Dynamic Noisy Rational Expectations Equilibrium with
Information Production and Beliefs-Based Speculation
Jerome Detemple and Marcel Rindisbacher˚
November 19, 2015
Abstract
A continuous time economy with information production and belief-based speculation is studied. All agents, informed, uninformed and speculators, have constant absolute risk aversion.
Equilibrium is in closed form, except for a coefficient satisfying a Riccati equation. Costly
information production generates asynchronous private and public information flows. Private information dissemination reduces price volatility. Intertemporal hedging amplifies this decrease.
Asynchrony between information flows increases volatility over time. A decomposition of exante and interim utilities, highlighting the sources of welfare, is obtained. Conditions for Pareto
efficiency of equilibrium are derived. Speculation is shown to be socially beneficial.
JEL Classification: JEL Classification : G14
Keywords: Intertemporal noisy rational expectations equilibria, costly information production,
asynchronous information flows, competition, speculation, beliefs, constant absolute risk aversion,
hedging, volatility, market price of risk, welfare, Pareto optimality, heterogeneous risk tolerances,
differential information, dividend surprises.
˚
Graduate School of Management, Boston University, 595 Commonwealth Ave Boston, MA 02215. Jerome Detemple: [email protected], Marcel Rindibacher : [email protected]
1
Introduction
Information and speculation are fundamental motives underlying securities trading. Individuals
with superior information have different assessments of returns, therefore find it beneficial to trade.
Individuals with different beliefs resulting from fundamental differences in views about the future,
also have their own assessments of returns. They find it optimal to speculate on the difference
between their beliefs and those of others. These motives for trade have long been sources of interest
as well as concerns. Regulatory limits on informed trading have been enacted in a variety of countries
to curb the rents extracted by certain types of informed agents. Current debates also question the
role of speculation in the recent financial crisis and regulatory proposals seeking to limit such activity
have been circulated. This paper seeks to shed light on these issues in a setting with endogenous
information flows. It examines the effects of information and speculation on the dynamic properties
of financial market equilibria. In particular, it attempts to identify the welfare benefits and costs
associated with regulations banning information-based trading or speculation.
The model under consideration has two key features, costly information production infrastructure
and speculation based on noise trading. Costly information infrastructure implies that information
arrives at different frequencies depending on its nature. Public information is costless. Its arrival rate
is therefore dictated by the frequency of news releases by the exogenous underlying source. Private
information, in contrast, is under the control of the acquirer. It is usually difficult to extract and does
not update automatically. It requires meeting the managers of a company or poring over documents
and data to extract a few bits of useful information regarding future prospects. The costly nature
of the information gathering technology implies that the rate of arrival is dictated by decisions of
the acquirer. This paper captures this basic asymmetry between public and private information
flows. It shows, in particular, that the endogenous asynchrony in news arrivals has profound effects
on the dynamic properties of equilibria. It also reveals that the information infrastructure plays an
important role in shaping the welfare properties of equilibria.
Speculation is a powerful motive underlying a fraction of trades in financial markets. Beliefsfueled speculation has long been recognized as an important ingredient for market models. This
paper develops a model where some individuals speculate by trading on noise (Black (1986)).1 Such
1
“Noise trading is trading on noise as if it were information. People who trade on noise are willing to trade even
though from an objective point of view they would be better off not trading. Perhaps they think the noise they are
1
individuals have the same conditional beliefs as informed agents, but instead of conditioning on
factual private information, they base their views on irrelevant noise. In other respects, they behave
rationally and maximize preferences by choosing the best allocation among the assets available. This
noise trading model enables us to endogeneize the noise trading demand and conduct a meaningful
welfare analysis. Speculative behavior is found to shape the informational properties of equilibrium
and the incentives to collect private information. It also plays a critical role for the welfare of society.
Our study is based on a continuous time setting in which the firm pays a liquidating dividend at
the terminal date and information flows are asynchronous due to costly acquisition. Public information, regarding the fundamental underlying the dividend, arrives continuously. Private information,
about the terminal dividend payment, arrives at discrete dates determined by the endogenous acquisition technology. In the base model, it arrives once, at the initial date, which can be rationalized
as the endogenous outcome of the infrastructure selection process. This asynchrony is the source
of multiple effects. When private information disseminates, expectations become less responsive to
public news, which lowers volatility. The subsequent frequent arrival of public information exerts
conflicting pressures on the local value of private information, the market price of risk and the
incentives of investors. A prominent effect is that it reduces the usefulness of the private information collected by the informed at the outset and of the endogenous information extracted from
equilibrium by the uninformed. This basic mechanism puts upward pressure on volatility as time
passes. As private information becomes more stale, the sensitivity of the stock price to public news
increases, leading to an increase in volatility. This effect is the sole effect when investors are myopic.
Investors with CARA utility, however, care about fluctuations in the opportunity set and therefore
hedge intertemporally (Merton (1971)). Hedging tames the response of optimal portfolios to shocks,
thereby reducing the sensitivity of the equilibrium price to shocks. It is therefore the source of a
further reduction in the level of volatility, uniformly over time.
The dissemination of private information has two effects on the welfare of agents. The first
effect is due to the immediate change in the price of the stock (price impact), the second is due
to the gains from trade associated with the improved information sets of agents (trading impact).
Asynchrony has multiple effects on these components. One set of effects arises through volatility,
which influences both the price impact and the trading impact. Another effect arises through the
trading on is information. Or perhaps they just like to trade.” Black (1986).
2
hedging demand of an agent, distinct from the effect of hedging on volatility. This hedging effect is
an important determinant of the trading impact.
In the absence of hedging, welfare improves uniformly across agents upon dissemination of private
information, when risk tolerance is either sufficiently low or sufficiently high. The immediate decrease
in volatility is accompanied by an increase in the stock price which raises the value of the initial
allocation of shares, leading to a positive price impact. At the same time, it reduces the market price
of risk, therefore the size of stock holdings and the resulting gains from trade. The sum of these
two effects is positive. The immediate increase in informational efficiency works in the opposite
direction as far as gains from trade are concerned. It fine-tunes the risk taking behavior of an agent,
leading to an increase in gains from trade. Volatility effects are tamed by (inversely related to)
risk tolerance. Trading effects are enhanced by (positively related to) risk tolerance. When risk
tolerance is low, volatility effects dominate, leading to a welfare improvement. At the other end of
the spectrum, when risk tolerance is high, the informational efficiently gains dominate, also leading
to a welfare gain. In both cases, these universal benefits associated with the dissemination of private
information offset the informational disadvantage of uninformed agents.
Hedging affects volatility as well as gains from trade. As explained, it further decreases volatility, hence magnifies the welfare improvements for low risk tolerance. But, it also curbs the agents’
risk-taking behavior, hence reduces the gains from trade associated with the dissemination of information. Nevertheless, when risk tolerance is high, the hedging motive weakens so that the previous
conclusions remain valid. Informational efficiency gains dominate, leading to a welfare improvement.
In these instances, it is Pareto optimal to permit trading based on private information.
Speculation plays a critical role for the welfare of market participants. In its absence, incentives
to collect information vanish. An equilibrium with homogeneous information flow, determined by
the fundamental, then prevails. In this equilibrium, volatility is driven by the volatility of the
fundamental, hence exceeds the volatility in the NREE. Information is also less precise than in the
NREE, worsening investment decisions. Both effects are sources of welfare reductions. Under the
conditions for Pareto optimality, prohibiting speculation leads to a Pareto dominated equilibrium.
When the skilled (informed) investor selects the information gathering infrastructure and the
cost is an increasing function of the number of signals produced, it is optimal to choose a finite
sampling frequency under mild conditions. Two opposite forces determine the outcome. On the
3
one hand, an increase in the number of signals produced increases the precision of the information
collected, which improves decision-making and welfare. On the other hand, the cost increases. As
long as the asymptotic precision of the private information is finite and the marginal cost of sampling
is bounded away from zero, the second effect will eventually dominate. In this instance, public and
private informations flow at different frequencies. Asynchrony ensures that the fundamental effects
and trade-offs documented in the base model will arise. Moreover, at private information arrival
times, the stock volatility experiences downward jumps. This phenomenon reinforces the stabilizing
role of private information documented in the base model with an endogenously single private signal.
Aside from the questions discussed above, the model also has practical implications for the
fund management industry. Academic debates regarding the skills of fund managers have figured
prominently in the literature.2 Evidence seems to suggest that a number of funds lack skill. Our
model of speculative noise traders captures one possible view of unskilled management, namely
the implicit suggestion of knowledge and skill through the promotion of sophisticated investment
strategies, but the actual lack of true information extraction skill and the absence of superior
information in the implementation of these strategies. Speculative noise traders, in our setting,
behave exactly as truly informed (skilled) agents because they seek to be identified as skilled, but
they operate on the basis of noise. Our model shows that this behavior serves useful purposes. The
existence of an active unskilled sector effectively clouds the price system, thus provides incentives for
information collection. The information conveyed by the price system furthermore allows uninformed
(retail) investors to improve their decision-making, which is a source of welfare.
The paper also contributes on the methodological front. The construction of equilibrium is
based on the private information price of risk (PIPR).3 The approach can be summarized in three
steps. The first determines the structure of the PIPR and the candidate equilibrium information
flows. This step can be carried out without knowledge of the hedging demands and the stock price.
The second derives a system of equations for the hedging demands and the stock price, conditional
on the information flows identified. The final step solves the system of equations and verifies the
informational content of equilibrium. This step-by-step approach is constructive in nature. It relies
on the fact that the PIPR is the ultimate source of all effects in the model.
2
See, for instance, Jensen (1968), Ferson and Schadt (1996) and Jagannathan, Malakhov and Novikov (2010).
The notion of PIPR is introduced in Detemple and Rindisbacher (2013) in the context of a portfolio selection
problem with private information.
3
4
1.1
Related Literature
The paper relates to various strands of literature on information and speculation.
Classical studies pertaining to informational efficiency are based on static models. Seminal
articles, identifying the determinants of efficiency in competitive markets, are those of Grossman
(1976, 1978) and Grossman and Stiglitz (1980). They demonstrate the possibility, as well as the
limits, of informationally efficient markets. Non-competitive behavior is examined by Hellwig (1980),
Kyle (1989) and Leland (1992). The first study argues that informed investors who are aware of
their price impact should not behave competitively. It resolves this problem, by showing that agents
can no longer affect the price, in the limit competitive equilibrium, as the number of informed
investors becomes large. The second considers informed investors who explicitly account for the
impact of their demands on the equilibrium price. It shows that imperfect competition reduces
the informational content of prices. The third focuses more specifically on insider trading and
properties of equilibrium in a static model with production and monopolistic insider. It finds that
private information increases the average stock price, decreases the return’s expectation and variance
for the uninformed, reduces the liquidity of the market and can increase or decrease welfare.
Dynamic models with asymmetric hierarchical information and competitive behavior were pioneered by Wang (1993, 1994). In these settings, the stock is an infinitely-lived asset that pays
dividends continuously/periodically through time. Informed investors observe the state variable
driving the expected future dividend. Uninformed investors do not, but they learn through dividends and prices. Noise trading injects supply stochasticity and prevents full revelation. Wang
(1993) derives a competitive noisy rational expectations equilibrium (NREE). This equilibrium is
stationary as the coefficients of the price process are constant. Asymmetric information is shown
to increase the stock’s long run risk premium. It can also increase the price volatility and enhance
negative serial correlation. Asymmetric information can therefore have a destabilizing effect. Wang
(1994) focuses on issues pertaining to the volume of trade in a similar setting. The article highlights
the relation between volume and price changes. Further insights are provided by He and Wang
(1995) in a model with terminal dividend, differential private information signals across agents,
hence non-hierarchical information flows, and a common frequency of private and public information arrivals as well as trades. They construct a competitive NREE and study the relation between
5
information flow and trading volume. The common frequency implies that there are no dynamic
effects between arrival times of private information signals.4,5
Albuquerque and Miao (2014) extend Wang (1994) by allowing for private advance information
about future dividends. They also allow for a private investment opportunity. Time is discrete and
advance information pertains to the temporary component of the dividend paid at the next date.
Agents derive utility from next period wealth, ignoring future consumption. They solve for the
stationary equilibrium, which is obtained up to a system of non-linear equations. The paper shows
that “good” advance information increases the stock price and the risk premium. It also shows that
informed (uninformed) investors behave as trend chasers (contrarians).
Brennan and Cao (1996) develop the only model with (exogenous) asynchronous information
flows. There is a private signal at the initial date followed by a sequence of public signals at
subsequent dates. Information arrival dates are also trading dates. Agents are assumed to behave
myopically. In other respects, their setting is similar to He and Wang (1995) They examine the
impact of an increase in the number of trading dates and the effects of financial innovation.
The present model builds on this literature. The main economic differences are twofold, (i) the
private information infrastructure is endogenous and (ii) noise trading is motivated by speculation.
Costly information infrastructure implies that information flows endogenously arrive at different
frequencies. Public information arrives continuously, whereas private information materializes infrequently. The asynchrony in information flows is the source of novel effects documented. Speculation
driven by beliefs differences enables us to endogeneize the demand of noise traders and conduct a
comprehensive welfare analysis. It opens up the possibility of Pareto ranking equilibria. The specific
beliefs model developed reflects Black’s (1986) idea that some investors trade on noise.
Other differences pertain to the scope of the analysis or the nature of the results. In that
regard, it could first be noted that the analysis is not restricted to stationary equilibria. Second,
equilibrium is in closed form, with the exception of a single coefficient that solves a Ricatti equation.
4
Effects of imperfect competition and asymmetric information on dynamic properties of prices and liquidity are
examined in Vayanos and Wang (2012). In a three-period model, they show that asymmetric information and imperfect
competition can have opposite effects on ex-ante expected returns.
