Cov(v,(x+m)) = ²Therefore

E[v| x + m ] = w +( x+m ) ²/(()² + ²m) = = w +(x + m) = p(x+m)

Step (2)The insider trader maximization problem is

Maxx E[x(v – (w +(x +m) )]=

= Maxx [x(v – w) – x²)]that implies

x* = (v – w )/2 = (v – w)

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Model 4 (Brokers and Traders)(Foucault and Lovo (2006))

The economyA group of 5 types of agents trades a risky asset for a risk-less asset.• One large liquidity trader;• N brokers;• One insider trader;• Liquidity (or noise) traders; Kyle economy• Risk neutral dealers.

We focus on the relation between the brokers and the large liquidity trader:

• A large liquidity trader with no private information needs to trade a non-negligible amount y of the risky asset.

• Each broker i has a pending transaction coming from other customers to trade an amount xi of the risky asset. Pending transactions xi are i.i.d. across the N brokers and privately known by each broker.

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The large trader has several options for trading the quantity y:

Option (M): Send his market order y to the market.In this case the expected transaction cost will be

C(M) = E[ y(v – p ) ]

where p is determined as in Model 3 (Kyle 1985):

p = w +(x + m +y)where• x = (v – w) is the insider trader’s market order;• m is the net order coming from other liquidity traders;• x + m +y is the aggregate market order.

Considering that the large trader does not have information on v and m, it results:

C(M) = E[y(v – p )] = E[y(v- w –(x + m +y)] =

= E[y(v – w – ((v – w) + m +y)] =

= – y2

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Option (Bi) Transfer his order to a broker i that will execute the quantity y at price

p = E[v] = w in exchange for a fee ci .

In this case the expected transaction cost will be

C(Bi) = E[y(v – p )] – ci

As the broker guarantees p = E[v] = w we have that

C(Bi) = – ci

• Shall the trader pass through a broker?• If yes which is the broker that shall be chosen by the seller?• How brokers transaction fees ci are determined?• How shall the trader proceed if he does not know the brokers fees ci ?

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Auction mechanism:

• the trader asks each of the N brokers the commission fee ci for executing an order of size y at price p =w.

• If ci = min {c1 ,c2, …cn} > y2 , then the trader refuses all brokers’ offers and trades directly in the market at expected cost y2

• If ci =min {c1 ,c2, …, cn} < y2 , then trader trades with the broker i .

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How much does a broker value the opportunity to trader an additional amount y?

Result 6: The minimum fee that Broker i is willing to ask to execute the order y is

Vi = y2 +2 yxi Remarks: • The minimum fee increases with y xi

• The minimum fee increases with |y|

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Proof: Recall that brokers cannot take positions in the asset. Broker i has a pending transaction of size xi. i.e., he has to trade an amount xi at

price w for other customers that paid him a fee gi.

If Broker i does not conclude a further deal with the large trader then he will have to trade the quantity xi in the market and his expected profit will be

i 0 = gi +xi E[(w – p)]

Where E[(w – p)] is the expected difference between the price granted to the customers and the execution price in the market.

Following the same argument used for the large trader we have

i 0 = gi – xi 2

If Broker i does conclude the deal with the large trader then he will have to trade the quantity xi + y in the market and his expected profit will be

i

1 = gi +ci+ (xi +y) E[(w – p)] = gi +ci – (xi +y) 2

Thus i1 i 0 only if ci >Vi = y2 +2 yxi .

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How brokers fees are determined?A broker knows that he will execute the trader

market order y only if:• he proposes the lowest fee among all

brokers;• his fee ci < y2

The broker maximization problem when he chooses ci is

Max c (c – Vi)Pr(Min-i{c-i} > c)1{c < y2}

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Let G( z ) = Pr(Vi < z) = Pr( (y² +2 yxi ) < z)

andF( z ) = Pr ( min {V-i} > z) = (1 – G(z))N-1

Then if c(V) is the symmetric equilibrium bidding function Broker i maximization problem becomes

Maxx (c(x) – Vi)F(x)1{c(x) < y²}

Result 7: Let Vi = (y² + 2yxi) then in equilibrium we have:

if Vi y²

and

c(Vi) = Vi if Vi > y²Remarks:• Expected transaction cost is smaller than y²• Transaction cost

– decrease with the number of brokers– decrease with the market deepness– increases with the covariance between xi and y.

