%Paper: ewp-fin/9507009
%From: back@wuolin.wustl.edu
%Date: Mon, 17 Jul 95 11:43:14 CDT

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%% The paper begins now. 
\title{Long-Lived Information and Intraday Patterns\thanks{This paper comprised
part of the second author's Ph.D. thesis at Washington University in St.\
Louis.  He gratefully acknowledges support from a Society of 
Actuaries Ph.D. Grant.  We are grateful for numerous
helpful comments from participants at seminars at 
the University of Iowa, the University of Michigan, Vanderbilt University,
Northwestern University, and the Workshop on Mathematical Problems in Finance
at the Institute for Advanced Studies.}
}
\author{}
\date{}
\maketitle
\thispagestyle{empty}
\begin{center}
Kerry Back\\
Olin School of Business\\
Washington University in St.\ Louis\\
St.\ Louis, MO  63130\\
back@wuolin.wustl.edu
\vskip .5in
Hal Pedersen\\
Faculty of Management\\
181 Freedman Crescent\\
University of Manitoba\\
Winnipeg, Manitoba R3T 5V4\\
CANADA\\
pedersn@bldgdrake.lan1.umanitoba.ca

\vskip 1in

First draft:  July, 1994\\
Current draft: \today
\end{center}
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\begin{center}
{\Large \bf Long-Lived Information and Intraday Patterns}
\vskip 2in
{\bf Abstract}
\end{center}
\vskip .3cm\noindent
This paper studies the effect of clustering of liquidity trades on
intraday patterns of volatility and market depth
 when private information is long-lived.
The assumption of long-lived information allows us to distinguish
between the patterns of information arrival and information use.
Our results are: (i) volatility follows the same pattern as
liquidity trading, (ii) there are no systematic patterns in the price impacts
of 
orders, and 
(iii) the timing of information arrival is
is unimportant.
Result (i) is the same as that obtained by
Admati and Pfleiderer (1988) in a model of short-lived private
information, but (ii) and (iii) are different.
\thispagestyle{empty}
\newpage\pagestyle{plain}\pagenumbering{arabic}

It is well documented that there are systematic intraday patterns in 
the volatility of security prices and the volume of trading. 
In a widely cited paper, 
Admati and Pfleiderer (1988) offer an explanation of these
patterns: clustering in time of liquidity-motivated trades induces 
clustering of information acquisition and informed trading, which
generates the same clustering of information-driven price changes.
The clustering of information acquisition takes the form of more
traders collecting information in some parts of the day than in
others.  When more traders collect (and use) information, the order
process is more informative; thus, price volatility is greater. 

It seems implausible to us that
the number of agents collecting information varies systematically
over the day.  It seems more plausible that the intensity with which
agents trade on their information may vary systematically.
However, this
distinction is impossible to make within the Admati-Pfleiderer
model, because private information has a short lifespan in that model.
Admati and Pfleiderer assume multiple public announcements occur each day, 
each of which reveals
all the information collected privately earlier. 

The purpose of the present paper is to examine the effect of clustering of
liquidity trades in a model of long-lived private
information, to see whether such clustering generates the same
clustering of information use as opposed to information acquisition,
and to determine the implications of such a model for volatility and market
depth.
In the model we investigate, which is a generalization of the continuous-time
model studied
in Kyle (1985) and Back (1992), there is only a single informed trader.  The
absence of competition gives this trader a great deal of flexibility
regarding the timing of his trades.  Thus, in a sense, our model is 
at the opposite extreme of the Admati-Pfleiderer
model, in which informed traders have no discretion regarding when they trade
(because their information lasts only one period). 

Our results can be summarized as follows: (i) the informativeness of
orders and the volatility of prices follow the same pattern as
liquidity trading, (ii) there are no systematic patterns in the price impacts
of orders, 
and (iii) given the total amount of private information, the pattern of
information 
flow to the informed trader is irrelevant.
The first result is the same as Admati and Pfleiderer's,
but the second and third are different.
A particular consequence of (i) and (iii) is that clustering of
liquidity trades induces clustering of information use (and hence
volatility), without implying the same clustering of information
acquisition. 

We specify exogenously the
information flow to the informed trader.
One can think of the informed trader as committing to information collection
at the beginning of the model. 
An implication of (ii) and (iii) is that the trader will be indifferent among
information arrival processes
that provide the same total amount of information.   Market depth, volatility,
and the costs of liquidity traders are also
independent of the pattern of information arrival.

We also take the pattern of liquidity trading to be exogenously
specified.  However, result (ii) implies that the expected execution
costs of liquidity traders do not depend on the
timing of their trades.  Hence, the liquidity traders can be
interpreted as either discretionary or nondiscretionary. 

Result (ii) is 
the implication of our model that differs most significantly from 
Admati and Pfleiderer's.  In the Admati-Pfleiderer model, both liquidity
traders and informed traders are motivated to concentrate their trades in
certain periods because execution costs -- the price impacts of
trades -- are low in those periods. 
Given the U-shaped
patterns of volume and volatility found on the New York Stock
Exchange,\footnote{See Wood, McInish, and Ord (1985) for the
volatility pattern and Jain and Joh (1988) for volume.} this
would imply an inverted U-shape for execution costs. 
However, bid-ask spreads have a U-shaped
pattern (Wood and McInish, 1992). 
Accounting for the probability of transacting within the
spread, Madhavan, Richardson and Roomans (1994) find that execution costs 
rise steadily over the day.  Neither the Admati-Pfleiderer model
nor our model can generate either a U-shaped pattern or an upward
sloping pattern.  Our result is
consistent with 
Ferguson, Mann, and Schneck (1993), who 
find no relationship in foreign exchange futures markets between
volume and volatility on the one hand and
execution costs on the other.  Our result also has some support
from Foster and Viswanathan (1993), who find no association between
volume and the adverse selection component of trading costs for 
two of the three sets of stocks they examine (the least actively
traded and moderately actively traded deciles).
However, Foster and
Viswanathan find a positive association between volume and the
adverse selection component for the most actively traded stocks,
which is inconsistent with both our result and Admati and
Pfleiderer's.  Apparently, neither
our model nor Admati and Pfleiderer's can explain the empirical patterns
in execution costs.\footnote{For an approach based on imperfect
competition in market making, see
Brock and Kleidon (1992), who, however, do not attempt to explain
volatility patterns.}