5
A vast microstructure literature also deals with non-competitive informed trading. Fundamental contributions
are in Kyle (1985) and Glosten and Milgrom (1985). In these models, risk neutral market makers extract private
information from the aggregate order flow and set the price so as to break even on average. This pricing rule does not
account for the endogenous interactions between risk, price appreciation and price level. The absence of diversification
benefits implies that trading is purely informational. Moreover, the price evolution is typically determined by the
exogenous noise trading behavior and is locally orthogonal to fundamental risk.
6
Other coefficients are explicit functions depending on the horizon. The timing effects identified are
driven by the asynchrony in information flows and the hedging motives of agents. Third, the
nature of the informational advantage of the informed differs. Relative to Wang (1993, 1994),
this advantage is finitely lived in our setting. Relative to Albuquerque and Miao (2014), it is
non-transitory. This follows, because the time between the reception of the private signal and
the resolution of the uncertainty pertaining to the dividend payment is a finite interval. This has
a critical impact on optimal behavior and pricing. Fourth, a detailed welfare analysis is carried
out. A novel decomposition of the welfare of each agent into two parts, one associated with the
value of the initial share endowment, the other with the dynamic gains from trade, is derived.
This decomposition is particularly useful to analyze the impact of regulation governing private
information usage and speculation.6 Fifth, a novel solution method introduced. The approach relies
on the construction of the PIPR in the equilibrium under consideration. The PIPR isolates the
effects of private information. Its structure can be used to formulate natural conjectures about
the informational content of the price. More significantly, it does not depend on the intertemporal
hedging behavior of agents.
Recently, Banerjee and Green (2015) develop a model where noise traders also trade on a signal
believed to be informative. Uninformed investors, unsure about whether they face an informed or
a noise trader, learn over time. Uncertainty about the identity of the counterparty generates a
nonlinear equilibrium price. The dynamic version of their model considers successive generations
of mean-variance investors, each living for two dates. They show numerically that the model can
produce expected return predictability and volatility clustering. Our framework shares the noise
traders’ specification. It differs as the identity of parties is common knowledge. It also differs in
that we consider long-lived agents with CARA utility. As shown, the implied hedging behavior has
significant effects on equilibrium properties. The endogeneity of information flows and the focus on
welfare also differ. Finally, the explicit nature of equilibria enables us to derive analytical results.
Our analysis also builds on a literature dealing with dynamic portfolio selection and costly
information acquisition. Detemple and Kihlstrom (1987) develop a model where the precision of the
information collected can be continuously selected. Huang and Liu (2007) examine a setting with two
6
The dynamic welfare results obtained extend the static analysis in Leland (1992), as they identify a new and
significant source of welfare gains. The analysis is comprehensive in that it encompasses all agents in the model.
7
types of news sources, continuous and periodic. At the initial date, the investor selects the optimal
precisions of the two types of news and the optimal frequency of the periodic source. The paper
shows that infrequent news arrival is optimal, which rationalizes the inattention puzzle (e.g., Hong,
Torous and Valkanov (2007)). Hasler (2012) and Andrei and Hasler (2014) use the continuous
acquisition framework to study optimal choices for constant relative risk aversion. They apply
expansion methods to calculate optimal policies and show that attention to news fluctuates with
economic conditions. The present paper considers a related costly information production problem,
but in an equilibrium setting with heterogeneously informed agents. It examines the interplay
between the endogenous price/volatility, the asynchrony of news and the informational efficiency
of equilibrium. It also addresses welfare questions pertaining to the use of private information for
trading and the relevance of speculation. Huang (2015) treats information acquisition in equilibrium
as well, but in the setting with myopic agents of Brennan and Cao (1996). As in Grossman and
Stiglitz (1980), he endogeneizes the initial population of informed, but not the frequency at which
private information is produced. The focus on derivatives is also different.
The notion of speculation based on differences in beliefs has been widely discussed. Foundations
appear in Working (1962) and Hirshleifer (1975). Black (1986) introduces the view that “noise
trading is trading on noise as if it were information.” Beliefs of noise traders therefore differ from
those of truly informed individuals. Beliefs-based speculation has received recent attention in the
context of financial reforms pertaining to derivatives. Posner and Weil (2012) argue for regulation,
on the ground that such an activity reduces consumption smoothing and is therefore socially harmful.
Duffie (2014) discusses various challenges to beliefs-based regulation and stresses the importance to
base policies on social welfare principles. The analysis carried out here sheds light on some of these
issues in the context of a stock market model. It shows that a ban on beliefs-based speculation could
increase price volatility and reduce informational efficiency, leading to welfare losses. It provides
explicit conditions under which stock market speculation is Pareto optimal.
Section 2 describes a benchmark model with asynchronous information flows and endogenous
acquisition at the initial date. Section 3 presents the demand functions. Section 4 studies the noisy
rational expectations equilibrium. Section 5 examines welfare properties. The optimal information
infrastructure with endogenous frequency is analyzed in Section 6. Extensions are in Section 7 and
conclusions follow. Proofs are in the appendix.
8
2
The Economy
This section describes benchmark model. The financial market is presented in Section 2.1, agents and
their information sets in Section 2.2 and candidate stock price processes in Section 2.3. Preferences
and optimal demands are in Sections 2.4 and 2.5. Equilibrium is defined in Section 2.6.
2.1
Assets and Markets
A riskless riskless asset and a risky stock are available for trade. The riskless asset is a money market
account paying interest at the rate r. In the absence of intertemporal consumption, which will be
assumed, the interest rate can be set at zero (r “ 0). The risky stock pays a liquidating dividend DT
at the terminal date T . The dividend is the terminal value of the process dDt “ µD dt ` σ D dWtD ,
where µD is a constant drift coefficient and σ D is a constant and positive volatility coefficient. W D
`
˘
D , defined on a probability space Ω, F D , P . The
is a Brownian motion process with filtration Fp¨q
process D can be viewed as a fundamental factor that eventually determines the terminal dividend.
One share of stock, perfectly divisible, is outstanding. It trades at an endogenously price S.
Trading takes place in continuous time. There are no restrictions on stock holdings or borrowing.
2.2
Agents, Noise and Information Signal
Three groups of investors operate in the financial market, informed, uninformed and noise traders.
The respective fractions of the three groups in the population are ω i , ω u and ω n , with ω i `ω u `ω n “
1. Each group is treated as a homogeneous entity with a representative individual.
The (representative) informed investor is a skilled individual, able to extract at cost k information
about the future stock payoff DT . In the benchmark model, information extraction is carried out
´ ` ˘ ¯
2
. Skill
at the initial date t “ 0 and generates the noisy signal G “ DT ` ζ, where ζ „ N 0, σ ζ
` ζ ˘2
` ζ ˘´2
increases, precision falls and
of the signal. When σ
is measured by the precision vζ “ σ
` ˘2
the informational content of the signal decreases. Thus, skill decreases. In the limit σ ζ Ñ 8, the
signal becomes pure noise and skill vanishes. The “informed” investor effectively becomes unskilled
(uninformed). The optimality of an information infrastructure producing a single signal at t “ 0 is
discussed in Section 6 (see Remark 21).
The uninformed investor does not have extraction ability. He/she observes prices and other
9
m be the public information filtration. The
quantities that are in the public information set. Let Fp¨q
u “ F m . The last investor trades on noise. His/her
uninformed filtration is the public filtration Fp¨q
p¨q
beliefs are similar to the beliefs of the informed, but depend on an independent random variable φ
as opposed to true private information. A precise description is provided below.
The three groups are identically endowed at the outset. The distribution of initial stock shares
`
˘
in the population is ω i , ω u , ω n .
2.3
Stock Price and Information Flows
The opportunity set of investors depends on the stock price structure. In this environment, there are
two sources of uncertainty, W D associated with fundamental information and φ with noise trading
speculation. Standard arguments can be invoked to write any candidate price process as,
dSt “ µSt dt ` σtS dWtS ,
ST “ DT .
(1)
m . It is
In this structure W S is a Brownian motion relative to the public information filtration Fp¨q
endogenous and, ultimately, relates to the underlying source of fundamental uncertainty W D . The
`
˘
m . The uninformed
coefficients µS , σ S of the price process are also endogenous and adapted to Fp¨q
observes the stock price, hence can retrieve the volatility coefficient from its quadratic variation.
` ˘´1 `
˘
The Brownian motion dWtS “ σtS
dSt ´ µSt dt is an innovation process in their filtration. The
S generated by S is in the public information flow F m . That is, F S Ď F m .
information filtration Fp¨q
p¨q
p¨q
p¨q
The information flow of the informed is augmented by the private signal G. Private information is
i ” F m _ σ pGq. As private information modifies the perception
carried by the enlarged filtration Fp¨q
p¨q
of the risk-reward trade-off, the fundamental source of risk W D is no longer Brownian motion relative
to the enlarged filtration. Instead, the translated process,
G|m
dWtG “ dWtS ´ θt
pGq dt
where
G|m
becomes a Brownian motion. The translation θt
G|m
θt
ˇ ‰
“
pGq dt ” E dWtS ˇ Fti
pGq is the private information price of risk
(PIPR), which is a function of the private signal G. Relative to private information, the stock price
¯
´
G|m
evolution is dSt “ µSt ` σtS θt pGq dt` σtS dWtG . The superior information is reflected in the
10
G|m
private information premium σtS θt
m is endogenous, the
pGq. Given that public information Fp¨q
private information premium is endogenous as well.
2.4
Informed and Uninformed Preferences
Throughout the paper, superscripts i and u are used to distinguish the informed piq from the
uninformed puq investor. Let Xtj denote the wealth of investor j at time t, j P ti, uu. Conditional
preferences have the von Neumann-Morgenstern representation,
” ´ ¯ˇ ı
´ ¯
ˇ
U j F0j “ E u XTj ˇ F0j for j P ti, uu
(2)
where the utility function has constant absolute risk aversion, u pXq “ ´Γ exp p´X{Γq. The parameter Γ ą 0 is the common absolute risk tolerance coefficient. Preferences of the informed (resp.
uninformed) are conditional on private (resp. public) information. For welfare comparisons, it is
useful to consider expected utility conditional on public information at date zero, called interim
‰
“ ` ˘ˇ
expected utility. Interim expected utility is U i “ E U i F0i ˇ F0m for the informed agent and
U u “ E r U u pF0u q| F0m s “ U u pF0u q for the uninformed. The last equality holds because F0u “ F0m .
Let N j be the number of shares held. Investors maximize (2) subject to the dynamics of wealth,
$
¯
¯
´´
’
& N i µS ` σ S θ G|m pGq dt ` σ S dW G
for j “ i
t
t
t
t
t
t
dXtj “
`
˘
’
% Ntu µSt dt ` σtS dWtS
for j “ u
(3)
j
and the informational constraint that N j be adapted to Fp¨q
for j P ti, uu.
2.5
Noise Trading: Beliefs and Preferences
The decisions of the informed are based on the conditional distribution of states given the realization
of the private signal,
P i pdωq ” P p dω| G “ xq|x“G “
P p G P dx| ωq
P pdωq .
P pG P dxq|x“G
Informed beliefs are based on the conditional Wiener measure P p dω| G “ xq. As the unconditional
and conditional distributions of the signal, P pG P dxq and P p G P dx| dωq, as well as the distribution
11
of states P pdωq are common knowledge, the conditional measure P p dω| G “ xq is known.
The noise trader is aware of the availability of private information and can calculate the conditional beliefs P p dω| G “ xq, but does not observe the realization of the private signal. Instead,
he/she relies on a forecast of this realization, x “ φ, and evaluates conditional beliefs at φ. The
noise trader’s beliefs are,
n
P pdωq ” P pdω|G “ xq|x“φ “
P p G P dx| ωq
“ exp
P pG P dxq
ˆż T
0
ˆ
P p G P dx| ωq
P pG P dxq
G|m
θt pxq dWtS
1
´
2
żT
0
˙
|x“φ
P pdωq
G|m
θt pxq2 dt
˙
.
The distribution of the forecast φ can be arbitrary. In the sequel, φ is assumed to be independent and
` ˘2
normally distributed with mean µφ and standard deviation σ φ . If µφ “ E rGs and σ φ “ V AR rGs,
the forecast is an unbiased and identically distributed estimate of G. Irrespective of the distributional
structure, φ is noise.
Ultimately, the noise trader is an agent with bounded rationality who attempts to replicate the
behavior of the informed, but without the benefit of observing the private signal. The noise trader
and the informed share conditional beliefs, therefore have the same demand structures. The realized
demands nevertheless differ, because one is truly informed, while the other is not. The noise trader,
effectively, speculates by trading on noise (Black (1986)).
The noise trader’s conditional preferences are U n pφq “ E n r u pXTn q| F0m s “ E0 r u pXTn q| G “ xs|x“φ ,
where the expectation E n r ¨| F0m s is with respect to the beliefs P n evaluated at a given realization of the forecast φ and the utility function is CARA with absolute risk tolerance parame¯
´
G|m
ter Γ. Under the beliefs P n , the stock price evolves as dSt “ µSt ` σtS θt pxq dt `σtS dWtx ,
ş¨ G|m
m -Brownian motion under the conditional Wiener measure
where W¨x “ W¨D ´ 0 θt pxq dt is a Fp¨q
G|m
P pdω|G “ xq, evaluated at x “ φ. The associated stock price of risk is θtφ ” θtm` θt
pxq. Interim
expected utility is U n “ E r U n pφq| F0m s “ E r U n pGq Lφ,G pG|F0m q| F0m s, where,
Lφ,G p x| F0m q ”
P p φ P dx| F0m q
P p G P dx| F0m q
is a density measuring the beliefs distortion relative to the informed investor. If µφ “ E rGs and
12
`
σφ
˘2
“ V AR rGs, then Lφ,G p x| F0m q “ 1 and the beliefs distortion vanishes.
m -progressively measurable numGiven x “ φ, the noise trader maximizes U n pφq over the Fp¨q
ber of shares N n , subject to the dynamic budget constraint under conditional beliefs, dXtn “
¯
¯
´´
G|m
Ntn µSt ` σtS θt pxq dt ` σtS dWtx . Note that even in the absence of a beliefs distortion, i.e.,
when Lφ,G p x| F0m q “ 1, the noise trader exhibits bounded rationality as the optimal policy is cho-
m , given a fixed realization φ, rather than adapted to
sen adapted to the public information flow Fp¨q
Ž
m
σ pφq. Full rationality fails, as the joint information conveyed by the
the enlarged filtration Fp¨q
m is ignored.7
realized forecast φ and the public information flow Fp¨q
2.6
Equilibrium
A rational expectations equilibrium (REE) for the economy under consideration is a triplet of
`
˘
demands N u , N i , N n and a price process dSt “ µSt dt `σtS dWtS , ST “ DT , such that (i) Individual
rationality: N j is optimal for agent j P tu, i, nu, and (ii) Market clearing: ω u N u `ω i N i `ω n N n “ 1.
u Ă Fi .