2 1

)(1

)(1)(

y

V

N

ii

i

dxvG

xGVVc

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Financial markets’ informational efficiency

• Does the price system aggregate all the pieces of information that are dispersed among investors?

• How does the trading technology affect financial markets informational efficiency?

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Market informational efficiency

Weak form efficiency: Trading prices incorporate all past common information.

You can’t beat the market knowing the past.

Semi-strong efficiency: trading prices incorporate all public information (past and present) .

You can’t beat the market using publicly available information

Strong form: Trading prices incorporate all information available in the economy (public and private).

No information of any kind can be used to beat the market

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Anticipated response to “bad news”

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Empirical evidence:

Financial market is weak form efficient (no use of technical analysis)

Financial market is semi-strong form efficient

Financial market is not strong form efficient

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THE ECONOMY  

A risky asset is sequentially exchanged for a risk-less asset among • Uninformed market makers; • Traders (risk neutral informed traders and liquidity traders).

V

V

Informed trader i receives a signal that is partially informative on the realization of V  

si { l, h} Pr(s=l| V= ) = Pr(s=h| V= ) = r (1/2,1)

where < E[V | s = l] < E[V] < E[V | s = h] <V

V

The liquidation value of the risky asset is v = V +

where V { , } E[] = 0 , > 0

 Remark: Traders’ signals are conditionally i.i.d. and not correlated with

V V

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Trading mechanism Trade is sequential and in each trading period:  •A trader is randomly selected

The trader submits a market order

• Market makers set the trading price efficiently observing the trader’s order and ignoring the trader’s portfolio composition and signal. 

• The trader leaves the market

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t t+1

A trader arrives and he submits a market order Q.

Market Makers observe Q and compete in price to execute the order.

Trading round t

Exchange takes place and the trader leaves the market.

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What do traders and market makers observe?

All agents observe the past history of trades (volume and prices) but they do not know the identity of past traders.  

Agents update beliefs using Bayes’ rule  Given a trading history Ht the public belief is t = Pr(V = | Ht )

Traders add to the public belief the information of their private signal s {l, h}  

ts = Pr (V = | Ht, s = s )

V

V

rr

r

tt

tl

)1()1(

)1(

)1)(1( rr

r

tt

th

22

t

E[v|Ht , s]

V

V

1

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What can traders and market makers learn?

  Informed traders’ private signals distributions only depend on the realization of

V. Any trading history can provide information about the realization of V but not about the realization of

Definition: the market is informational efficient in the long run if all private information is revealed, i.e., the random variable E[V|Ht] converges to V.

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A TOY MODEL

Each Trader can BUY one unit  of the asset, SELL one unit  of the asset or retain from trading. There are informed traders with utility function

UT(v, x, m) = v x + m

 There are (1- ) liquidity traders that buy and sell with probability k 0.5 and do not trade with probability (1- 2k)

Market makers utility function is

UM(v, x, m) = (v + C) x + m  

with 0 At time t, bid and ask prices are:  

Bt = E[v| Ht , trader sells ] + C

 At = E[v| Ht , trader buys ] + C

    

 

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Glosten and Milgrom

If = 1 and C = 0, then market makers and informed traders are risk neutral speculators.

Informed trader will buy (resp. sell) if and only if their signal is h (resp. l)

The bid price is Pr(V= |Ht , sell) + (1- Pr (V= |Ht , sell))

Pr(V= |Ht , sell) =

The ask price is Pr(V= |Ht , buy) + (1- Pr (V= |Ht , buy))

Pr(V= |Ht , buy) =

In the long run prices converge to fundamentals and market is efficient.

V VV

VVV

V

V

V

V

krr

kr

tt

t

)1())1()1((

))1()1((

krr

kr

tt

t

)1())1)(1((

))1((

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t

E[v|Ht , s]

V

V

1

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If = 0 and C ( , ) then the price is constant

There exist ’< ’’ such that

If t < ’ informed traders always sells

If ‘< t < ’’ then an informed trader will buy (resp. sell) if and only if their signal is h (resp. l)

If t > ’’ informed traders always buy

The history of trade cannot revel the realization of V.

Bikhchandani Hirshleifer and Welch (JPE 1992)

VV

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t

E[v|Ht , s]

V

V

1

C

’ ’’

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vh() 

vl() 

Bid

Ask

Bid and ask

1

V

* **

V

 Result: Market is informational efficient if and only if = 1 and C = 0.