Bruno Biais

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Asymmetric information the bid-ask spread

• Bruno Biais
• Toulouse School of Economics
• Rotman Schools Distinguished lecture Series
• Toronto University
• January 2008

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Simple model Market order placed by risk averse
investor (ARA ?) trading both for liquidity
(inventory shock L) and informational motives
(signal s on value v). vpse, where p is
a constant, E(s)0, E(e)0, and s² V(e). s,
e, and L jointly normal and independent. For
simplicity, normalize E(L) to 0. Competitive
market makers/liquidity suppliers with ARA ?,
average inventory I They dont know L, for them
it is a random variable.
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Market structures First, we consider the case
where market order placed then limit orders
placed trade conducted at uniform price
(crossing supply demand curves). In this
context, we first assume informed agent is
competitive (in the line of Grossman and
Stiglitz, AER 1980), then well consider
strategic informed agent. Second, we turn to the
case where liquidity suppliers first place
schedules of limit orders then are hit by
market order discriminatory pricing applies
each limit order filled at its own price.
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Differences with Kyle (1985) Glosten Milgrom
(1985) In these papers, market orders placed by
informed /or noise traders purely random
trade. Present model rational agents, noise in
prices due to random inventory shock L. In these
papers, risk neutral market makers. Here risk
averse. As in Glosten Milgrom (1985), market
makers face single trader contrast with Kyle
(1985) batched order flow from informed noise
gt informed must cope with random noise component
of trade (with linear prices risk neutrality as
in Kyle 1985, this does not matter.) As in Kyle
(1985), distributions are normal projection
theorem used contrast with two-point distribtion
in GM (1985).
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Objective of the informed agent Choose optimal
quantity, given rationally anticipated
price maxQ (LQ)(ps)-Qp-((?s²)/2)(LQ)²
First order condition Q (ps-p)/(?s²)-L
(ps-?s²L-p)/(?s²) (?-p)/(?s²) ? valuation
of the strategic informed trader for the
asset increasing in private signal, and
decreasing in inventory. Q affine in ?. Hence,
Q reveals ? when quoting prices, the liquidity
suppliers will ake this into account. ? can be
written as an affine function of Q ?p?s²Q
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Joint normality s, e, and L are jointly normal,
and ? is affine in s and L Hence, ?, e and s are
jointly normal. Hence, conditionally on ?, the
final wealth of the liquidity suppliers is
normal. Hence their objective function is still
mean variance.
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Objective of market maker i maxqi(p)
(Iiqi)E(v?)-qip-(?/2)(Iiqi)²Var(v?) Having
observed Q, liquidity suppliers have inferred ?.
They use it in their information set. First
order condition qi(p)(E(v?)-p)/(?Var(v?))-(?
Var(v?))/(?Var(v?))Ii qi(p)(µi-p)/(?Var(v
?)) µi E(v?)-?Var(v?)Ii valuation of
market maker i
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Market clearing ?i qi(p) Q 0
N((E(v?)-p-?Var(v?)I)/(?Var(v?)))(?-p)/(?s²)0
E(v?)-p-?Var(v?)I(?Var(v?))/(N?s²)(?-p)0
p(1(?Var(v?))/(N?s²)) ?
(?Var(v?))/(N?s²) E(v?)-?Var(v?)I.
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Conditional expectation Since ?, e and s are
jointly normal we can use the projection
theorem) E(v?) E(v) cov(v,?)/V(?)
(?-E(?)) p (cov(ps?,s-?s²L)/(V(s-?s²L))
(?-p) p (V(s))/(V(s)(?s²)²V(L)) (?-p)
pd(?-p) d?(1-d)p dV(s)/(V(s)(?s²)²
V(L)) measures information content of order
flow increases in informativeness of the insider
signal decreases in non--informational
component of the trade. d0 no private
information.
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Conditional variance Var(v?)
V(v)(1-corr(v,?)) Since random variables jointly
normal, conditional variances are constant, only
function of parameters do not vary with
different realizations of conditioning variable.
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Equilibrium price Substitute value of E(v?)
into market clearing to get equilibrium price as
a function of ? p(1(?Var(v?))/(N?s²)) ?
(?Var(v?))/(N?s²) d?(1-d)p-?Var(v?)I
Equilibrium price linear in ?
paß? a((1-d)p-?Var(v?)I) / (1
?Var(v?)/(N?s²) ) ß ( d ?Var(v?)/(N?s²)) /
(1 ?Var(v?)/(N?s²)) Since dlt1, ßlt1. The
greater the informational asymmetry, the greater
d, the greater ß the more the price reacts to ?.
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Equilibrium price as a function of Q Since
equilibrium price affine in ?, itself affine in
Q, equilibrium price linear in Q p aß?
aßp?s²Q p a/(1-ß) (ß/(1-ß))?s²
Q. Denote it pM?Q.
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Graphic representation
p
pM?Q.
Ask
M
Bid
Q
Q gt 0 market order to buy
Q lt 0 market order to sell
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Equivalence with simultaneous moves Here,
market order Q placed observed, then liquidity
suppliers quote prices (similar to open
outcry/floor or its computerised version).
Alternatively, market orders limit orders
placed simultaneously, as in call auction. In our
simple model the two are strategically
equivalent Since, price reveals Q (pM?Q), when
posting order to be filled iff price is p,
liquidity supplier can condition on information
content of this price. Conditioning quantity
offered on price is equivalent to observing Q.
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Empirical implication (1)