The model is described in Part~\ref{model}.  Part~\ref{results} 
presents the results, and Part~\ref{conclusion} concludes the paper.

\section{The Model}\label{model}

Trading occurs over an interval $[0,1]$.  This period can be
interpreted as a single day, a part of a day, or several days.
The risk-free rate is taken to be zero.
During the period $[0,1]$, orders for a risky asset
are submitted by a single informed trader and
uninformed ``liquidity traders'' to risk-neutral competitive market makers, who
set prices and clear the market.  An announcement at time 1 reveals the
asset value, which is a finite-variance random variable $\tilde{ v}$. 
Prior to that time, competition between the market 
makers forces the price to equal the conditional expectation of $\tilde{ v}$,
given the market makers' information.  Their information consists of the
history of combined informed and liquidity-motivated orders.  We want to find
the optimal trading strategy for the informed trader and the equilibrium
price-adjustment rule for the market makers. 

The model is more general than that studied by Kyle (1985) and Back (1992) 
in two respects: (i) the informed trader learns over time about $\tilde{ v}$,
and (ii) the volatility of liquidity trading is time-varying.

To model (i), assume there is a sufficient statistic $S(t)$ for the informed
trader's
information at each time $t$, in the sense that his
conditional expectation of $\tilde{ v}$ at time $t$ equals
$f(t,S(t))$ for some function $f$. 
Take $f$ to be strictly monotone in $S$,
so increases in $S$ represent good news for the asset.
Assume $S(0)$ is normally
distributed with mean zero and that 
$S$ follows a Gaussian process:
\begin{equation}\label{S}
dS(t)=\sigma_s(t)dW_s(t),
\end{equation}
where $\sigma _s$ is a deterministic and continuous function of time,
and $W_s$ is a Wiener process.  A special case is $\sigma_s \equiv 0$,
in which case our information structure is the same as Kyle's 
(1985).\footnote{Actually, we should also assume $S(0)=\tilde{ v}$ to be
fully consistent with Kyle's presentation, but whether 
the informed trader actually observes $\tilde{ v}$ or only has a signal 
about $\tilde{ v}$ 
is a trivial distinction, because 
the solution in any case only depends on his expectation of $\tilde{ v}$.
It is harmless to interpret this expectation as the true value, 
since all agents in the model are risk neutral.}
The deterministic volatility $\sigma_s^2(t)$ is a feature of any
Gaussian filtering model.

When $\sigma_s^2(t)$ is large, the agent is learning a lot. 
To understand this, note that the variance of $S(1)$
is $\var S(0) + \int _0^1 \sigma_s^2(t)\,dt$.  This variance
represents the uncertainty the market would have regarding the
informed agent's signal at time 1, if the market learned nothing
before time 1.  When
$\sigma_s^2(t)$ is large, the uncertainty is increasing at a high rate,
reflecting the fact that the agent is learning at a high rate.

To model (ii), denote the 
cumulative orders of liquidity traders through time $t$ by
$Z(t)$, and assume the process $Z$ is 
a Gaussian process:
\begin{equation}\label{Z}
Z(0)=0 \quad\text{and}\quad dZ(t)=\sigma_z(t)dW_z(t),
\end{equation}
where $\sigma _z$ is a deterministic, strictly positive, continuous function 
of time, and
$W_z$ is a Wiener process independent of $W_s$ and $\tilde{ v}$.\footnote{
A trivial generalization of the model would be to
allow $S$ and $Z$ to have deterministic drifts.  A deterministic drift
would add nothing to the informativeness of $S$; hence, one might as well
work with the ``detrended'' process, as we are doing here.  As for a 
drift in $Z$, the market here is so ``deep'' that a predictable
component of liquidity trades has no
effect on prices:  market makers are willing to take the opposite side
of any trade that is known to have no information content, without
requiring a price concession.  Again, one can work with the detrended
liquidity-trade process, as we are implicitly doing here.  Of course, if 
market makers were 
risk-averse, or imperfectly competitive, or had position limits (such as
margin requirements), then even
predictable liquidity trades could affect prices.}

We now introduce an
assumption regarding the parameters of the model.
Assume that
\begin{equation}\label{assumption}
\var S(0)>\sup_t 
\frac{ \int _t^1\sigma^2_s(u)\,du}{\int _t^1\sigma ^2_z(u)\,du}
\cdot \int_0^1 \sigma_z^2(u)\,du - \int_0^1\sigma_s^2(u)\,du.
\end{equation} 
Particular cases in which (\ref{assumption}) is
satisfied are when $\sigma^2_s \equiv 0$ 
or when $\sigma^2_s$ and
$\sigma^2_z$ are constants.  In each of these cases, the right-hand side of
(\ref{assumption}) is zero.
Each of these particular cases is a
generalization of Kyle (1985).
Assumption (\ref{assumption}) does not seem too 
restrictive, but we will comment on its role in the concluding section of the
paper.