The REE is noisy (NREE) if the informed and uninformed filtrations differ, Fp¨q
p¨q
3
Optimal Stock Demands and Residual Demand
The next two propositions describe the stock demands of the various agents in the economy.
Proposition 1 The optimal number of shares held by the uninformed and informed investors are,
Ntu “ Γ
θtm ` hut σ D
σtS
G|m
and
Nti “ Γ
θtm ` θt
pGq ` hit pGq σ D
σtS
where θ m is the price of risk for the uninformed and, with Et r¨s ” Et r ¨| Ftm s,
hut
hit pxq
7
“
hut
1
´
2
żT
t
1
“´
2
żT
t
ı
”
m
pθvm q2 dv
BDt Et ξt,v
´
¯ı
”
G|m
m
θvG|m pxq2 ` 2θvm θvG|m pxq dv ” hut ` ht pxq .
BDt Et ξt,v
The beliefs distortion and the absence of Bayesian updating is similar to the bounded rational behavior studied in
Dumas, Kurshev and Uppal (2009). Alternatively, under the assumption of uncertainty about the counterparty and
learning, the uncertainty structure becomes similar to the one in Banerjee and Green (2015).
13
The terms Γhut σ D {σtS , Γhit σ D {σtS are the intertemporal hedging demands for the uninformed and
informed. The informed holds more shares than the uninformed if and only if the private information
¯
´
G|m
G|m
premium exceeds the adjusted difference in hedging terms, σtS θt pGq ` ht pGq σ D ą 0.
An investor with CARA utility seeks to hedge stochastic fluctuations in the opportunity set
(Merton (1971)). The optimal stock demand has a mean-variance component as well as a dynamic
hedging component. The demands in Proposition 1 have this basic structure in common.
The fundamental difference between the two investors resides in their evaluation of the expected
stock return. The informed evaluates the return on the basis of private information as well as public
information. The resulting expected return has two components. The first one, µSt “ σtS,D θtm , is the
G|m
expected return based on public information. The second one, σtS θt
pGq, is the additional premium
G|m
calculated on the basis of private information. This premium is affine in the PIPR θt
pGq, i.e., the
private information price of risk (Detemple and Rindisbacher (2013)). The PIPR is the incremental
price of risk assessed in light of information that is not revealed by public information sources. It
represents the private information price of risk conditional on public information. Thus, the informed
mean-variance component has a public information part, Γθtm{σtS , and a private information part,
G|m
Γθt
pGq {σtS . The uninformed mean-variance demand has only a public information part, Γθtm {σtS .
The difference in assessed expected returns induces a difference in hedging behavior. The informed agent hedges stochastic fluctuations in the market price of risk as well as in the PIPR. His/her
G|m
total hedging demand can be decomposed by writing hit pGq “ hut ` ht
G|m
pGq, where hut , ht
pGq
capture the two separate motives. The uninformed hedging demand stems entirely from the stochastic behavior of the price of risk. It is also of interest to note that the hedging demand depends on
the information set of the agent considered. For the informed, hedges are conditional on the private
information filtration, hence parametrized by the private signal. For the uninformed, the hedge
depends on the public filtration.
The hedging demand formulas are valid for any given, but arbitrary, diffusion opportunity set.
The specific functional form depends on the structure of the latter.
Proposition 2 The optimal number of shares held by the noise trader is,
G|m
Ntn “ Γ
θtm ` θt
G|m
pφq ` hnt pφq σ D
θtm ` θt
“
Γ
σtS
14
pφq ` hit pφq σ D
σtS
(4)
G|m
where θ m is the uninformed price of risk, θt
pφq is a speculative premium/discount reflecting the
departure from rationality and Γhnt σ D {σtS is the intertemporal hedging demand. The hedging demand
of the noise trader has the same functional form as that of the informed, but evaluated at φ instead of
G. The noise trader holds more (resp. less) shares than the uninformed if and only if the speculative
¯
´
G|m
G|m
premium exceeds the adjusted difference in hedging terms, σtS θt pφq ` ht pφq σ D ą 0.
The optimal noise trading demand has a mean-variance component and a hedging component.
The mean-variance component has two parts. The first part, Γθtm {σtS , is the usual mean-variance
demand of an uninformed rational agent. It reflects a demand based on public information. The secG|m
ond part, Γθt
pφq {σtS , is a speculative demand associated with an informational signal consisting
of pure noise. The hedging component can also be split in two parts by writing hnt pφq “ hn,mpr
pφq `
t
hn,pipr
pφq, reflecting hedging motives stemming from stochastic fluctuations in the MPR and in the
t
PIPR. Because the noise trader and the informed seek to hedge the same objects, the structures
of their hedges are the same. The only difference is the conditioning factor, consisting of the true
signal G for the informed and the independent random variable φ for the noise trader. Thus,
G|m
pφq “ ht
pφq “ hut and hn,pipr
hnt pφq “ hit pφq, hn,mpr
t
t
pφq. In the end, the noise trader demand
mimics the demand of the informed. It effectively corresponds to the demand of an investor with
randomized beliefs, i.e., an unskilled active investor.
Remark 3 The combined demand of the informed and noise trader, i.e., the residual demand, is,
Nt ” ω i Nti ` ω n Ntn “ Γ
¯
´
¯
´
G|m
G|m
ωθtm ` ω i θt pGq ` hit pGq σ D ` ω n θt pφq ` hnt pφq σ D
σtS
where ω “ ω i ` ω n . The residual demand is an affine function of the weighted average price of
`
˘
G|m
G|m
risk (WAPR) Θt G, φ; ω i , ω n ” ω i θt pGq ` ω n θt pφq and of the weighted average normalized
`
˘
hedge (WANH) ht G, φ; ω i , ω n ” ω i hit pGq ` ω n hnt pφq. If the PIPR and hedges are also affine
G|m
functions, the residual demand depends on Θt pZ; ωq “ θt
pZ; ωq and ht pZ; ωq “ hit pZ; ωq, which
are functions of the signal Z ” ω i G ` ω n φ and the combined population weight ω “ ω i ` ω n .
15
4
Competitive Noisy Rational Expectations Equilibrium
The competitive NREE is described in Section 4.1. Properties of the PIPR and the WAPR are
examined in Section 4.2. Price and return properties are discussed in Section 4.3.
4.1
Competitive Equilibrium Structure
In order to present the main result, define the combined share of the informed and the noise trader
ω “ ω i ` ω n and the functions of time,
α̂ ptq ” α ptq ` αh ptq ,
β̂ ptq ” β ptq ` β h ptq ,
γ̂ ptq ” γ ptq ` β h ptq
(5)
`
˘
where αh , β h , γ h are associated with the aggregate hedging behavior of the agents and defined in
the Appendix, and
α ptq “
1 ´ κt ω D
σ ,
H ptq
β ptq “ ´ω
1 ´ κt ω i D
σ ,
H ptq
`
˘
1 ´ κt ω i µD pT ´ tq ´ ω n κt µφ D
σ ,
γ ptq “ ´ω
H ptq
´ ¯2
` ˘2
H ptq “ σ D pT ´ tq ` σ ζ ,
κt “
ω i H ptq
M ptq
(6)
` ˘2
ω i σ D ps ´ tq
λ pt, sq “
, s P rt, T s
M ptq
´ ¯2
` ˘2
M ptq “ ω i H ptq ` pω n q2 σ φ .
(7)
(8)
`
˘
The function H ptq “ V ar G|FtD is the conditional variance of the private signal G given fundamen`
˘
tal information at t. The function M ptq “ V ar Z|FtD is the conditional variance of an endogenous
signal Z ” ω i G ` ω n φ given fundamental information at t. The coefficients κt “
λ pt, sq “
COV pDs ,Z|FtD q
V ARpZ|FtD q
COV pG,Z|FtD q
V ARpZ|FtD q
and
are regression coefficients. The next proposition presents the NREE.
Proposition 4 A competitive NREE exists. The equilibrium stock price is,
St “ Â ptq Z ` B̂ ptq Dt ` F̂ ptq
where
Z “ ωiG ` ωnφ
˙ ˆ
˙
ş
D T h
H pT q ω M pT q 1´ω
, B h ptq “ eσ t β pvqdv
B̂ ptq “ B ptq B ptq , B ptq “
H ptq
M ptq
ˆż T
´
¯ ˙
D
Âptq “ λ pt, T q ` σ
B̂ psq α̂ psq ` β̂psqλpt, sq ds
h
ˆ
t
16
(9)
(10)
(11)
`
σD
F̂ ptq “ B̂ ptq µ pT ´ tq ´
Γ
D
˘2 ż T
t
B̂ psq2 ds ` σ D
Iˆ ptq “ λ pt, T q ` σ D
żT
t
żT
t
B̂ psq γ̂ psq ds ´ ω n Iˆ ptq µφ
B̂ psq β̂ psq λ pt, sq ds
(12)
(13)
´
¯
and α̂, β̂, γ̂, λ as in (5)-(8). The coefficients of the equilibrium stock price process (1) are,
µSt
`
σS
“ t
Γ
G|m
where Θt pZ; ωq ” ω i θt
˘2
`
˘
D
´ σtS Θt pZq ` hm
,
t pZq σ
σtS “ B̂ ptq σ D
(14)
Θt pZ; ωq “ α ptq Z ` β ptq Dt ` γ ptq
(15)
h
h
h
hm
t pZ; ωq “ α ptq Z ` β ptq Dt ` γ ptq
(16)
G|m
pGq ` ω n θt
i
u
pφq is the endogenous WAPR and hm
t pZ; ωq ” ht ` ht pZ; ωq
is the endogenous aggregate hedging. The innovation in the uninformed filtration is dWtS “ dWtD ´
D|m
θt
m “ F D,Z -Brownian motion, where,
dt, an Fp¨q
p¨q
D|m
θt
“
‰
¯
`
˘
E dWtD |Ftm
ωi σD ´
Z ´ ω i Dt ` µD pT ´ tq ´ ω n µφ .
“
“
dt
M ptq
(17)
The competitive equilibrium price in (9) is an affine function of the fundamental D and of the
random variable Z. This random variable is a noisy translation of the private information signal G.
It provides information about the terminal dividend, but is less informative than the private signal.
m . It follows that Z
Both the price S and the fundamental D are in the public information set Fp¨q
m and F D,Z Ď F D,S Ď F m . Conversely, the pair pD, Zq
is publicly observed as well. Thus, Z P Fp¨q
p¨q
p¨q
p¨q
S Ď F D,Z . Thus, F D,S “ F D,Z Ď F m .
reveals the price S, i.e., Fp¨q
p¨q
p¨q
p¨q
p¨q
In equilibrium, the uninformed extracts the noisy signal Z from the pair pD, Sq. The uninformed
also observes the residual aggregate demand function ω i Nti ` ω n Ntn , described in Remark 3. At
equilibrium, the residual demand is also affine in D and Z. If therefore fails to reveal any information
beyond what is already contained in pD, Sq. In the end, the equilibrium public information set
D,S
D,Z
m . The equilibrium uninformed filtration
consists of the pair pD, Zq. That is, Fp¨q
“ Fp¨q
“ Fp¨q
u “ F m “ F D,S “ F D,Z . The equilibrium informed filtration is strictly more informative,
is Fp¨q
p¨q
p¨q
p¨q
i “ F m _ σ pGq Ą F m “ F u . The equilibrium is a noisy rational expectations equilibrium.
Fp¨q
p¨q
p¨q
p¨q
17
The specific impact of intertemporal hedging on equilibrium can be identified by comparing to
`
˘
the solution above with that for αh , β h , γ h “ p0, 0, 0q. This benchmark case corresponds to a model
in which all agents behave myopically and have traditional mean-variance demands. As revealed
by Proposition 4, intertemporal hedging modifies the coefficients of the equilibrium relationships,
not their overall structure. A surprising aspect is that the information content of equilibrium does
not change. Indeed, the endogenous signal Z “ ω i G ` ω n φ is the same as in the benchmark pure
mean-variance setting. This follows, because neither the PIPR, nor the WAPR, are affected by the
hedging activities of agents. Both only depend on the public and private information structure.