• The adverse-selection model predicts that the
  informational price impact of trades should be
  commensurate with the private signal underlying
  the informed trade.
• Consistent with this, Seppi (1992) finds positive
  correlation between price changes associated with
  block trades and subsequent innovations in
  earnings announcements.

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Empirical implication (2)

• Both inventory and adverse-selection theories
  predict trades impact prices, but the former
  predicts this impact should be transient, while
  the latter predicts it should be permanent.
• Permanent impact due to impact of unexpected
  trades on expectations
• Hasbrouck (1991) analyzes joint process of trades
  quote revisions using a VAR approach. He finds
  trades have permanent impact.

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Permanent versus transient impact
Long term price in pure adverse selection
model (?c0, dgt0)
Transaction price
Mix of the two (?gt0, dgt0)
Midquote
Long term price in pure inventory cost model (? gt
0, d0)
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Empirical implication (3)

• ? in equation is a measure of the depth of the
  market (as ? goes up, depth is reduced).
• As the informational motivation of trades becomes
  relatively more important, ? goes up.
• Lee, Mucklow and Ready (1993)
• Around earnings announcements (when adverse
  selection likely) depth is reduced and spreads
  wide on the NYSE.
• More pronounced for announcements with larger
  subsequent price changes.

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Equilibrium with strategic informed With
competitive informed agent no need to start with
conjecture on equilibrium price function, with
strategic agent, conjecture needed. Competitive
informed agent doesnt need to form beliefs on
relationship between trade size and price.
Strategic informed agent must do so, to take into
account the impact of his trade on the price.
Above we obtained a linear price function, here
posit linear price function pM?Q. As
shown below, this is consistent with rationality.
The conjecture that prices are linear in trade
size is self-fulfilling. But there could be
other, nonlinear, equilibria, in contrast with
competive case where unique equilibrium is
linear.

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Optimality for informed agent Given her rational
expectation of the price function, informed agent
chooses Q to maximize (LQ)(ps)-Q(M?Q)-((?s²)/
2)(LQ)² , First order condition (ps) -
(M2?Q)-?s²(LQ) 0 (ps-?s²L)-M
(?s²2?)Q Q (?-M)/(?s²2?). Lower
sensitivity of Q to ?, than if competitive, to
reduce price impact. FOC can be rewritten as ?
M(?s²2?)Q ? affine in Q.