The scale of the signal process $S$ is arbitrary, so we will choose a scale
that simplifies the notation.  Given a signal process $S$ as defined above, 
set $\hat{ S} = b_zS/b_s$, 
where $b_s^2  \equiv 
\var S(0)+\int_0^1\sigma_s^2dt$ and 
$b_z^2 \equiv \int_0^1\sigma_z^2dt.$ 
Note that $b_s^2$ is the variance of $S(1)$ and hence a
measure of the total amount of private information. 
Likewise, $b_z^2$ is the
variance of $Z(1)$ and hence a measure of the total amount of
liquidity trading.\footnote{More precisely, $b_z^2$ 
represents the amount of liquidity trading only after netting out
all offsetting trades occurring during $[0,1]$.  The gross amount of
liquidity trading is infinite: 
if one adds up the
absolute changes in $Z$ over increasingly finer partitions of $[0,1]$,
the sum is unbounded (and even this unbounded sum is net of
offsetting instantaneous trades, because $dZ(t)$ represents only the net
liquidity trades at time $t$).}
For the rescaled signal process $\hat{ S}$, we have 
$\var\hat{ S}(1)\equiv \hat{ b}_s^2=b_z^2$.  This rescaling 
also affects the function $f$. Specifically,
$$\hat{ f}(x)\equiv E[\tilde{ v}|\hat{ S}(1)=x] = E[\tilde{
v}|S(1)=b_sx/b_z]=f(b_sx/b_z).$$
Assumption (\ref{assumption}) holds for $\hat{ S}$ if and only if it
holds for $S$. 
Henceforth, we will work with $\hat{ S}$ and $\hat{ f}$, but drop the
``hats.'' 
Note that this convention means we have $b_s^2=b_z^2$ (which allows
us to avoid writing a factor $b_s/b_z$ that would otherwise appear throughout
the
definition of the equilibrium).


The remainder of the model description essentially follows Back (1992).
Let $X(t)$ denote the number of shares acquired by the informed trader during
$[0,t]$, and set $Y(t)=X(t)+Z(t)$. 
Market makers observe $Y$, which is the combined informed and liquidity trades,
so their information structure is the
filtration ${\bf F}^Y = \{{\cal F}^Y(t)\}$, where
${\cal F}^Y(t)=\sigma\{Y(u)|0 \leq u \leq t\}.$ 
Define a pricing
rule to be a function $(t,y) \mapsto P(t,y):[0,1]\times \Re \rightarrow
\Re$.  Given a particular trading strategy for the informed trader, 
call a pricing rule $P$
{\em competitive\/} if
the condition
\begin{equation}\label{competitive}
P(t,Y(t))=E[\tilde{v}|{\cal F}^Y(t)]
\end{equation}
is satisfied for all $t$. 
Note that we are taking the price at each
time $t$ to depend only on $Y(t)$ rather than
on the entire history of $Y$ through time $t$. 
We will prove that there is a rule of this form satisfying (\ref{competitive}).

The informed trader observes the asset price and signal
process.  For technical reasons, it is convenient to suppose that he
also observes the liquidity trades.  We will
justify this later by showing that the asset price can be inverted to
compute the liquidity trades.\footnote{Our approach is the same as looking
at the artificial symmetrically-informed economy to prove the existence
of fully revealing rational expectations equilibria; see Grossman (1981).}
The information structure of the informed trader is therefore
assumed to be the filtration
${\bf F} = \{{\cal F}(t)\}$,
where ${\cal F}(t)=\sigma\{S(u),Z(u)|0 \leq u \leq t\}.$ 
In order to simplify the analysis we will restrict the informed trader to 
trading strategies that are absolutely continuous; i.e., trading in 
rates.\footnote{  As Back (1992) has demonstrated, there is no gain in 
generality from allowing 
the informed trader general semimartingale trading strategies.}  Consequently, 
we assume that $X(t)=\int_0^t\theta(u)\,du,$ for some process $\theta$
adapted to  ${\bf F}$.
We need to impose some constraint on the 
informed agent's strategy in order to rule out the analogue of the
``doubling strategies'' that exist
in competitive models (Harrison and Kreps, 1979). 
A constraint that suffices is:
\begin{equation}\label{double}
\int _0^1 EY(t)^4\,dt < \infty \quad\text{and}\quad \int _0^1
Ef(X(t)+Z(1))^4\,dt
< \infty.
\end{equation} 

Let $P$ denote an arbitrary 
pricing rule.  The end of period wealth accruing to the informed trader 
from an application of a trading strategy $\theta$ in the face of this 
pricing rule is
\begin{equation}
W(1)=\int_0^1[\tilde{v}-P(u,Y(u))]\theta(u)\,du,
\end{equation}
where, as explained before,
\begin{equation}\label{Y}
Y(u) = Z(u) + \int _0^u \theta (t)\,dt.
\end{equation} 
Given a price rule $P$, a trading strategy for the informed agent is 
{\em optimal} if it maximizes the expected value of $W(1)$ over all 
trading strategies satisfying (\ref{double}). 
An {\em equilibrium} is a pair $(P,\theta)$ such that $P$ is a competitive
pricing rule given $\theta$ and such that $\theta$ 
is an optimal trading strategy given P.