Remark 5 (Limit economy with small informed) Consider the limit economy with an infinitesimal
informed population (ω i Ñ 0 and ω n Ñ 1´ω u “ ω). The limit equilibrium price is Stsi “ Âsi ptq Z si `
B̂ si ptq Dt ` F̂ si ptq, where Z si “ ω n φ and,
“
µS,si
t
¯2
´
σtS,si
Γ
´
¯
` si ˘
m,si
D
,
´ σtS,si Θsi
Z
;
ω
`
h
pZ;
ωq
σ
t
t
` si ˘
si
si
si
si
Θsi
t Z ; ω “ α ptq Z ` β ptq Dt ` γ ptq ,
σtS,si “ B̂ si ptq σ D
hm,si
pZ; ωq “ αh,si ptq Z si ` β h,si ptq Dt ` γ h,si ptq
t
` si ˘
G|m,si
with coefficients defined in (41)-(52). The limit WAPR is Θsi
pφq. Innovations in
t Z ; ω “ ωθt
D|m
the uninformed filtration vanish dWtS “ dWtD because θt
Ñ 0 when ω i Ñ 0. The limit equilibrium
fails to reveal any private information. If, in addition, there is no noise trader (ω i , ω n , ω Ñ 0), the
equilibrium collapses to a no-trade equilibrium where,
Stsi,0
“ Dt `
˜
`
σD
µ ´
Γ
D
˘2 ¸
pT ´ tq ,
σtS,si,0
D
“σ ,
µS,si,0
t
`
σD
“
Γ
˘2
.
In the absence of hedging, price volatilities in the two limit economies and the economy of Proposition
4 rank as σtS ă σtS,si ă σtS,si,0 “ σ D for t ă T . Intertemporal hedging can alter this ranking.
Indeed, with a small informed, the market price of risk becomes more sensitive to the fundamental
(´β si ą ´β ą 0). Hedging amplifies this sensitivity (´β h,si ą β h ą 0). Volatility, which is
negatively related to the sensitivity of the market price of risk, decreases. Thus σtS ż σtS,si depending
¯
´
S,si
S
ă σtS,si,0 “ σ D for t ă T . As the payment date
on parameter values. In all cases, max σt , σt
approaches, the volatilities converge, limtÑT σtS “ limtÑT σtS,si “ limtÑT σtS,si,0 “ σ D . Informed
18
trading increases the informational efficiency of the market. It stabilizes the market by reducing the
stock’s exposure to fundamental shocks if the hedging effect does not dominate (i.e., B̂ ptq ă B̂ si ptq).
Remark 6 (Limit economy with small uninformed) Consider the limit economy with an infinitesimal uninformed population (ω u Ñ 0 and ω i Ñ 1 ´ ω n ). The limit equilibrium price is Stsu “
Âsu ptq Z su ` B̂ su ptq Dt ` F̂ su ptq, where Z su “ p1 ´ ω n q G ` ω n φ and,
µS,su
“
t
¯2
´
σtS,su
Γ
`
˘
m,su
su
pZ; 1q σ D ,
´ σtS,su Θsu
t pZ ; 1q ` ht
su
su
su
su
su
Θsu
t pZ ; 1q “ α ptq Z `β ptq Dt `γ ptq ,
σtS,su “ B̂ su ptq σ D
hm,su
pZ; 1q “ αh,su ptq Z su `β h,su ptq Dt `γ h,su ptq
t
with coefficients defined in (53)-(66). If, in addition, there is no noise trader (ω i Ñ 1, pω u , ω n q Ñ 0),
the equilibrium collapses to a no-trade equilibrium where,
Stsu,0 “ Âsu,0 ptq G ` B̂ su,0 ptq Dt ` F̂ su,0 ptq ,
µS,su,0
“
t
¯2
´
σtS,su,0
Γ
,
σtS,su,0 “ σtS,su “ B̂ su ptq σ D ,
Z su “ G
m,su
su
Θsu
pZ; 1q “ 0
t pZ ; 1q “ ht
´
¯
`
˘
with Âsu,0 , B̂ su,0 , F̂ su,0 as in (67)-(72). The pair D, S su,0 , in the limit economy, is fully reveal-
ing. Stock price volatilities in the three equilibria rank as σtS,su “ σtS,su,0 ă σtS ă σ D for t ă T . As
the payment date approaches, limtÑT σtS,su “ limtÑT σtS,su,0 “ limtÑT σtS “ σ D . Equilibrium prices
in economies with small uninformed (large informed) populations are less sensitive to fundamental
shocks and have lower volatility.
Remark 7 (Limit economy with small noise trader) Consider the limit economy with an infinitesimal noise trader population (ω i Ñ 1 ´ ω u and ω n Ñ 0). The limit equilibrium price is Stsn “
Âsn ptq Z sn ` B̂ sn ptq Dt ` F̂ sn ptq, where Z sn “ G and,
µS,sn
“
t
¯2
´
σtS,sn
Γ
`
`
˘ ˘
` sn i ˘
´ σtS,sn Θsn
Z; ω i σ D ,
Z ; ω ` hm,sn
t
t
`
˘
Θsn
G; ω i “ αsn ptq Z sn `β sn ptq Dt `γ sn ptq ,
t
σtS,sn “ B̂ sn ptq σ D
`
˘
Z; ω i “ αh,sn ptq Z sn `β h,sn ptq Dt `γ h,sn ptq
hm,sn
t
with coefficients defined in (73)-(83). The pair pD, S sn q, in the limit economy, is fully revealing.
19
Stock price volatilities in the three equilibria rank as σtS,sn ă σtS ă σ D for t ă T . As the payment
date approaches, limtÑT σtS,sn “ limtÑT σtS “ σ D . The equilibrium price in the economy with small
noise trader population is less sensitive to fundamental shocks and has lower volatility.
4.2
PIPR and WAPR Properties
To provide further insights about the structure of equilibrium, it is instructive to start with the
PIPR. The PIPR is the (negative of the) instantaneous volatility of the growth rate of the conditional
density of the private information signal given public information. In equilibrium, with Ftm “ FtD,Z ,
G|m
θt pGq
“ vol
ˆ
dpG
t pGq
G
pt pxq
˙
G|D,Z
¯ G ´ µG|D,Z `
´
˘ D
G ´ µt
G|D,Z
t
i
“ ´
“ ´
¯2 vol µt
¯2 1 ´ κt ω σ .
G|D,Z
G|D,Z
σt
σt
In the model under consideration, given the linearity of the endogenous signal Z revealed, the
conditional density is normal. The conditional mean alone depends on the dividend. The conditional
variance is a function of time. The PIPR therefore reduces to the volatility of the conditional mean
suitably normalized. It is affine in the private signal. As noted in Remark 3, it follows that the
˘
`
WAPR becomes Θt G, φ; ω i , ω n ” Θt pZ; ωq and that the residual demand is an affine function of
Θt pZ; ωq. The equilibrium risk premium inherits this affine structure. Moreover, the equilibrium
residual demand, being affine in Θt pZ; ωq, also reveals the signal Z “ ω i G` ω n φ.
The next corollary describes the behavior of the endogenous PIPR.
Corollary 8 The equilibrium PIPR is,
G|m
θt
G|D,Z
`
˘ D
G ´ µt
i
pGq “ ´
¯2 1 ´ κt ω σ “ α1 ptq G ` α2 ptq Z ` β0 ptq Dt ` γ0 ptq
G|D,Z
σt
α1 ptq ”
σD
,
H ptq
α2 ptq ” ´
κt σ D
ωiσD
“´
,
H ptq
M ptq
β0 ptq “
β ptq
,
ω
γ0 ptq “
γ ptq
ω
where ω “ ω i ` ω n and β ptq , γ ptq are defined in (6)-(8). The coefficients α1 ptq , α2 ptq and β ptq
are the sensitivities with respect to the private signal G, the endogenous public signal Z and the
fundamental Dt . The coefficient γ ptq is a translation factor. The following properties hold,
(i) Sensitivity to information: α1 ptq ą 0, α2 ptq ă 0, β ptq ă 0.
20
(ii) Dynamic behavior:
(ii-1)
Bα1 ptq
Bt
(ii-2)
Bγptq
Bt
ą 0,
Bα2 ptq
Bt
ă 0,
Bβptq
Bt
ă0
ą 0 if and only if H ptq ă H ` with H ` as defined in (84)-(85).
The reaction of the equilibrium PIPR to news is intuitive. Indeed, a larger private signal indicates
a greater terminal dividend, thus provides more valuable information. In contrast public information,
be it endogenous or exogenous, reduces the local value of private information.
The evolution of these sensitivities is also intuitive. The reaction to private information α1 ptq
is tamed by the unconditional variance of the signal H ptq in the denominator. Over time, the
informed observes the fundamental and updates the content of the private signal. Effectively, the
residual private information is G ´ Dt . This residual signal becomes more informative over time, as
uncertainty resolves, thereby enhancing the value of information. For the same reason, the precision
of the endogenous public signal increases. This reduces the (negative) sensitivity of the PIPR to the
endogenous signal, which decreases the value of private information. The reaction to fundamental
information reflects the same effect. Its decrease further reduces the value of information.
The WAPR is closely related to the PIPR and inherits most of its properties.
Corollary 9 The equilibrium WAPR is given by (15). The coefficients α ptq and β ptq are the
sensitivities with respect to the endogenous public signal and the fundamental information. The
coefficient γ ptq is a translation factor. The properties of pβ ptq , γ ptqq are the same as those of
pβ0 ptq , γ0 ptqq in Corollary 8. The behavior of α ptq differs in the following respects,
` ˘2
(i) Sensitivity to information: α ptq ą 0 if and only if σ φ ą sH ptq.
(ii) Dynamic behavior: α ptq increases with time if and only if κ2t ă 1{ω i ω.
The behavior of α ptq “ α1 ptq ` α2 ptq ω is more intricate because α1 ptq , α2 ptq have different,
sometimes opposite properties. The evolution of α ptq is especially noteworthy. If ω i ωκ20 ă 1, the
coefficient increases over time. If ω i ωκ20 ą 1 and ω i ωκ2T ă 1, it initially decreases, then increases. If
ω i ωκ20 ą 1 and ω i ωκ2T ą 1, it decreases throughout. The possibility of a U -shaped pattern reflects
conflicting effects on α1 ptq and α2 ptq. Under the conditions stated, the decrease in α2 ptq dominates
early on, then is overtaken by the increase in α1 ptq. An illustration is in Figure 1.
21
4.3
Price and Return Properties
Fundamental information accumulates with the passage of time, providing more precise estimates
of the next dividend payment. Information accumulation affects the properties of equilibrium. The
next corollary describes these dynamic aspects of the price and return components. For transparency,
it first presents the pure mean-variance case, then describes the incremental effects of hedging.
Corollary 10 (i) Suppose that αh ptq “ β h ptq “ γ h ptq “ 0 for all t P r0, T s (pure mean-variance
case). The stock price sensitivity to the fundamental increases over time. The volatility of the stock
price, σtS “ B ptq σ D , increases over time. The minimal and maximal volatility values are obtained
at the initial and terminal dates,
σ0S
“ B p0q σ
D
“
ˆ
H pT q
H p0q
˙ω ˆ
M pT q
M p0q
˙1´ω
σD ,
lim σtS “ B pT q σ D “ σ D .
tÑT
The stock’s price of risk µSt {σtS “ B ptq σ D {Γ ´ pα ptq Z ` β ptq Dt ` γ ptqq becomes more sensitive to
the fundamental over time (i.e., ´β ptq ą 0 increases for all t P r0, T s).
(ii) Suppose that αh ptq , β h ptq , γ h ptq are given by their equilibrium values. Relative to case (i),
the stock price is less sensitive to the fundamental (B̂ ptq ă B ptq), the stock’s price of risk is more
sensitive (β̂ ptq ą β ptq) and volatility decreases (B̂ ptq σ D ă B ptq σ D ). The coefficient β̂ ptq can
increase or decrease over time. The coefficient B̂ ptq and the volatility σtS “ B̂ ptq σ D increase over
time. Volatility converges to the volatility of the fundamental, limtÑT σtS “ σ D , and can reach a
minimal value at some intermediate time t˚ P p0, T q.
In order to understand the behavior of the price and its characteristics, it is useful to start with
the benchmark pure mean-variance case (i). At the initial date, the uninformed extracts the noisy
signal Z from the price. In the absence of a hedging motive, this information is most valuable
when there is no other source of information, i.e., at the initial date. In the early stages of the
economy, the price is heavily influenced by this initial information and, for this reason, does not
react significantly to fundamental information. Over time, fundamental information accumulates,
reducing the usefulness of the initial piece of information extracted. The impact of fundamental
information (resp. the endogenous noisy signal) on the stock price grows (resp. decreases), thereby
increasing the stock’s volatility. The same phenomenon applies to the price of risk. Its sensitivity
22
to fundamental information increases, implying that the price of risk volatility increases over time.
Intertemporal hedging, in case (ii), has intricate effects on some of these patterns. The positive
correlation between the stock’s price of risk and the fundamental induces agents to hedge so as
to reduce the sensitivities of their demands to the fundamental. The aggregate demand inherits
this property. Market clearing implies that the sensitivity of the equilibrium market price of risk is
negatively related to the sensitivity of the aggregate hedging demand,
θtm “
Bθtm
BΘt pZq Bhm
pZq D
σtS
D
´ Θt pZq ´ hm
“´
´ t
σ .
t pZq σ ùñ
Γ
BDt
BDt
BDt
The sensitivity of the MPR therefore increases. So does the sensitivity of the risk premium. The
price, inversely related to the premium, becomes less sensitive. Volatility therefore decreases relative
to the mean-variance case. Moreover, hedging incentives have a cumulative effect on volatility, which
becomes stronger as the horizon recedes. The volatility discount B h ptq “ B̂ ptq {B ptq is therefore
smaller at longer horizons (B h ptq decreases as T ´ t increases). When the time to dividend payment
is sufficiently long, volatility can become a small fraction of the volatility of the fundamental and
the volatility of the equilibrium mean-variance price. In such a scenario, private information, which
is the ultimate source of all effects, is a dominant stabilizing force. At the opposite end, when the
payment date approaches, the hedging motive weakens, to eventually vanish, and volatility behaves
as in the pure mean-variance setting. Figure 2 illustrates typical volatility patterns.