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Liquidity suppliersfirst order
condition qi(p)(µi-p)/(?Var(v?))
(unchanged)
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Market clearing N (E(v?)-p-?Var(v?)I)/(?Var(v
?))(?-M)/(?s²2?) 0 E(v?)-p-?Var(v?)I(?Var(
v?))/(N(?s²2?)))(?-M) 0 d?(1-d)p-p-?Var(v
?)I(?Var(v?))/(N(?s²2?)))(?-M) 0 p
(1-d)p-?Var(v?)I-(?Var(v?))/(N(?s²2?)))M
d?Var(v?)/(N(?s²2?)) ?. Equilibrium price
affine in ?.
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Equilibrium price affine in Q Substituting
?M(?s²2?)Q from the first order condition of
the informed agent, the equilibrium price is p
(1-d)p-?Var(v?)I-?Var(v?)/(N(?s²2?)))M
d?Var(v?)/(N(?s²2?)) M
d?Var(v?)/(N(?s²2?))Q(?s²2?) p
(1-d)pdM-?Var(v?)I d(?s²2?)?Var(v?)/NQ
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Closing the equilibrium loop To close the
rational expectations loop, we identify this
price function with the conjectured one
pM?Q Thus M p-(?Var(v?)I)/(1-d). ?
d(?s²2?)?Var(v?)/N ?(1-2d)d?s²?Var(v?)/N
?(d?s²?Var(v?)/N)/(1-2d). For this to be
positive we impose dlt(1/2).
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Analysing welfare Profits of the informed
losses of uninformed. From utilitarian
perspective with CARA utilities, this transfer
has no impact on social welfare. Yet
information asymmetries do reduce social welfare,
by reducing risk--sharing gains from trades.
For simplicity assume liquidity suppliers risk
neutral (?0). To maximize gains from trade,
risk--averse agent should entirely trade out of
his endowment shock. First--best trade
Q-L. Is this gain from trade realised in
equilibrium ?
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Welfare Information asymmetry strategic
behavior gt equilibrium trade (Q) less responsive
to inventory shocks than first--best
trade ?Q/?L1-2dlt?Q/?L1. The greater
the adverse--selection problem (d) the lower the
second--best welfare. Informed agent scales
back trade to reduce price impact (? Q) Price
impact of trades ? ?s²d/(1-2d), increases in
adverse selection (d) Reduction in social
welfare // reduction in market liquidity caused
by information asymmetries strategic behaviour.
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Consequences of strategic informed trading

• Same information revelation (?)
• But informed agent trades less aggressively, to
  avoid impact on prices.
• Hence less gains from trade.
• Observationally equivalent to competitive
  informed trading ?

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Discriminatory pricing We now turn to the second
market structure we consider First, limit
orders are placed, then market order placed,
trade at discriminatory price. This is in the
line of Glosten (JF 1994). To simplify liquidity
suppliers are risk neutral.
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Price paid for each of Q shares in uniform price
auction
Uniform price p(Q) E(vQ)
p(Q)
Q
Price p(Q) quoted by liquidity suppliers knowing
total demand Q Total price paid for Q shares
Q p(Q)
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Price paid with discriminatory prices
Price Schedule Limit Order Book
p(Q) marginal price of Qth unit
p(q) marginal price of qth unit
p(0) marginal price of first unit
q
Q
Total price paid for Q shares Integral from 0 to
Q of p(Q) When quoting price for qth share,
liquidity supplier does not know how much further
the order will walk the order book. He only knows
order size (Q) at least as large as q.
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Upper lower-tail expectations In this market,
liquidity suppliers cant condition on total
market order quantity Q What the liquidity
suppliers know is that if a limit order to sell
at price p(q) is hit, the market order is at
least as large as the cumulated depth of the book
up to that price Q gt q Expectation of value
given that this order has been hit "upper--tail
expectation" EvQgtq. If liquidity suppliers
risk neutral competitive, ask prices equal
upper--tail expectations, bid prices equal
lower--tail expectations.
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Graphically
Price Schedule Limit Order Book
Marginal price of qth unit EvQgtq
q
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Small trade spread Infinitesimal trades have a
discrete impact on prices (price schedule
discontinuous at 0). Contrasts with
uniform--price where price impact commensurate
with size of trade (price continuous at 0).
p
Marginal price with Discriminatory Pricing
q
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Intuition Small--trade spread because ask price
for infinitesimal buy order (e) impounds
non--infinitesimal information content
conditioning set, in EvQgt e includes case
where total quantity small (Q close to e) (in
this case E(vQ) is low) case where it is
large (Qgtgte) (in that case E(vQ) is
large). Contrasts with uniform price market
p(0) quoted knowing Q0.
p
q
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Equilibrium trade The greater his valuation for
the asset (?), the more the trader buys. For
valuations close to p, no trade. Difference in
valuation not worth paying the small trade spread.
q(?)
?
Partial market break-down for agents with
relatively small gains from trade
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Competitive market makers the cost of trades