\section{Results}\label{results}

The equilibrium is defined in the following theorem.  It is unique within a
certain class, which we will not describe explicitly here [see Back (1992)].
In this theorem, and henceforth, we write $f(s)$ for 
$f(1,s) \equiv E[\tilde{ v}|S(1)=s]$. 

\begin{nonum-theorem}
Define 
\begin{eqnarray} 
\Sigma(t)&=&\int_t^1[\sigma^2_z(u)-\sigma_s^2(u)]\,du, \label{sigma}\\
\theta(t)&=&\sigma_z^2(t) \frac{S(t)-Y(t)}{\Sigma(t)}. \label{theta}
\end{eqnarray} 
The trading strategy $\theta$ is well defined by (\ref{theta}) in the
sense that there exists a unique solution $(\theta,Y)$ to the system
consisting of (\ref{theta}) and (\ref{Y}). 
Given the trading strategy (\ref{theta}), the distribution of $S(t)$
conditional on ${\cal F}^Y(t)$ is normal with mean $Y(t)$ and variance
$\Sigma(t)$. 
Let $\pi(t,y,\cdot)$ denote the density 
function for the normal distribution with mean $y$
and variance $\int_t^1\sigma_z^2(u)\,du$.
Define
\begin{equation}\label{P}
P(t,y)=\int _{-\infty}^{\infty}f(s)\pi(t,y,s)\,ds.
\end{equation} 
The pair $(P,\theta)$ is an equilibrium.
In this equilibrium, the price evolves as
\begin{equation}\label{dP}
dP(t,Y(t)) = \lambda(t,Y(t))\,dY(t),
\end{equation} 
where
\begin{equation}\label{lambda}
\lambda(t,y)\equiv \frac{ \partial} {\partial y} P(t,y).
\end{equation} 
If $f$ is continuously differentiable with 
$E[f'(S(1))]<\infty,$
then the process 
$\lambda(t,Y(t))$ is a martingale relative to the market
makers' information structure ${\bf F}^Y$.
\end{nonum-theorem}

Notice that $P(t,y)$ defined by (\ref{P}) is strictly monotone in $y$, 
by virtue of the monotonicity of $f$. 
Therefore, the informed trader can
invert $P(t,Y(t))$ at each time $t$ to compute $Y(t)$.  Since he
knows his own orders $X(t)$, this reveals $Z(t)$, justifying the
assumption made in Part~\ref{model} regarding his information.

The proof of the theorem
is in the appendix, but we will explain the essence of the argument
here. 
Market makers at each time $t$ are trying to estimate $f(S(1))$. 
Since $S(1)$ equals $S(t)$ plus an independent increment, market makers need to
estimate $S(t)$.  It turns out that, when the informed agent trades according
to (\ref{theta}), the distribution of $S(t)$ conditional on the market makers'
information at time $t$ is normal with mean $Y(t)$ and variance $\Sigma(t)$.
This is proven via the Kalman filter.  Hence, given the market
makers' information, $S(1)$ is distributed normally
with mean $Y(t)$ and variance $\int _t^1 \sigma^2_z(u)\,du$. 
Thus, the price defined by (\ref{P}) is
$$P(t,Y(t)) \equiv \int _{-\infty}^{\infty}f(s)\pi(t,Y(t),s)\,ds =
E[f(S(1))|{\cal F}^Y(t)],$$
which implies the pricing rule satisfies the equilibrium condition. 
As for the optimality of the trading strategy (\ref{theta}), the key fact is
that $Y(t)-S(t) \rightarrow 0$ as $t \rightarrow 1$, when (\ref{theta}) is
followed.  This implies $P(t,Y(t)) \rightarrow f(S(1))$, so the price is
``right'' by time 1.  We can show that this is the only requirement for
optimality.  This fact is implicit in Kyle's (1985) argument and is made
explicit in Back (1992).  Given the risk neutrality, it is not unexpected
that many optima exist simultaneously.

To interpret the equilibrium, we begin by observing that, as in
Admati-Pfleiderer (1988), clustering of liquidity trades leads to clustering of
informed trades, information flow to the market, and price volatility. 
That informed trades follow the same pattern as liquidity trades is
evident from the factor $\sigma^2_z(t)$
in the trading strategy (\ref{theta}). 
The information flow to the market can be seen by looking at the change in the
market's uncertainty about $S(t)$.  As stated in the theorem, the conditional
variance of $S(t)$ is $\Sigma(t)$, and 
from (\ref{sigma}) we have
$$\Sigma'(t)=\sigma^2_s(t)-\sigma^2_z(t).$$
The two components of $\Sigma'(t)$ capture the two reasons that the
market's uncertainty regarding $S(t)$ changes
over time. If no information were communicated to the market via orders,
the uncertainty would grow over time at rate $\sigma^2_s(t)$, because of
the change in $S(t)$ itself.  Since $\Sigma$ actually changes
at rate $\sigma^2_s(t)-\sigma^2_z(t)$, the term $\sigma^2_z(t)$ must
represent the rate at which information is communicated via orders.
Thus, it is the time pattern of liquidity trading
that determines the rate at which
information is communicated to the market. 
Furthermore, the rate of information flow to the market determines the
volatility of prices. 
Indeed, the volatility of prices is given by
$$dP\,dP = \lambda^2\,dY\,dY=\lambda^2\,dZ\,dZ=\lambda^2\sigma^2_z\,dt.$$
Therefore, Admati and Pfleiderer's result that concentration of liquidity
trades
leads to the same concentration of volatility holds in our model also.