5
Welfare Analysis
This section examines the welfare properties of equilibrium when trading based on private information and/or speculation are permitted. In order to carry out a meaningful comparison of welfare
across equilibria with and without private information, expected utilities conditional on initial public information, i.e., interim expected utilities, are calculated and compared. The equilibrium with
private information is interim Pareto optimal if and only if the interim welfare of all agents improves.
23
5.1
Welfare of Informed and Uninformed Agents
The certainty-equivalent (CE) value of private information is an important component of the welfare
of the informed. It captures the utility gain from the use of private information for trading. It equals,
i
I pG|Zq “
“
„ żT
¯ ˇˇ 
´
1
G|m
G|m
m
i
E ξT
θt pGq θt pGq ` 2θt dtˇˇ F0i
2
0
„

żT
¯ ˇˇ
´
1
G|m
G|m
m
m
m
ˇ
E ξT
θt pGq θt pGq ` 2θt dtˇ F0
2
0
(18)
where the second equality follows because the informed state price density depends on the inverse of the density process of the conditional Wiener measure, i.e., ξTG “ ηTG ξTm where ηTG ”
pPT pG P dxq {P pG P dxqq´1 .8 In the NREE, the value of private information quantifies the difference between the welfare of the informed and that of the uninformed investor.
In order to compare welfare across economies with and without private information, the equilibrium public information structure matters. In the economy without private information, information is homogeneous and generated by the fundamental. In the NREE, the uninformed extracts the endogenous signal Z from equilibrium. The CE value of this signal for the uninformed
stems from the associated trading gains. It is a quadratic function of the hedge-adjusted WAPR
ϑ pZ|Dt q ” Θt pZ; ωq ` hm
t pZ; ωq, given by,
ˇ 
„ żT
„ żT

ˇ
1
1
2
2
m
m
ϑt pZ|Dt q dt “ E ξT
ϑt pZ|Dt q dtˇˇ F0u
I pZq ” E0 ξT
2
2
0
0
u
(19)
where we recall that E0 r¨s ” E r ¨| F0m s. The weighted average structure of the WAPR implies that
it can also be written as a linear function of the value of private information for the informed.
Proposition 11 In the equilibrium without private information, all agents are uninformed. Initial
(interim) utilities and wealth levels are,
U
8
j,ni
“u
´
X0j
¯
ˆ
1
exp ´ 2
2Γ
żT´
¯2 ˙
S,ni
σt
dt ,
0
j P tu, iu
(20)
Note also that the expectation of a random variable based on the conditional Wiener measure, E r ¨| G “ xs, is the
same as the unconditional expectation
“
‰ of the random variable scaled by the density process of the conditional Wiener
measure, E r ¨| G “ xs “ E pηTx q´1 ¨ .
24
X0j
“
N0j
˜
D0 `
˜
` D ˘2 ¸ ¸
σ
T ,
µ ´
Γ
D
σtS,ni “ σ D .
(21)
Optimal share allocations are equal to the initial share endowments, Nvi “ Nvu “ Nvn “ 1 for all
v P r0, T s.9 Consider the NREE with initial share endowments N
j
0
“ 1 for j P tu, i, nu. Interim
“
`
˘‰
“ ‰
expected utilities differ by the interim value of private information, E0 U i “ U u E0 exp I i pG|Zq ,
where I i pG|Zq ď 0 is given by (18) and,
u
U “
u pX0u q exp
ˆ
1
´ 2
2Γ
żT
0
` S ˘2
1
σt dt `
Γ
żT
0
σtS ϑt pZ|E0 rξtm Dt sq dt
´
¯
u p
p p0q D0 ` Fp p0q ,
p0q Z ` B
X0u “ N 0 A
with ϑ pZ|Dt q ” Θt pZ; ωq ` hm
t pZ; ωq.
˙
´ I pZq ă 0
u
(22)
p ptq σ D
σtS “ B
(23)
Interim utilities, in the NREE, can be decomposed as,
u
U ”
“
E0 U
i
‰
E0 ru pXTu qs
ˆ
˙
X0u
“ ´Γ exp ´
exp p´E0 rξTm log ξTm sq
Γ
ˆ
˙
ˇ ‰˘‰
“ ` i ˘‰
“
`
“
X0i
” E0 u XT “ ´Γ exp ´
E0 exp ´E ξTG log ξTG ˇ F0i
Γ
with X0u “ X0i “ S0 . The first term, in each decomposition, is the value of initial stock holdings,
which is measurable relative to public information. These values are the same because initial endowments correspond to the equilibrium holdings in the economy without private information and are
therefore the same across representative agents.10 The second term, related to the value of the gains
from trade, captures the benefits of trading. The value of the gains from trade for the informed is,
ˇ ‰
ˇ ‰
ˇ ‰
“
“
“
E ξTG log ξTG ˇ F0i “ E ξTG log ξTm ˇ F0i `E ξTG log ηTG ˇ F0i “ E r ξTm log ξTm | F0m s`E r ξTm log ηTx | F0m s|x“G
where E r ξtm log ξTm | F0m s is the value of the gains from trade for the uninformed. The increment,
„ żT
¯ 
´
1
θvG|m pxq θvG|m pxq ` 2θvm dv “ I i pG|Zq
E r ξtm log ηTx | F0m s|x“G “ E0 ξTm
2
0
ř j j
j
Optimal demands are identical Ntj “ Γθtm {σtS . In equilibrium
` i u n ˘j ω Nt “ 1 so that Nt “ 1 for j P ti, u, nu.
By assumption, the distribution of initial shares is ω , ω , ω . The representative individual of each group
therefore owns 1 share at the outset. Aggregate endowment is ω i 1 ` ω u 1 ` ω n 1 “ 1, where 1 is the supply of shares.
9
10
25
corresponds to the value of private information (18). It is positive, implying that the informed is
always better off than the uninformed.
It is also instructive to examine the constituents of the value of the gains from trade for the
uninformed. As shown by (22), this value splits in three parts,
´E0 rξTm log ξTm s
1
“´
2
żT
0
”
E0 ξtm pθtmq2
ı
1
dt “ ´ 2
2Γ
żT
0
`
˘2
1
σtS dt`
Γ
żT
0
σtS ϑt pZ|E0 rξtm Dt sq dt´I u pZq .
˘2
`
To shed light on this expression, note that θtm “ σtS {Γ ´ ϑt pZ|Dt q so that pθtm q2 “ σtS {Γ ´
şT ` ˘2
2σtS ϑt pZ|Dt q {Γ ` ϑt pZ|Dt q2 . The first term, 2Γ1 2 0 σtS dt, therefore reflects the variance impact
on the endogenous market price of risk. This term is positive because a greater riskiness induces an
ı
”
şT
increased equilibrium price of risk. The last term, I u pZq “ 12 0 E0 ξtm ϑt pZ|Dt q2 dt, is the value
of the endogenous information signal extracted from equilibrium adjusted by the corresponding
hedging term. It is also positive, because information improves the efficiency of the pricing of
şT
risk and the resulting gains from trade. The middle term, Γ1 0 σtS ϑt pZ|E0 rξtm Dt sq dt, captures the
interaction between the risk and information components of the price of risk. This specific form
emerges because the stock volatility is deterministic and the hedge-adjusted WAPR is an affine
function of the fundamental (implying E0 rξtm ϑt pZ|Dt qs “ ϑt pZ|E0 rξtm Dt sq). In the equilibrium
without private information, investors are symmetric in all respects. The market price of risk is
then entirely determined by the riskiness of the stock. The value of the gains from trade are
completely driven by the stock’s variance.
The relation between informed and uninformed utilities in the NREE simplifies welfare comparisons across equilibria. Both agents are better off if the welfare of the uninformed improves.
Proposition 12 The uninformed is better off in the NREE, ∆u ” U u ´ U u,ni ě 0, if and only if
∆Pp u pZq
Γ
` ∆Tpu pZq ě 0, where ∆P u pZq “ S0 ´S0ni is the gain/loss from the valuation of initial share
endowments (price impact) and ∆T u pZq is the gain/loss from dynamic trading (trading impact).
The price and trading impacts are,
żT
´
¯`
˘
∆Vp
u
D
D
p
p
p
p psq γ
∆P pZq “ B p0q ´ 1 D0 ` µ T ` A p0q Z ´
`σ
B
p psq ds ´ ω n Ip p0q µφ
Γ
0
∆Tpu pZq “
1
∆Vp
´
2Γ2
Γ
żT
0
σtS ϑ pZ|E0 rξtm Dt sq dt ` I u pZq
26
(24)
(25)
¯
` ˘2 ş T ´
p ptq2 ´ 1 dt ă 0 is the reduction in the realized variance of the price. The
B
where ∆Vp “ σ D
0
welfare of the uninformed improves, in particular, if risk tolerance is sufficiently large (limΓÑ`8 ∆u “
´
¯
p ˆ 8 where ∆H
p ”
`8). For sufficiently small risk tolerance, limΓÑ0 ∆u “ ´sgn ∆Vp ` ∆H
p S ´ ∆H
p T , defined in (86)-(87), captures the net effect of hedging on the price impact (2∆H
p S)
2∆H
p T ).
and the trading impact (∆H
Proposition 12 identifies the sources of welfare gains and losses for the uninformed when private
information trades are allowed. The first effect, ∆Ppu pZq, captures the price impact on the initial
stock holdings of the uninformed. It can be positive or negative depending on parameter values. It is
positive if the partial dissemination of private information in the NREE causes a sufficient reduction
in risk. The second effect, ∆Tpu pZq, captures the change in the gains from trade. This component
splits into three parts, a riskiness effect (∆V {2Γ2 ), an informational efficiency effect (I u pZq) and
şT
a noise trading beliefs effect (´ 0 σtS ϑ pZ|E0 rξtm Dt sq dt{Γ). Allowing private information trading
şT
p ptq2 dt ă T ). As a result, the market price of risk decreases,
reduces the volatility of the stock ( 0 B
which also reduces welfare. The first part is then negative. It also disseminates private information
and increases the informational efficiency of the market. Better information improves investment
allocations and leads to welfare gains. The second part is positive. Finally, permitting the use
of private information will prompt the emergence of noise traders, whose activity limits efficiency
gains. The bias induced by their activities can be a source of welfare gains or losses. The third
part can be positive or negative. It is null when noise trading beliefs are unbiased, i.e., µφ “ E rGs.
Overall, when risk tolerance is large, the positive informational efficiency effect dominates and the
welfare of the uninformed improves. When risk tolerance is small, the riskiness effect dominates. It
increases (decreases) the stock price when hedging does not (does) dominate, implying a positive
(negative) price impact. It also reduces (increases) the price of risk, generating a negative (positive)
trading impact. The price impact dominates leading to an overall welfare gain (loss).
The next corollary provides further insights. To simplify notation, define,
J”
`
˘
∆u 2
K1,T
u
u
´
∆u ∆u
4K0,T
K2,T ,
Γ˘ “
u
∆
2K0,T
u
∆ ˘
´K1,T
? ,
J
Γ0 “ ´
∆
K1,T
u
∆
K0,T
∆ for j P t0, 1, 2u are defined in (88)-(91). With this notation,
where the coefficients Kj,T
27
Corollary 13 The uninformed is as well off in the NREE as in the equilibrium without private
information under the following conditions,
u
u
u
∆ ą 0, or if K ∆ “ K ∆ “ 0.
(i) uniformly in Γ P r0, `8q if K2,T
2,T
1,T
u
∆ ă 0.
(ii) for Γ ě max tΓ` , 0u if K2,T
u
u
∆ “ 0 and K ∆ ‰ 0.
(iii) for Γ ě max tΓ0 , 0u if K2,T
1,T
¯
´
¯
´
u
∆
p .
“ ´sgn ∆Vp ` ∆H
Furthermore, sgn K2,T
p ă 0, the
Corollary 13 shows that the welfare of the uninformed depends on hedging. If ∆Vp `∆H
p ě 0, the reverse can hold. An improvement is
uninformed is better off in the NREE. If ∆Vp ` ∆H
p ą 0 or Γ ě Γ0 if ∆Vp `∆H
p “ 0.
nevertheless assured for Γ sufficiently large, i.e., Γ ě Γ` if ∆Vp `∆H
Moreover, a necessary condition for the dominance of the equilibrium without private information
p ą ´ ∆Vp ą 0.
is that the net effect of hedging be strictly positive and sufficiently large, ∆H
Ultimately, the corollary identifies parameter regions where banning the use of private information
reduces the welfare of the uninformed. Figure 3 illustrates the possible configurations.
5.2
Welfare of Noise Trader
Let fφ|Z p x| Zq (resp. fG|Z p x| Zq be the Gaussian density of φ (resp. G) conditional on Z. The
likelihood ratio Lφ,G p x| Zq ” fφ|Z p x| Zq {fG|Z p x| Zq captures the beliefs divergence between the
noise trader and the informed. Explicit formulas are in the proof of Corollary 15 in Appendix B.
“
`
˘
‰
Corollary 14 The noise trader’s interim utlity is E0 rU n pφqs “ U u E0 exp I i pG|Zq Lφ,G pG|Zq
where Lφ,G pG|Zq “ fφ|Z pGq {fG|Z pGq. The noise trader is as well off as the uninformed trader if
“
`
˘
‰
and only if E0 exp I i pG|Zq Lφ,G pG|Zq ď 1. The noise trader is as well of as the informed if
ω i “ ω n , V AR rGs “ V AR rφs and E rφs “ E rGs, so that Lφ,G pG|Zq “ 1.
The corollary expresses the interim welfare of the noise trader relative to that of the uninformed.
The interim utility differential stems from the difference in the gains from trade, reflected in the CE
I i pG|Zq, and the difference in beliefs, reflected in the likelihood ratio Lφ,G pG|Zq. If the noise trader
happens to have the same conditional beliefs as the informed, i.e., if the conditional distributions of
φ and G given public information coincide, then Lφ,G p G| Zq “ 1. In this case, the interim utilities
“
‰
“
`
˘‰
coincide, E0 rU n pφqs “ U u E0 exp I i pG|Zq “ E0 U i pGq .
