Continuous Auctions & Insider Trading
Albert S. Kyle (Econometrica 1985)
Presented by Jay Li
Mar. 2007
I. Main Questions and Answers
How quickly is new private information about the value of t incorporated into market
prices?
 Incorporated gradually in the trading process and fully incorporated by the end of
trading in continuous time.
How valuable is private information to an insider?
 Insider profits are proportional to the amount of prior inside information.
How does noise trading affect the price volatility?
 No effect if the insider is risk-neutral
What determines market liquidity?
 Noise trading contributes to the depth of the market by camouflaging the insiders.
 Market depth is inversely proportional to the amount of private information not
yet incorporated into prices.
 Market resilience is determined by the trading of the insider, not the noise trader.
II. Single Auction Equilibrium
Players:
Insider, noise traders, and market makers
Variables:
Ex post liquidation value of the risky asset v~ ~ N ( p0 ,  0 )
Quantity traded by noise traders u~ ~ N (0,  u2 ) , u~  v~
Quantity traded by insiders ~
x
Price set by market makers ~p
p)~
x
Profits of the insider ~  (v~  ~
The rule of the game
Step 1:
Exogenous values of v~ and u~ are realized and the insider chooses the quantity he
trades ~
x when observing v~ but not u~ .
x  X (v~ ) .
The insider’s trading strategy X is a function such that ~
Step 2:
The market makers determine the price ~p at which they trade the quantity necessary
to clear the market when observing ~
x  u~ but not ~
x or u~ (or v~ ) separately.
p  P( ~
x  u~ ) .
The market makers pricing rule P is a function such that ~
When maximizing his profit, the insider takes into account the effect the quantity he
chooses to trade in step one is expected to have on the price established in step two,
taking the rule the market makers use to set prices in step two as given.
The Equilibrium:
Proof
Suppose the linear function of the market makers’ pricing rule and insider’s trading
strategy are as follows:
~
p  P( ~
x  u~)     ( ~
x  u~) and ~
x  X (v~ )    v~
(2.4)
Noting E[u~ ]  0 , insider’s profit is then
Maximizing profit yields the FOC
v    2x  0
Comparing with (2.4), we get
Plugging (2.4) into (2.2) yields
   (  v~  u~)  E[v~ |   v~  u~]
Applying the conditional expectation of normally distributed variables, i.e.,
1
E[ X 2 | X 1 ]   2   2111
( X 1  1 ) , we get
Solving (2.6) and (2.8) s.t. SOC   0 yields the results together with   p 0 and
   p 0
Properties of the Equilibrium
Informativeness of prices:
Applying the conditional variance of normally distributed variables, i.e.,
1
var[ X 2 | X 1 ]   22   2111
12 yields
~
~
1  var[ v | p ]   0 / 2
Thus one half of the insider’s private information is incorporated into prices and the
volatility of prices is unaffected by the level of noise trading  u2
Market depth, i.e., the order flow necessary to induce prices to rise or fall by one dollar,
measured by 1 /  , is proportional to the amount of noise trading, and inversely
proportional to the amount of private information.
Maximized profit, (v  p 0 ) 2 / 4 , is proportional to the depth of the market. A horizontal
expansion of the supply curve induces the insider to trade a proportionately larger
quantity without affecting prices, and this makes his profits correspondingly larger as
well.
Expected profit E[(v  p 0 ) 2 / 4 ]  (var[ v  p 0 ]  ( E[v  p 0 ]) 2 ) / 4 
proportional to standard deviation of both v~ and u~ .
1
( 0 u2 ) 1 / 2 ,
2
III. Sequential Auction Equilibrium
Features in addition to those of the single auction game
Trading consists of N auctions, which take place over one trading day beginning at t=0
and ending at t=1. That is
u~ (t ) is a Brownian motion process with instantaneous variance  u2 .
.
2
~
Thus u n ~ N (0,  u t n ) , where t n  t n  t n 1 .
~
xn  ~
xn  ~
x n 1
x n is the aggregate position of the insider after the nth auction, so that ~
denotes the quantity traded by the insider at the nth auction.
~
p n is the market clearing price set by the market makers at nth auction.
Trading strategy
Pricing rule
Profits of the insider
The Equilibrium
Properties of the Equilibrium
The informativeness of prices,  n , declines monotonically, reflecting the fact that private
information of the insider is gradually incorporated into prices (see (3.19)). It is shown
that  N goes to 0 in a continuous time setting, so private information is fully
incorporated into prices by the end of the trading.
The market depth, again, is negatively proportionate to the amount of information not yet
incorporated into prices, and proportional to the amount of noise trading (see (3.18)).
The expected profits of the insider are proportional to ( 0 u2 )1 / 2 , as in the single auction
model.
If  u doubles,  n halves;  n ,  n , and  n doubles;  n is unaffected. That is, increasing
the amount of noise trading increases market depth proportionately, increases
proportionately the profits of the insider by encouraging him to trade more, and leave the
informativeness of prices unchanged.
Outline of the Proof
1st step:
Using a backward induction argument to obtain the insider’s trading strategy and
expected profits as a function of the market makers’ pricing rule. Since the pricing
rule is characterized by the market depth parameter  n , the insider’s problem is to
decide how intensely (measured by  n ) to trade on the basis of his private
information, given the market depth expected at current and future auctions. If future
market depth is greater, he is going to trade more intensively in the future and vice
versa.
The backward induction argument also shows that the insider’s profit function is
quadratic and the linear equilibrium is recursive. In the meantime, the intensity with
which the insider trades,  n , is derived as a function of the current market depth  n
and the measure of the value of private information at future auctions  n , which itself
is a function of market depth at those auctions.
2nd step:
Market efficiency condition is used to derive the market depth  n and the
informativeness of prices  n from  n and  n 1 . Intuitively, given the level of noise
trading, the depth of the market at the nth auction depends negatively on how much
private information that is not yet incorporated into prices after the (n-1)th auction,
and how intensively the insider trades upon his private information. This also
determines how much of the insider’s remaining private information is incorporated
into prices at the nth auction, and how much remains private. The market makers, by
setting price in response to insider trading, reduce the market liquidity.
3rd step:
The uniqueness of the equilibrium is shown.