We now turn to the properties that distinguish our model from Admati and
Pfleiderer's.  The first difference is that the timing of information flow to
the informed trader is irrelevant in our model. 
Notice that the pricing rule is completely determined by the joint
distribution of $\tilde{ v}$ and $S(1)$, acting through $f$, and by the
function $\sigma^2_z$, acting through $\pi$. 
Hence, given the joint distribution of $\tilde{
v}$ and $S(1)$, the pricing rule does not depend on any other
characteristics of the information process $S$. 
The same is therefore true for $\lambda(t,Y(t))$, which is the reciprocal of
what Kyle terms ``market depth,'' and for the variance of price changes.
Furthermore, the same must be true for the expected profits of the informed
trader and the expected execution costs of liquidity traders.
These quantities depend on the total amount of information but do not depend
on the pattern of information arrival. 
The basis for this irrelevance result is
that in equilibrium the informed trader ``smooths'' his use of information.
The factor $(S(t)-Y(t))/\Sigma(t)$ in the trading strategy (\ref{theta})
represents the private information of the informed trader, normalized to have
unit conditional variance.  By virtue of this normalization, the trading of the
informed agent does not depend on the amount of private information, as
measured by the conditional variance.
Thus, the
pattern of information communication to the market does not depend on
the pattern of information acquisition by the informed trader.

The second difference is the fact that in our model $\lambda(t,Y(t))$ is a
martingale.  In Admati and Pfleiderer, $\lambda$ follows a pattern
opposite to that of volume and volatility.  The martingale property here
follows
from the fact that the
pattern of informed trading precisely tracks that of liquidity trading, due to
the factor $\sigma^2_z(t)$ in the trading strategy (\ref{theta}). 
Intuitively, the
probability that an order is informed in our model does not change
systematically
over time, so the sensitivity of prices to orders does not change
systematically.

A final difference is that, in the equilibrium of our model, liquidity traders
are indifferent about when they trade.  The aggregate execution cost of the 
liquidity traders is defined as
$$\int _0^1 dZ\,dP = \int _0^1 \lambda \,dZ\,dY = \int _0^1 \lambda\,dZ\,dZ =
\int _0^1 \lambda \sigma^2_z\,dt.$$
The expected aggregate execution cost is
$$E \left[  \int _0^1 \lambda(t,Y(t))\sigma^2_z(t)\,dt \right] = \bar{
\lambda}\var Z(1),$$
where $\bar{ \lambda}$ is the constant $E[\lambda(t,Y(t))]$.
Therefore, the expected aggregate execution cost depends only on
the total amount of liquidity trading, as measured by $\var Z(1)$,
and not on the pattern of liquidity trading.

If $\tilde{ v}$ and $S(1)$ have a joint normal distribution,
then market depth is constant, as in Kyle (1985).  This is illustrated in the
following.

\begin{nonum-example}
Assume $\tilde{ v}=\bar{v}+\tilde{\varepsilon}$, where $\tilde{\varepsilon}$ is
distributed normally with mean zero and variance $\sigma^2_{\varepsilon}$.
Assume the signal process reveals $\tilde{\varepsilon}$ at time 1.  Then, as
defined in Part~I, the normalized signal at time 1 is $S(1)= b_z
\tilde{\varepsilon}/\sigma_{\varepsilon}$, and $f(s)=\bar{v}+
\sigma_{\varepsilon}s/b_z$.  Set $\lambda=\sigma_{\varepsilon}/b_z$. 
Then the equilibrium pricing rule (\ref{P}) is
$$P(t,y)=\bar{v}+\lambda y.$$
Since $\lambda$ is constant,
actual as well as expected execution costs are independent of the
pattern of liquidity trading.  Furthermore, volatility is a constant
multiple of $\sigma^2_z$.  Let $\Omega(t)$
denote the conditional variance of the random variable $E[\tilde{
v}|{\cal F}(t)]$ given ${\cal F} ^Y(t)$.  In this model, we have
$E[\tilde{ v}|{\cal F}(t)]=\bar{v} +\lambda S(t)$, so $\Omega(t)
=\lambda^2\Sigma (t)$.  Therefore,
$$\frac{ 1}{\lambda} =\frac{\sqrt{\Sigma(t)} }{\sqrt{\Omega(t)}}.$$
As in Kyle (1985, p.~1317), market depth (the reciprocal of $\lambda$) is
``inversely proportional to the amount of private information (in the
sense of an error variance) which has not yet been incorporated into
prices.''  In Kyle, ``market depth is proportional to the amount of
noise trading,'' but here it is proportional to 
$$\sqrt{\Sigma(t)}=\sqrt{\int _t^1[\sigma^2_z(u)-\sigma^2_s(u)]\,du}.$$
\end{nonum-example}

\section{Conclusion}\label{conclusion}

We have shown that in a dynamic Kyle model with a single risk-neutral informed
trader,
the volatility of prices
follows the same pattern as liquidity trading and is unaffected by
the pattern of information arrival.  The reciprocal of market depth
(Kyle's $\lambda$ parameter) is
a martingale.  The model delivers the observed
coincidence of high volatility with high volume without requiring the
number of traders collecting information to vary systematically over
the day. 

We assumed there is nontrivial private information at the beginning
of the trading period [see (\ref{assumption})].  The reason for this is as
follows.  The informed trader will want
to make unbounded trades unless the $\lambda$ parameter is a
martingale, so in equilibrium there cannot be systematic changes in
$\lambda$ over time.  In order for this to be consistent with
equilibrium, the
information content of orders must be constant over time.  This is
apparently impossible if there is no private information, or too
little private information, at the beginning of the trading period. 