28
5.3
Pareto Optimal NREE
The relations between interim utilities in the NREE simplify welfare comparisons across equilibria.
Pareto dominance of the NREE over an equilibrium where investors are symmetric is ensured if the
interim welfare of the uninformed and the noise trader improves.
Corollary 15 The NREE is (weakly) interim Pareto optimal if and only if ∆u ě 0 and the certainly
“
`
˘
‰
equivalent gain of the noise trader gain is sufficiently high, ´Γ log E0 exp I i pG|Zq Lφ,G pG|Zq ą
´∆Pp pZq ` Γ∆Tpu pZq. If µφ “ E rGs, ω i “ ω n and V AR rφs “ V AR rGs, then the interim and
therefore also the ex-ante utilities of the noise trader and the informed are identical, E0 rU n pφqs “
“
‰
“
‰
E0 U i pGq and E rU n pφqs “ E U i pGq . In this case, the NREE is (weakly) Pareto optimal under
the conditions of Corollary 13.
The conditions for weak Pareto optimality ensure that all agents are as well off. When risk
tolerance converges to zero, the uninformed utility eventually becomes at least as large in the
NREE because of the price impact (Proposition 12). At the same time, if beliefs are unbiased, the
differential trading impact vanishes, ensuring that the noise trader attains the same ex-ante utility
as the uninformed. The NREE becomes Pareto optimal. If the noise trader happens to have the
same beliefs as the informed, he/she reaches the same ex-ante utility. The NREE is then Pareto
optimal under the conditions ensuring that the uninformed agent is as well off.
Corollary 15 has ramifications for market regulation. Permitting private information trades is
Pareto efficient when risk tolerance is sufficiently low, or sufficiently large and 0 ď ∆u `I n `
∆T n holds. In those cases, either the informational efficiency gains or the decrease in the riskiness
of the stock market dominate, leading to a welfare improvement. Scope for regulation exists in
intermediate cases. In these cases, factors such as the behavior of the noise trader, the properties
of dividends and the weights of the various investor populations matter, and have to be evaluated
to determine the relevance of regulatory constraints.
Corollary 16 Suppose that condition (i), (ii), or (iii) of Corollary 13 holds and ∆Pp pZq ` Γ∆Tpu pZq
“
`
˘
‰
ąΓ log E0 exp I i pG|Zq Lφ,G pG|Zq . The NREE is then (weakly) Pareto optimal.
Under the conditions of the corollary, regulation banning the use of private information is welfare
reducing.
29
Remark 17 The results above extend Leland’s (1992) analysis to a dynamic competitive setting. In
a static framework, the dynamic trading components p∆T n , ∆T u q are absent. So are intertemporal
aspects of the price impacts p∆P n , ∆P u q, such as volatility (∆V ). As conjectured by Leland, some
dynamic effects, e.g., trading effects, can dampen the price impacts.11 However, for sufficiently low
risk tolerance, the price impacts continue to dominate.
5.4
Speculation and Pareto Optimality
In order to pinpoint the role of speculation, suppose that the noise trader behaves as an uninformed
agent. The population weights are then ω n “ 0 and ω u “ 1 ´ ω i . Let E ns denote the resulting
equilibrium without speculation.
Corollary 18 Under the conditions of Corollary 16, the NREE E nree Pareto dominates the equilibrium without speculation E ns . Dominance holds whether informed agents act independently or
coordinate and act as a group.
In the absence of speculation, the stock price conditional on information acquisition is fully
revealing. In this equilibrium, all individuals observe the signal G which becomes public information.
The CE value of private information is then null, I i pG|Zq “ 0. The absence of a reward for
information acquisition combined with its cost imply that the representative informed is worse off
than the uninformed. There are no incentives to acquire.
Recall now that the representative informed is actually composed of a continuum of identical
informed agents. If these individuals act independently, they will each choose not to acquire information (free rider problem). The resulting equilibrium is a no information equilibrium in which all
agents are identical and common information flows are generated by the fundamental.
If individuals coordinate and act as a group, they compare their ex-ante utilities with and without
acquisition. In both cases equilibrium is fully revealing. The informational content, however, differs
across equilibria. With acquisition, the private signal G leaks to the market and the common
Ž
i ” FD
D . Again,
information flow is Fp¨q
σ pGq. Without acquisition, the common flow is Fp¨q
p¨q
11
“Insider trading “moves up” the resolution of uncertainty. This one time benefit may be relatively more important
in a two-period model than in a multiperiod model. If so, my results may overestimate the benefits from insider trading.
But we must await the development of multiperiod rational expectations models to answer this question definitively.”
(Leland (1992), p. 885).
30
information acquisition is not rewarded. The corresponding ex-ante CEs for the informed are,
CE
i,wa
CE
“E
i,na
rS0wa s
“E
1
1
V AR pS0wa q `
´
2Γ
2Γ
rS0na s ´
E
1
“ D0 ` µ T ´
Γ
D
0
1
1
V AR pS0na q `
2Γ
2Γ
where,
rS0wa s
żT
żT
0
`
˘2
σvS,wa dv,
ż
1 T ` S,na ˘2
E
“ D0 ` µ T ´
σv
dv,
Γ 0
` D ˘2
` ˘2
T
σ
wa
V AR pGq “ σ D T,
V AR rS0 s “
H p0q
rS0na s
D
` S,wa ˘2
σv
dv ´ C
żT
0
`
σvS,na
σtS,wa
˘2
dv
`
˘2
σζ
σD
“
H ptq
σtS,na “ σ D
V AR rS0na s “ 0.
The gain for acquiring GA ” CE i,wa ´ CE i,na is therefore,
1
GA “ ´
2Γ
„ż T ´
`
0
˘2
σvS,wa

` S,na ˘2 ¯
wa
na
´ σv
dv ´ pV AR rS0 s ´ V AR rS0 sq ´ C “ ´C
where the right hand side follows from
şT ´ S,na¯2
şT ´ S,wa ¯2
dv ` V AR rS0wa s. This gain is
dv
“
σ
v
0 σv
0
strictly negative. The optimal choice under coordination is again to forego information acquisition.
In summary, in the absence of speculation, the equilibrium is non-informative irrespective of
the behavior of the informed. With speculation, equilibrium is the NREE described in previous
sections. The conditions in Corollary 16 ensure that the NREE is weakly Pareto optimal, hence
Pareto dominates the non-informative equilibrium. Under these conditions, regulation banning
beliefs-based speculation is welfare reducing (see Figure 4).
6
Optimal Information Production Infrastructure
This section endogeneizes the frequency at which private information is produced.
!
)
Let ℑ pnq ” tnj : j “ 0, ..., n be an equidistant partition of the time interval r0, T s, with tn0 “ 0
and tnn “ T . The last date tnn is the dividend payment date. Each other date tnj represents an
information arrival (production) time. The interarrival time is tnj ´ tnj´1 “ ∆n . It represents
the time required to produce the next signal. The partition ℑ pnq is an information production
31
infrastructure with n arrival dates. Such an infrastructure has an upfront cost k pnq, an increasing
function of the number of arrival dates. Once the upfront cost has been paid, the infrastructure
ˆ ´ ¯ ˙
2
n
produces a proprietary signal Gj “ DT `ζj at time tj , j “ 0, ..., n ´ 1, where ζj „ N 0, σjζ
and the noise process ζ “ tζj : j “ 0, ..., nu is independent of all other uncertainty (but ζj , ζk could
be correlated for j ‰ k). The ex-ante utility associated with the infrastructure ℑ pnq is V i pnq. The
informed selects n so as to maximize ex-ante utility subject to his/her budget constraint.12
Once the information infrastructure is set up, it enters production and generates the sequence of
private signals described above. Let Nt be a deterministic counting process that tallies the number of
signals received up to time t. The vector of private signals received by time t is G1t “ rG1 , . . . , GNt s.
This vector increases in size every time a new signal arrives.
The noise trader emulates the behavior of the informed, but based on the sequence of irrelevant
forecasts φ1t “ rφ1 , . . . , φNt s instead of Gt . Noise trading beliefs are identical to informed beliefs,
except that they are parametrized by φ instead of G. The rest of the economy remains unchanged.
The certainty equivalent of the information infrastructure ℑ pnq is,
«
˜
Sn
CE pnq “ ´Γ log E exp ´ 0 ´ K0n ´ E
Γ
i
where S0n ” E
”ś
ˇ
ˇ
n´1 m
n DT ˇ Z1
i“0 ξtn
i ,ti`1
ı
«
n´1
k
ÿź
k“0 i“0
ˇ
ˇ
ˇ m,n
n ˇF
ξtmni ,tni`1 ∆DtKL
0
k
ˇ
ff¸ff
is the change in the Kullback-Leibler divergence
and ∆DtKL
n
k
measure between the signal density conditional on initial information F0m,n and information Ftm,n ,
n
“
∆DtKL
k
ı
”
¨
˛
m,n
m,n
m,n
V
AR
G
|
F
n
k`1
tk`1
1
1 V AR r Gk`1 | F0 s
V AR r Gk`1 | F0 s ‚
ı` ˝
ı´
ı
”
”
”
log
m,n
m,n
m,n
2
2 V AR G | F n
V AR Gk`1 | Ftn
V
AR
G
|
F
n
k`1
k`1
tk`1
tk
k
˛2 ¨ ”
¨¨ ”
ı
ı
E
1 ˚˚
˚
` ˚
2 ˝˝
Gk`1 | Ftm,n
n
k`1
c
”
´E
r Gk`1 | F0m,n s ‹
V AR Gk`1 | Ftm,n
n
k`1
ı
˚E
‹ ´˚
‚ ˝
Gk`1 | Ftm,n
n
k
c
”
´
˛2 ˛
m,n
E r Gk`1 | F0 s ‹ ‹
V AR Gk`1 | Ftm,n
n
k
ı
‹ ‹
‚‚
Finally, denote by ζNt the Nt ˆ 1 vector of signals before time t.
Proposition 19 The optimal information structure has n˚ information production dates, where
12
The cost of modifying the infrastructure once it is built is assumed to be prohibitive.
32
n˚ ě 1 solves the inequality,
¨
„
ˆ
˚
S m,n
´ 0Γ
˚
K0n
„
ˇ
ˇ
˚
KL ˇ m,n
m
i“0 ξtn˚ ,tn˚ ∆Dtn˚ `1 ˇ F0
i
i`1
k
ř n ˚ śk
´E
´
k“0
˚ E exp
˚
ˆ
„
´Γ log ˝ „
˚
řn˚ ´1 śk
˚
S m,n
m
E exp ´ 0 Γ ´ K0n ´ E
k“0
i“0 ξ n˚
ti
˙ ˛
‹
˚
˚
ˇ
˙ ‹
‚ ď k pn ` 1q´k pn q .
ˇ m,n˚
KL ˇ
˚ ∆D n˚ ˇ F0
,tn
tk
i`1
` ˘2
` ˘2
Furthermore, if 1{11Nt V AR rζNt s´1 1Nt Ó σ ζ for some constant σ ζ P p0, `8q (no-singularity
condition) and the cost increment ∆kn “ k pn ` 1q ´ k pnq is bounded away from zero (boundedness
condition), then n˚ ă 8.
The choice of an optimal information production infrastructure trades off the marginal utility
benefit with the marginal cost. The marginal utility benefit depends on the incremental CE value
of the information structure. When the latter falls below the marginal cost, raising the frequency
at which information is produced becomes suboptimal. The optimal frequency is the solution of the
equality that attains the highest expected utility.
The no-singularity (NS) condition prevents an asymptotic arbitrage. As shown in Proposition
22, an increase in the number of signals is isomorph to a reduction of the variance of the signal error.
If the signal error disappears, there is an asymptotic arbitrage opportunity because the limiting risk
neutral probability measure is not absolutely continuous. Condition NS guarantees that information
remains noisy in the limit.
Remark 20 Condition NS is a weak restriction on the information infrastructure. For instance, it
holds when the noise process is ζtnk “ ζ ` ζ̄ tn where ζ is a common component. In this case,
k
” ı
V AR r
ζt “
´ ¯2
“ ‰
V AR rζNt s “ σ ζ 1Nt 11Nt ` V AR ζ̄ Nt
´ ¯2
´ ¯2
1
1
ζ
Ó
σζ
“
σ
`
“ ‰´1
´1
1
1
1Nt V AR rζNt s 1Nt
1Nt
1Nt V AR ζ̄ Nt
as Nt Ñ 8. Information is asymptotically incomplete and it is therefore optimal to acquire a finite
number of signals when the cost increment is bounded away from zero.
The boundedness (B) condition on the cost increment also represents a mild restriction. In
practice, it becomes prohibitively costly to increase the private information frequency beyond some
33
level. Information production methods, which are typically time intensive, are simply unable to
generate arbitrarily large numbers of non-redundant signals.
Under conditions NS and B, the optimal information structure produces signals at a finite
frequency. The endogenous arrival frequency of private news is therefore lower than the arrival
frequency of public news. The asynchrony of information flows is an endogenous phenomenon.
Remark 21 The optimal infrastructure consists of a single signal at the initial date if the maximal
solution of the inequality in Proposition 19 is n˚ “ 1. This outcome arises, in particular, if the cost
of producing multiple signals is sufficiently large.
The next result describes equilibrium in this multi-signal environment.