The willingness to use information at a constant rate is one
manifestation of the patience of the informed trader in this model.
If the trader faced competition from other informed traders, 
or if he were risk averse,
it might be consistent with equilibrium for the $\lambda$ parameter
to vary in systematic ways.  This is an interesting issue for future research.
\vfil\eject

\noindent{\Large \bf Appendix}
\vskip .8cm
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}

The purpose of this appendix is to prove the theorem.
Throughout the proof, we will use the notation 
$\xi(t)=S(t)-Y(t)$.

\begin{lemma}\label{lemma0}
For each $t$, $\Sigma(t) >0$.  Furthermore, $\sigma^2_z(1) >
\sigma^2_s(1)$.
\end{lemma}

\begin{proof}
Invoking the normalization $b^2_z=b^2_s$, assumption
(\ref{assumption}) can be restated as
\begin{equation}\label{new-assumption}
1 > \sup_t \frac{ \int_t^1 \sigma_s^2(u)\,du}{\int_t^1
\sigma^2_z(u)\,du}.
\end{equation}
This implies $(\forall t)\;\Sigma(t)>0$.  Furthermore, by the
mean-value theorem, the ratio of
integrals in (\ref{new-assumption}) equals
$$\frac{ \sigma_s^2(a(t))}{\sigma_z^2(b(t))}$$
for some $t < a(t),b(t) < 1$.  Taking the limit as $t \rightarrow 1$
(\ref{new-assumption}) implies
$$1 > \frac{\sigma_s^2(1)}{\sigma_z^2(1)}.$$

\end{proof}

Consider the claim that $\theta$ is well
defined by (\ref{Y}) and (\ref{theta}).  Obviously, (\ref{theta})
defines $\theta$ uniquely if $Y$ is well defined, so the issue is the
existence of a solution $Y$ to the stochastic differential equation
defined by (\ref{Y}) and (\ref{theta}).
This is a linear stochastic
differential equation with coefficients that are locally bounded on
$[0,1)$ and adapted to ${\bf F}$.  Hence, it has a unique solution on
$[0,1)$ (Karatzas and Shreve, 1988, \S 5.6.C).  We can define $Y$ by
continuity at $t=1$ if $\lim_{t \rightarrow 1} Y(t)$ exists a.s.  This
is established in the following lemma, which will also be used
subsequently.

\begin{lemma}\label{lemma1}
Given the trading strategy (\ref{theta}), $\lim_{t \rightarrow 1}Y(t) =S(1)$
a.s.
\end{lemma}

\begin{proof}
The claim is that $\lim_{t \rightarrow 1}\xi(t)=0$ a.s. 
The process $\xi$ satisfies the stochastic differential equation
$$d\xi(t)=\frac{-\sigma_z^2(t)}{\Sigma(t)}\xi(t)\,dt
+\sigma_z(t)\,dW_z(t)
-\sigma_s(t)\,dW_s(t).$$
This may be rewritten as
\begin{equation}\label{sde}
d\xi(t)=\frac{-\sigma_z^2(t)}{\Sigma(t)}\xi(t)\,dt+\sigma(t)\,dW(t),
\end{equation}
where $\sigma^2(t)\equiv \sigma_z^2(t)+\sigma_s^2(t),$
and $W$ is a standard Wiener process. 

The solution of the stochastic differential equation (\ref{sde}) 
with initial condition $\xi(0)=S(0)$ is 
(Karatzas and Shreve, 1988, \S 5.6C)
$$\xi(t)=
\phi^{-1}(t)\left[ S(0)+\int _0^t \sigma(u)\phi(u)\,dW(u)\right],$$
where
$$\phi(t)\equiv \exp \left( \int _0^t \sigma^2_z(u)\Sigma^{-1}(u)\,du
\right).$$
Here, and in the following argument, the inverse notation $^{-1}$
denotes the reciprocal.
The desired result $\xi(t) \rightarrow 0$ a.s.\ as $t \rightarrow 1$
will follow from showing
\begin{equation}\label{limit1}
\phi^{-1}(t) \rightarrow 0
\end{equation}
and
\begin{equation}\label{limit2}
\phi^{-1}(t)\int _0^t \sigma(u)\phi(u)\,dW(u) \rightarrow 0\text{a.s.}
\end{equation}
as $t \rightarrow 1$.
In the proofs of these results, we will use without further comment the
facts that $\sigma^2_z(\cdot)$ and $\sigma^2(\cdot)$
are continuous, strictly positive functions and hence are bounded above
and bounded away from zero.  Likewise, 
$\sigma^2_z(\cdot)-\sigma^2_s(\cdot)$ is bounded above, and it
is bounded away from zero on a neighborhood of $t=1$ by virtue of
continuity and Lemma~\ref{lemma0}.

To prove (\ref{limit1}), it suffices to show that $\int _0^t
\Sigma^{-1}(u)\,du \rightarrow \infty$.  An application of the
mean-value theorem gives, for each $u$ and some $u^* \in [u,1]$,
$$\Sigma(u) = -\Sigma'(u^*)(1-u)=[\sigma^2_z(u^*)-\sigma^2_s(u^*)](1-u)
\leq a(1-u)$$
for some constant $a$.  Therefore
$$\int _0^t \Sigma^{-1}(u)\,du \geq a^{-1}\int _0^t (1-u)^{-1}\,du
\rightarrow \infty.$$

It remains to establish (\ref{limit2}).  It suffices to show that
$$\phi^{-1}(t)\int _0^t \phi(u)\,dW(u) \rightarrow 0 \text{a.s.}$$
Applying the law of the iterated logarithm to the continuous 
local martingale $\int _0^t \phi(u)\,dW(u)$ (Durrett, 1984, p.~77) gives 
$$\limsup_{t\rightarrow1}
\frac{\int_0^t\phi(u)dW(u)}
{\sqrt{2\int_0^t\phi^2(u)\,du\cdot
\log\log\int_0^t\phi^2(u)\,du}}=1.$$
Hence, it suffices to show that
$$\phi^{-1}(t) \sqrt{2\int_0^t\phi^2(u)\,du\cdot
\log\log\int_0^t\phi^2(u)\,du} \rightarrow 0,$$
equivalently,
\begin{equation}\label{limit3}
\frac{ 2\int_0^t\phi^2(u)\,du\cdot \log\log\int_0^t\phi^2(u)\,du}
{\phi^2(t)} \rightarrow 0.
\end{equation} 
This follows from (\ref{limit1}) unless
\begin{equation}\label{limit4}
\int _0^t\phi^2(u)\,du \rightarrow \infty,
\end{equation}
so suppose (\ref{limit4}) holds.