Proposition 22 The information flow generated by the vector signal process G1t “ rG1 , . . . , GNt s
rt “ DT ` r
is equivalent to the information flow generated by the univariate process G
ζ t where
´1
1 V AR ζ
1
ζN
r Nt s Nt
r
. The noise trader using the conditional beliefs of the informed evaluated
ζt ” 1 t
´1
1N V ARrζNt s 1Nt
t
at the vector randomization process φ1t “ rφ1 , . . . , φNt s is identical to a noise trader using conrt and evaluated at the univariate randomization φ̃t
ditional beliefs based on the distribution of G
´1
φ1 V ARrφNt s 1Nt
“ 1Nt
. The resulting NREE has the same structure as in Proposition 4, but with
´1
1N V ARrζNt s 1Nt
t
´ r ¯2
” ı ` ˘
` ζ ˘2
2
“ V AR rζs replaced by the time-varying variance σtζ
σ
“ V AR r
ζ t , σ φ “ V AR rφs
´ ¯2
” ı
r.
r and Z replaced by Zrt “ ω i G
rt ` ω n φ
replaced by σtφ̃ “ V AR φ
t
t
Proposition 22 is an information aggregation result. It shows that the sequential observation of
the collection of noisy signals pertaining to the terminal dividend payment, Gt , is informationally
equivalent to the reception of a univariate signal with time-varying noise, G̃t . The filtrations generated by the two signal structures are effectively the same. Simple adjustments to the formulas in
Section 4, as described in the proposition, lead to the NREE in this multi-signal setting.
p ptq σ D jumps down at information arrival
Corollary 23 The volatility of the stock price σtS “ B
times t “ tj , j “ 1, ..., J and increases in between private information arrival times t ‰ tj , j “
1, ..., J. Private information has a stabilizing effect on price volatility.
Each time a new private signal arrives, it partially disseminates through the market. Dissemination is partial, because the activity of the noise trader hides the true value of the signal. Nevertheless,
34
instantaneously, all market participants acquire new information about the future dividend. They
immediately become less responsive to public news, such as information carried by the fundamental.
This instantaneous reaction lowers volatility at arrival times. Between arrival times, the information
related to past private signals does not change, whereas new fundamental information keeps materializing. Investors therefore become progressively more responsive to public news. The sensitivity
of demands to public news increases, ultimately leading to an increase in the stock price volatility.
In the limit, as the dividend date approaches, past private information becomes irrelevant and the
stock price volatility converges to the volatility of the fundamental. Overall, private information
has a stabilizing effect. It lowers the price volatility relative to an economy without private signals.
With multiple signals, it lowers volatility at each arrival time. As described in earlier sections, this
recurring reduction in volatility is the source of welfare benefits for all agents involved.
7
Extensions
Various extensions of the model are now considered. Section 7.1 studies heterogeneous risk tolerances. Section 7.2 deals with differential information. Section 7.3 allows for a dividend surprise. It
is shown that these extensions can all be mapped into the structure analyzed in Sections 2-4.
7.1
Heterogeneous Risk Tolerances
Suppose that risk tolerances are heterogenous, Γu ‰ Γi ‰ Γn . The NREE is described next.
Proposition 24
Consider the economy with heterogeneous risk tolerances Γu ‰ Γi ‰ Γn . A
NREE exists. It is given by the formulas in Proposition 4, where risk tolerance Γ is replaced by the
`
˘
ř
aggregate risk tolerance Γa ” jPtu,i,nu ω j Γj and where population weights ω i , ω n , ω are replaced
by risk-tolerance-adjusted weights pr
ωι, ω
rn, ω
r q with ω
r ι ” ω ι Γι {Γa for ι P ti, nu and ω
r”ω
ri ` ω
r n . The
endogenous public signal becomes Zr “ ω
riG ` ω
r n φ, thus depends on risk tolerance. The WAPR and
the stock price volatility also depend on risk tolerance.
The heterogeneity in risk tolerances has an effect on the informational content of equilibrium.
When the ratio Γn {Γi increases, the endogenous signal becomes less precise and informational efficiency decreases. The noise trader invests more aggressively, relatively speaking, which clouds the
35
information conveyed by the residual demand. In the limit, when the noise trader becomes infinitely
more risk tolerant than the uninformed so that Γn {Γi Ñ 8, the endogenous signal is overwhelmed
by noise and the private signal of the informed remains concealed. At the opposite end of the
spectrum, when Γn {Γi Ñ 0, equilibrium fully reveals the private information available.
The diversity in risk attitudes has multiple effects on volatility. The next corollary describes the
impact of an increase in the ratio of risk tolerances between the noise trader and the informed.
Corollary 25 Suppose that Γn {Γi increases, but Γa stays constant. Then, BB ptq ą 0, Bβ h ptq ă 0
and BB̂ptq ă 0 for all t P r0, T s. Thus, BσtS ă 0 for all t P r0, T s. Intertemporal hedging has a
taming effect on the reaction of volatility to the increase in the tolerance ratio Γn {Γi .
Under the conditions of the corollary, the endogenous information revealed becomes less precise.
In the absence of hedging, investors increase their reliance on fundamental information, which
increases the price volatility (BB ptq ą 0). Intertemporal hedging has a taming effect. It causes
agents to reduce the sensitivities of their individual demands to the fundamental, which reduces the
sensitivity of the aggregate demand (Bβ h ptq ă 0). To ensure market clearing, the sensitivity, hence
the volatility, of the equilibrium market price of risk must increase. The negative relation between
the stock price and its risk premium explains the taming effect on the stock price volatility.
7.2
Differential Information
Suppose that there is a continuum of informed investors, ι P r0, 1s , distributed on the unit interval
according to the measure µ pdιq. Informed agent ι receives the private signal Gι “ DT ` ζ ` ζ ι ,
where ζ i , ζ are mutually independent Gaussian random variable with mean zero. Assume further
that there is a continuum of noise traders, so that each informed agent has a follower. Noise trader
ι has the same beliefs as informed agent ι , but evaluated at φι “ φ ` ει . The random forecast φι
consists of a common component φ and an independent Gaussian random variable ει with mean
zero. All other components of the economy remain the same as before.
Let ςtι ” V AR rζ ι s {V ARt rGι s be the ratio of the idiosyncratic signal noise relative to the public
signal, i.e., the inverse signal-to-noise ratio.
The next proposition shows that the NREE is given by the formulas in Proposition 4 with
adjusted coefficients. The adjustments required depend on population moments involving the signal36
r1s
to-noise ratios, namely ̺t
”
ş1
µpdιq
0 1`ςtι
r2s
and ̺t
h
”
ş1 ´
0
1
1`ςtι
¯2
µ pdιq. They modify the coefficients of
the ODEs that determine the parameters β̌ , α̌h , γ̌ h of the aggregate hedging demand.
Proposition 26 Suppose that,
(i) The signal noise has a systematic component ζ and an idiosyncratic component ζ ι :
ř
i
limN Ñ8 N ´1 N
i“1 ζ “ 0.
ş1
0
ζ ι µ pdιq “
(ii) The speculative signal forecasts of noise traders have a systematic component φ and an idiosynş1
ř
i
cratic component ει : 0 ει µ pdιq “ limN Ñ8 N ´1 N
i“1 ε “ 0.
Under conditions (i) and (ii), the equilibrium with differentially informed investors is identical
to the NREE with a representative informed investor described in Proposition 4, but with coefficients
h
αh , β h , γ h replaced by α̌h , β̌ , γ̌ h where:
h
(a) β̌ solves the same Riccati ODE as β h in (34) but with κ0 , κ1 in (35) replaced by κ̌0 , κ̌1 in (92),
(b) α̌h , γ̌ h are as αh , γ h in (36), but with κ2 , κ3 , κ4 , κ5 in (37)-(40) replaced by κ̌2 , κ̌3 , κ̌4 , κ̌5 in
(93)-(96).
A NREE for an economy with purely differential information (non-hierarchical information
structure) is obtained by setting ω u ” 1 ´ ω i ´ ω n “ 0. The NREE in Proposition 4 without
differential private information is recovered by setting ςti “ 0.
Coefficients in the NREE with differential information are modified by the inverse signal-to-noise
ratio ςtι of the idiosyncratic components ζ ι . The assumption that the private information signals
differ by idiosyncratic noise components implies that the infinite regress problem, typically associated
with non-hierarchical information structures (see Townsend (1983)), does not arise. This assumption
ensures that only the systematic information component matters in the aggregate. It implies that
the equilibrium information flow is generated by D, Z, as in the equilibrium without differential
information. A NREE exists and is given by the formulas in Proposition 4, with suitably adjusted
parameters. All properties documented in Sections 4 and 5 continue to hold. In particular, welfare
implications and Pareto rankings of equilibria are as in the model without differential information.
The price impact and gains from trade impact are obtained by using the adjusted coefficients.
If ζ “ ω u “ 0, the model specializes to a setting with differential information similar to He and
Wang (1995), but with noise trading instead of stochastic supply.
37
7.3
Dividend Surprise
r T “ DT ´ ` ǫ where ǫ represents a surprise, i.e., a jump. The jump size ǫ is assumed
Suppose that D
to be normally distributed and independent of the signal noise ζ and of FTD . Furthermore, suppose
that private information pertains to the continuous part of the dividend, G “ DT ´ ` ζ.13
Proposition 27 The equilibrium stock price Srt is,
Srt “
$
’
& St ` E rǫs ´
V ARrǫs
Γ
’
% St ` ǫ
for t ă T
(26)
for t “ T
where St is the price in the absence of a surprise, described in Proposition 4. The price jump at T
is ∆SrT “ ǫ ´ E rǫs `
V ARrǫs
.
Γ
The volatility and risk premium are unaffected by the surprise. The
ı
”
risk premium converges to ET ´ ∆SrT “ V ARrǫs
as T approaches. The conditional variance of the
Γ
ı
”
terminal price jump converges to V ART ´ ∆SrT “ V AR rǫs. The equilibrium at T´ is a symmetric
no-trade equilibrium where NTι ´ “ 1 for ι P tu, i, nu.
The dividend surprise at the terminal date does not affect the informational content of equilibrium, prior to the payment date. It also leaves the structure and properties of equilibrium unchanged.
The reason for this irrelevance result is the absence of information regarding the eventual surprise.
The stock price volatility in this setting is still driven by the volatility of the fundamental.
Volatility, in the NREE, has the same properties relative to the equilibrium without private information. The welfare properties of equilibrium and the Pareto dominance of the NREE under the
conditions outlined in Section 5 continue to apply.
8
Conclusion
This paper studied the effects of information production and speculation on the structure and properties of a non-stationary noisy rational expectations equilibrium. Private information dissemination
was found to have a stabilizing effect, as it reduces the volatility of the stock price. Costly information production, resulting in asynchrony between private and public information flows, leads
13
The dividend paid, DT , differs from the terminal value of the fundamental DT ´ by the independent random
variable ǫ. Private information pertains to the fundamental value DT ´ .
38
to an increase in volatility in between information arrival dates. Private information, although
fundamentally detrimental to non-informed agents, has nevertheless welfare benefits for all market
participants. By lowering risk, it increases the value of the stock and therefore of initial allocations. By disseminating through the market, it improves the decisions of investors and the resulting
gains from trade. For economies with sufficiently high or low risk tolerances, the welfare benefits
offset the costs for all agents involved. Under these conditions, the NREE with private information
trades Pareto dominates the equilibrium without private information collection. Speculation plays
a central role. In its absence, incentives for information collection vanish and the potential welfare
benefits of information are lost.
The dynamic model developed in the paper is tractable and produces closed form solutions, with
the exception of a single coefficient. It therefore offers a useful platform to examine complex issues
related to information asymmetry in financial markets. For instance, it provides a natural setting
to further study policy questions. Should trades based on private information be banned? Beyond
issues of enforceability, the results in this paper suggest that an outright ban may not be best for
society. Should they be restricted in less stringent ways? If so, what are appropriate regulations?
What are the efficiency and liquidity effects of various types of restrictions? Likewise, should
speculation be restricted and if so in what ways? These questions are of fundamental importance
for the smooth functioning of financial markets and the welfare of market participants. Their
analysis requires extensions of the model incorporating the relevant regulatory constraints under
consideration and is therefore beyond the scope of the present study. Issues such as these could
prove interesting avenues for future research.
Appendix
Appendix A: Conditional Moment Formulas
The next lemma provides formulas for the conditional moments of the fundamental under a change of measure.
¯
´
¯
´
şT
şT
r ptq Dt ` γ ptq and
r ptq z ` β
Lemma 28 Define dPr z “ exp ´ 12 0 f pzq2 dv ´ 0 f pzq dWvD dP where f pzq “ ´ α
r γ are functions of time. The first two conditional moment of D under Pr z for s ě t are,
α
r , β,
”
ı
r z Ds | FtD “ a1 pt, sq z ` b1 pt, sq Dt ` g1 pt, sq
E
(27)
” ˇ
ı
r z Ds2 ˇ FtD “ a2 pt, sq z ` b2 pt, sq Dt ` c2 pt, sq Dt z ` d2 pt, sq z 2 ` b1 pt, sq2 Dt2 ` g2 pt, sq
E
39
(28)
where,
a1 pt, sq ” σ
D
żs
t
b1 pv, sq α
r pvq dv,
a2 pt, sq ” 2
żs
d2 pt, sq ” 2σ
D
żs
t
b1 pt, sq ” exp σ
D
żs
t
˙
r
β puq du ,
´
¯
b1 pv, sq2 k pvq a1 pt, vq ` σ D α
r pvq g1 pt, vq dv,
t
b2 pt, sq ” 2
ˆ
żs
t
b1 pv, sq2 k pvq b1 pt, vq dv,
2
b1 pv, sq α
r pvq a1 pt, vq dv,
c2 pt, sq ” 2σ D
g2 pt, sq ” 2
żs
t
żs
t
g1 pt, sq ”
żs
t
b1 pv, sq k pvq dv
k pvq ” µD ` σ D γ pvq
b1 pv, sq2 α
r pvq b1 pt, vq dv
b1 pv, sq
2
˜`
σD
2
˘2
¸
` k pvq g1 pt, vq dv.