The proof of (\ref{limit3}) now follows from repeated application of 
l'H\^{o}pi\-tal's rule.  A first application, combined with dropping
bounded factors and terms known to converge to zero, shows that it
suffices to establish
$$\frac{ \log\log\int_0^t\phi^2(u)\,du}{1/\Sigma(t)} \rightarrow 0.$$
A second application, again dropping bounded factors, shows that it
suffices to establish
$$\frac{ \phi^2(t)\Sigma^2(t)}{\int_0^t\phi^2(u)\,du \cdot
\log\int_0^t\phi^2(u)\,du} \rightarrow 0.$$
A third application shows that it suffices to establish
$$\frac{ \sigma^2_s(t)\Sigma (t)}{1+\log\int_0^t\phi^2(u)\,du}\rightarrow 0,$$
and this is clearly true.

\end{proof}

We now establish the claim regarding the ${\cal F} ^Y(t)$-conditional
distribution of $S(t)$. 

\begin{lemma}\label{theorem2'}
For each $t$,
the ${\cal F}^Y(t)$-conditional distribution of $S(t)$ is normal with mean
$Y(t)$ and variance $\Sigma(t)$.
\end{lemma}

\begin{proof} 
We have 
\begin{eqnarray*}
d\xi(t) & = & -\alpha(t)\xi(t)\,dt-\sigma_z(t)\,dW_z(t)
 +\sigma_s(t)\,dW_s(t),\\
dY(t) & = & \alpha(t)\xi(t)\,dt+\sigma_z(t)\,dW_z(t), 
\end{eqnarray*}
where
$\alpha(t)=\sigma_z^2(t)/\Sigma(t)$.
These are the state (or signal) and observation equations, in the terminology
of
filtering.
Conditional on ${\cal F}^Y(t)$, $\xi(t)$ is normally distributed.  We will
denote its conditional 
mean by $\hat{ \xi}(t)$ and its conditional variance by
$\Sigma^*(t)$.   Therefore, $S(t)$ is conditionally normal with mean $\hat{
\xi}(t)+Y(t)$ and variance $\Sigma^*(t)$.

The 
Kalman-Bucy filtering equation
for $\hat{\xi}(t)$ is\footnote{
See Kallianpur (1980, \S 10.2).  The state and observation equations
are written in Kallianpur's notation as 
$A_0(t)=0,$ $A_1(t)=-\alpha(t),$
$A_2(t)=0,$ $C_0(t)=0,$ $C_1(t)=\alpha(t),$ $C_2(t)=0,$
$B(t)=[-\sigma_z(t),\sigma_s(t)],$ and
$D(t)=[\sigma_z(t),0].$}
\begin{equation}\label{kalman-bucy}
d\hat{\xi}(t)=-\alpha(t)\hat{\xi}(t)dt+\frac{1}{\sigma_z(t)}
[\alpha (t)\Sigma^*(t) -\sigma_z^2(t)]\,d\nu(t),
\end{equation}
with initial condition $\hat{ \xi}(0)=E[S(0)]=0$.
In (\ref{kalman-bucy}),
$\nu$ denotes a Wiener process (the innovations process).
The variance process $\Sigma^*$ satisfies the differential equation
$$\frac{ d}{dt}\Sigma^*(t)  =  -2\alpha(t)\Sigma^*(t)+\sigma_z^2(t)+
\sigma_s^2(t)-\frac{1}{\sigma_z^2(t)}[\alpha (t)\Sigma^*(t) -\sigma_z^2(t)]^2 
$$ 
and initial condition
$$\Sigma^*(0) = \var\xi(0) = \sigma^2_0.$$
This Riccati equation is well known to have a unique solution.  It is
easy to check that it is solved by 
$\Sigma^*=\Sigma$, which shows that $\Sigma$ is the 
conditional variance of $S(t)$.

Substituting $\Sigma^*=\Sigma$ into (\ref{kalman-bucy}), it becomes
$$d \hat{ \xi}(t)=-\alpha(t)\hat{ \xi}(t)\,dt.$$
The unique solution of this with initial condition $\hat{ \xi}(0)=0$ 
is $\hat{ \xi} \equiv 0$.  Hence, the conditional mean of $S(t)$ is $Y(t)$.

\end{proof}

The next two lemmas establish that $(P,\theta)$ is an equilibrium.

\begin{lemma}\label{lemma3}
Given the trading strategy (\ref{theta}), the pricing rule (\ref{P}) is
competitive.
\end{lemma}

\begin{proof}
Using successively the facts that $Y(1)=S(1)$ a.s., the fact that 
$S$ has independent zero-mean increments, and Lemma~\ref{theorem2'}, we
obtain
\begin{eqnarray*} 
E[Y(1)|{\cal F} ^Y(t)]&=& E[S(1)|{\cal F}^Y(t)]\\
&=& E[S(t)|{\cal F} ^Y(t)]\\
&=&Y(t).
\end{eqnarray*}
Hence, $Y$ is a martingale relative to the filtration ${\bf F}^Y$.
L\'evy's theorem (Karatzas and Shreve,
1988, p.~157) therefore implies that the processes $(Y,{\bf F}^Y)$ and
$(Z,{\bf F}^Z)$ are equivalent in law.  Successively applying the law of
iterated expectations, the fact that $Y(1)=S(1)$ a.s., 
and this equivalence in law gives
\begin{eqnarray*}
E[\tilde{v}|{\cal F}^Y(t)]&=&E[f(S(1))|{\cal F}^Y(t)]\\
&=&E[f(Y(1))|{\cal F}^Y(t)]\\
&=&\int_{-\infty}^{\infty}f(y)\pi(t,Y(t),y)\,dy,
\end{eqnarray*}
which completes the proof.

\end{proof}

\begin{lemma}\label{lemma4}
Given the pricing rule (\ref{P}), the trading strategy (\ref{theta}) is
optimal.
\end{lemma}

\begin{proof}
Let $\theta$ denote a generic trading strategy for the informed trader. 
The proof depends on establishing a sharp upper bound on the expected
wealth of the informed trader.  Define
$$j(s,y)=\int_y^s(f(s)-f(x))\,dx.$$
Let $\phi (t,s,\cdot)$ denote the density function for the normal
distribution with mean $s$ and variance $\int _t^1
\sigma^2_s(u)\,du$.  For $t<1$, define
\begin{equation}\label{J}
J(t,s,y)=\int_{-\infty}^{\infty}
\int _{-\infty}^{\infty}j(s',y')\phi (t,s,s')\pi (t,y,y')
\,ds'\,dy'.
\end{equation}
Set $J(1,s,y)=j(s,y)$.  The function $J$ is continuous at $t=1$ 
(Karatzas and Shreve, 1988, p.~255).

In the following, we omit writing the
arguments of the various functions and use subscripts on $J$
to denote partial derivatives. 
Obviously, 
\begin{equation}\label{maximum}
(\forall \; y)\;\;J(1,s,y)>J(1,s,s)=0, 
\end{equation}
with equality holding if and only if $y=s$.  Also, one can
differentiate (\ref{J}) to obtain\footnote{See 
Karatzas and Shreve (1988, p.~254).  Note that their regularity
condition (4.3.3), which justifies differentiating (\ref{J}) through
the integral operator, follows from $Ef(Z(1))^2=Ef(S(1))^2=E \tilde{
v}^2 < \infty$ and the bound $j(s,y)\leq (s-y)(f(s)-
f(y))$.\label{regularity}}
\begin{equation}\label{pde}
J_t+\frac{1}{2}\sigma_s^2J_{ss}
+\frac{1}{2}\sigma_z^2J_{yy}=0.
\end{equation}
Applying It\^o's Lemma
and making use of equation (\ref{pde}) we find that
\begin{eqnarray*} 
\lefteqn{J(1,S(1),Y(1))-J(0,S(0),0)= }\\
& & \int_0^1J_s\sigma_s\,dW_s
+\int_0^1J_y\theta\,dt+\int_0^1J_y\sigma_z\,dW_z.
\end{eqnarray*} 

Upon applying the expectation operator, the two stochastic integrals
against the Wiener processes vanish,\footnote{As explained in
footnote~\ref{regularity}, we can calculate $J_y$
and $J_s$ by differentiating through the integral operator in
(\ref{J}).  The conditions $E \int
_0^1 J_y^2\,dt < \infty$ and $E \int _0^1 J_s^2\,dt < \infty$ follow
from the bound on $j$ in footnote~\ref{regularity} and the
constraint (\ref{double}), and these conditions imply that the
stochastic integrals are martingales.} and this equation becomes
$$E\left[-\int_0^1J_y\theta\,dt|{\cal F}(0)\right]=J(0,S(0),0)
-E \left[ J(1,S(1),Y(1))|{\cal F}(0) \right] .$$
A direct calculation shows that
$$-J_y= E[\tilde{v}|{\cal F}(t)]-P.$$
By iterated 
expectations, we can write the informed trader's optimization problem as
\begin{equation}\label{max}
\max_{\theta} \; E\left[\int_0^1\{E[\tilde{v}|{\cal F}(u)]-
P(u,Y(u))\} \theta(u)\,du\right].
\end{equation}
Therefore, the objective function takes the
value
$$J(0,S(0),0)-E[J(1,S(1),Y(1))|{\cal F}(0)] \,\leq\, J(0,S(0),0),$$
with, from (\ref{maximum}),
 equality obtaining if and only if $S(1)=Y(1)$ a.s.  Since,
by Lemma~\ref{lemma1}, this
equality holds for the strategy (\ref{theta}), that strategy
is optimal.

\end{proof}

Equation (\ref{dP}) follows from It\^o's Lemma, noting that
the $dt$ terms cancel (the explanation is the same as in
footnote~\ref{regularity}). 
We can write $P(t,y)$ as
$$\int _{-\infty}^{\infty}f(y+u)\,\pi (t,0,u)\,du.$$
The assumptions regarding the derivative of $f$ guarantee that we can
differentiate
this with respect to $y$ under the integral operator.
This gives
$$\lambda(t,y)=\int _{-\infty}^{\infty}f'(y+u)\pi(t,0,u)\,du.$$
It follows that $\lambda(t,Z(t))=E[f'(Z(1))|{\cal F}^Z(t)]$. 
Given the equality of the laws of
$(Z,{\bf F}^Z)$ and $(Y,{\bf F}^Y)$ (see the proof of Lemma~\ref{lemma3}), this
implies $\lambda(t,Y(t))$ is the martingale $E[f'(Y(1))|{\cal F} ^Y(t)]$.

\vfil\eject
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\vskip .5cm\noindent
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