“
‰
“ ˇ
‰
r z Ds | FtD “ b1 pt, sq and BDt E
r z Dv2 ˇ FtD “ b2 pt, vq ` c2 pt, vq z ` 2b1 pt, vq2 Dt .
Moreover, BDt E
ˇ D‰
ˇ
“
‰
“ D
“
‰
D
D
D rz
Dˇ
rz
r z WsD ´ WtD ˇ FtD “
Proof
ˇ
¯
” ş of´ Lemma 28. ¯ Using
ıE Ds | F
” t “ Dt ` µ ˇ ps ´
ı tq ` ´σ E Ws ´ Wt“ Ft , E
‰
r pvq E
r z Dv | FtD ` α
r z s θvm ´ θvZ|D pzq dv ˇˇ FtD and E
r z θvm ´ θvZ|D pzqˇˇ FtD “ ´ γ pvq ` β
r pvq z gives,
´E
t
żs
żs
”
”
ı
ı
r pvq E
r z Dv | FtD dv.
r z Ds | FtD “ Dt ` µD ps ´ tq ` σ D
β
E
r pvq zq dv ` σ D
pγ pvq ` α
t
t
¯ˇ
”´
ı
ˇ
“
‰
r z dWvD ˇ FtD “ ´E
r z θvm ´ θvZ|D pzq ˇˇ FtD dv,
Defining k pvq ” µD ` σ D γ pvq and solving gives (27). Also, as E
rz
E
„ż s
it follows that,
” ˇ
ı
r z Ds2 ˇ FtD
E
“
“
t
ˇ

„ż v
 żs
”
ı
´
¯ ˇˇ
ˇ
r z Dv | FtD dvµD ´ σ D E
rz
E
Dv θvm ´ θvZ|D pzq dv ˇˇ FtD
Dv dDv ˇˇ FtD “
t
t
”
r z Dv
E
´
¯ˇ
¯ˇ
ı
ı
”
´
ˇ
ˇ
r pvq Dv ` α
r z Dv γ pvq ` β
r z ˇ FtD
θvm ´ θvZ|D pzq ˇ FtD “ ´E
ˇ
„ż s
 ´ ¯
żs
”
ı
2
ˇ
rz
r z Dv | FtD dv
Dt2 ` 2E
Dv dDv ˇˇ FtD ` σ D ps ´ tq “ Dt2 ` 2µD
E
t
t
żs
¯ˇ
ı
´ ¯2
”
´
ˇ
D
D
z
r pvq Dv ` α
r Dv γ pvq ` β
r pvq z ˇ Ft dv ` σ D ps ´ tq
`2σ
E
Dt2
`2
żt s ´
t
żs
¯
”
ı
” ˇ
ı
´ ¯2
r pvq E
r z Dv | FtD dv ` 2σ D
r z Dv2 ˇ FtD dv ` σ D ps ´ tq .
k pvq ` σ D α
r pvq z E
β
t
Solving this linear equation gives
” ˇ
ı
r z Ds2 ˇ FtD
E
“
“
“
żs
´ ¯2 ż s
´
¯
”
ı
r z Dv | FtD dv
b1 pt, sq2 Dt2 ` σ D
b1 pv, sq2 dv ` 2
b1 pv, sq2 k pvq ` σ D α
r pvq z E
t
t
´ ¯2 ż s
2
2
D
2
b1 pt, sq Dt ` σ
b1 pv, sq dv
t
żs
´
¯
`2
b1 pv, sq2 k pvq ` σ D α
r pvq z pg1 pt, vq ` b1 pt, vq Dt ` a1 pt, vq zq dv
t
żs
´ ¯2 ż s
b1 pt, sq2 Dt2 ` σ D
b1 pv, sq2 dv ` 2
b1 pv, sq2 k pvq g1 pt, vq dv
t
t
ˆż s
˙
żs
żs
2
2
D
2
`2
b1 pv, sq k pvq b1 pt, vq dvDt ` 2
b1 pv, sq k pvq a1 pt, vq dv ` σ
b1 pv, sq α
r pvq g1 pt, vq dv z
t
t
t
żs
żs
`2σ D
b1 pv, sq2 α
r pvq b1 pt, vq dvzDt ` 2σ D
b1 pv, sq2 α
r pvq a1 pt, vq dvz 2
t
t
Defining the coefficients as in the lemma leads to the expressions stated.
40
Appendix B: Proofs
Proof of Proposition 1. The state price density (SPD) for the uninformed is,
ξtm
ˆ
1
“ exp ´
2
żt
0
pθvm q2
XTu
dv ´
u
żt
θvm dWvS
0
˙
ˆ ż¨
˙
m
S
” E ´ θv dWv
.
0
t
Γ log ξTm
Optimal terminal wealth equals
“ ´Γ log y ´
where the shadow price of initial wealth y u solves
“ m
‰
m
u
u
m
m
u
m
X0 “ ´Γlogy ´ ΓE rξT log ξT s. Intermediate wealth is Xt “ ´Γ log y u ´ Γ log
” ξt ş´ ΓEt ξt,Tı log ξt,T . Using
“
‰
`
˘
ş
ş
T
T
T m
m
m
m
log ξt,T
“ 21 Et ξt,T
pθvm q2 dv . If the condilog ξ m
dWvS ` θvm dv ` 21 t pθvm q2 dv gives Et ξt,T
t,T “ ´ t θv
t
u S
tional expectation
” isş Markovianı in Dt , which will be shown to hold in equilibrium, the optimal demand is Nt σt “
T
m
m
m 2
D
Γ
Γθt ´ 2 BDt Et ξt,T t pθv q dv σ . The formulas for the mean-variance and hedging components follow.
´ş
¯´1
t G|m
x
For the informed ξti “ ξtm ηtG where ηt,T
” P p G P dx| FTm q {P p G P dx| Ftm q´1 “ E 0 θv pxq dWvS
. Optimal
t
ˇ ‰
“
terminal wealth is XTi “ ´Γ log y i ´ Γ log ξTi , where y i solves X0i “ ´Γlogy i ´ ΓE ξTi log ξTi ˇ F0i . Thus,
ˇ ı
”
i
i
i ˇ
Xti “ ´Γ log y i ´ Γ log ξti ´ ΓE ξt,T
log ξt,T
ˇ Ft
ˇ ‰
“ i
i ˇ i
because log ξti “ log ξtm ` log ηtG and E ξt,T
log ξt,T
Ft “
„
´
¯2 ˇˇ 
i
E ξt,v
θvm ` θvG|m pGq ˇˇ Fti
“
“
1
2
şT
t
„
´
¯2 ˇˇ 
i
E ξt,v
θvm ` θvG|m pGq ˇˇ Fti dv, where,
„

´
¯2 ˇˇ
m
E ξt,v
θvm ` θvG|m pxq ˇˇ Ftm
Et
”
m
ξt,v
pθvm q2
ı
` Et
”
m G|m
ξt,v
θv
|x“G
pxq2
ı
|x“G
”
ı
m m G|m
` 2Et ξt,v
θv θv pxq
|x“G
.
”
´
¯ı
`
˘
şT
m G|m
θv pxq θvG|m pxq ` 2θvm dv|x“G . If the conditional expecThus, Xti “ Xtu ` Γ log y u {y i ´ Γ log ηtG ´ Γ2 t Et ξt,v
tation is Markovian in Dt , as will be shown to hold in equilibrium, the portfolio of the informed follows.
The proof of Proposition 4 relies on auxiliary Lemmas 29-37.
D,Z
m
Lemma 29 Suppose that public information is carried by the filtration Fp¨q
“ Fp¨q
. The PIPR is the affine function,
G|m
θt
¯
x ´ µG|D,Z
x ´ µtG|D,Z ´
i
D
t
pxq “ ´
σD
¯2 1 ´ κt ω σ “
H ptq
G|D,Z
σt
”
´
¯
ı
“ Dt ` µD pT ´ tq ` κt Z ´ ω i Dt ` µD pT ´ tq ´ ω n µφ
´
¯ 2 ˆ´ ¯ 2
´ ¯2 ˙ ´
¯
´
¯
σtG|D,Z “
σ D pT ´ tq ` σ ζ
1 ´ κt ω i ” H ptq 1 ´ κt ω i
G|D,Z
µt
κt “
ω i H ptq
,
M ptq
´ ¯2
´ ¯2
H ptq “ σ D pT ´ tq ` σ ζ ,
´ ¯2
´ ¯2
M ptq “ ω i H ptq ` pω n q2 σ φ .
“
‰
G|m
Proof of Lemma 29. The PIPR satisfies d log pG pxq , W S “ θt
pxq dt, where pG pxq is the conditional density
m
at time t of the signal G, given public information Fp¨q . To determine the conditional density, note that,
G “ DT ` ζ “ Dt ` µD pT ´ tq `
żT
t
σ D dWsD ` ζ
ˆż T
˙
´
¯
Z “ ω i G ` ω n φ “ ω i Dt ` µD pT ´ tq ` ω i
σ D dWsD ` ζ ` ω n φ.
t
˙
ˆ
G|D,Z
x´µt
D,Z
m
1
with parameters
n
Under the assumption Fp¨q
“ Fp¨q
, the conditional density is pG
t pxq “
G|D,Z
G|D,Z
σt
σt
¯
´ş
`
˘
T
G|D,Z
G|D,Z
µt
, σt
as defined in the lemma (M ptq is the variance of Z ´ ω i Dt ` µD pT ´ tq “ ω i t σ D dWsD ` ζ `
ω n φ). An application of Ito’s lemma establishes the result announced.
41
D,Z
m
Lemma 30 Suppose that Fp¨q
“ Fp¨q
. The WAPR is an affine function of Z, Θt pZ; ωq ” ω i θtG|m pGq`ω n θtG|m pφq “
α ptq Z ` β ptq Dt ` γ ptq where,
`
˘
1 ´ κt ω i µD pT ´ tq ´ ω n κt µφ D
1 ´ κt ω D
1 ´ κt ω i D
α ptq “
σ ,
β ptq “ ´ω
σ ,
γ ptq “ ´ω
σ ,
ω “ ωi ` ωn.
H ptq
H ptq
H ptq
Proof of Lemma 30. From Lemma 29 and the definition of the WAPR,
Θt pZ; ωq
”
“
“
`
˘ G|D,Z
G|D,Z
Z ´ ω i ` ω n µt
Z ´ ωµt
G|m
σD ”
σD
pφq “
pGq ` ω n θt
H ptq
H ptq
`
˘˘
``
˘`
˘
Z ´ ω 1 ´ κt ω i Dt ` µD pT ´ tq ` κt Z ´ ω n µφ
σD
H ptq
˜
¸
`
˘
1 ´ κt ω i µD pT ´ tq ´ ω n κt µφ
1 ´ κt ω
1 ´ κt ω i
Z´ω
Dt ´ ω
σ D ” α ptq Z ` β ptq Dt ` γ ptq .
H ptq
H ptq
H ptq
G|m
ω i θt
This establishes the claim.
D,Z
m
Lemma 31 Suppose that Fp¨q
“ Fp¨q
. The optimal hedging demand of the uninformed is Nth,u “ Γhut pZq σ D {σtS
u
u
u
where hut pZq “ ψ1t
Z ` ψ2t
Dt ` ψ3t
with,
u
“´
ψ1t
żT ˆ
t
α
p pvq b1 pt, vq `
u
ψ3t
“´
żT ˆ
t
˙
1p
β pvq2 c2 pt, vq dv,
2
u
ψ2t
“´
żT
t
p pvq2 b1 pt, vq2 dv
β
˙
1p
p pvq γ
pvq2 b2 pt, vq dv
β
p pvq b1 pt, vq ` β
2
r ptq ´
and the functions b1 , b2 , c2 are defined in Lemma 28 with γ ptq ” γ
p pvq ” β pvq ` β h pvq and γ
α pvq ` αh pvq, β
p pvq ” γ pvq ` γ h pvq.
σtS
Γ
. The remaining coefficients are α
p pvq ”
”
şT m 2 ˇˇ m ı
m
m
Proof of Lemma 31. Uninformed hedging demands are determined by Ht,T
” ´ 21 E ξt,T
pθv q dv ˇ Ft . As
t
ˇ
” ´ş
¯
ı
ˇ D
¨ Z|D
D,Z
D
m
m
E r K| Ft s “ E E t θv pzq dWv
K ˇ Ft
for arbitrary K (recall that Fp¨q ” Fp¨q ) and dWvm “ dWvD ´
T
|z“Z
θvZ|D dv it follows that,
m
Ht,T
“
“
ˇ

˙
ˆ ż¨
„ ˆż ¨
˙ żT
ˇ
1
E ´ θvm pzq dWvm
θvZ|D pzq dWvD
pθvm pzqq2 dv ˇˇ FtD
´ E E
2
t
t
T
T t
|z“Z
ˇ
ˇ

„ż T
„ ˆ ż¨´

˙ żT
¯
ˇ
ˇ
1
1 rz
θvm pzq2 dv ˇˇ FtD
´ E E ´
”´ E
θvm pzq ´ θvZ|D pzq dWvD
θvm pzq2 dsˇˇ FtD
2
2
t
t
T t
|z“Z
´ ş ´
¯
¯
¨
Z|D
where the last equality uses the definition dPr z {dP ” E ´ 0 θvm pzq ´ θv pzq dWvD , the next to last E pM q E pN q “
Z|D
E pM ` N ` rM, N sq and θt
T
pzq is the volatility of the conditional density of DT given Z,
Z|D
θt
αD ptq ”
ωiσD
,
M ptq
pzq “ αD ptq z ` β D ptq Dt ` γ D ptq
β D ptq ” ´ω i αD ptq ,
´
¯
γ D ptq ” ´αD ptq ω i µD pT ´ tq ` ω n µφ .
It follows from Propositions 1 and 2, and from market clearing that θtm “ σtS {Γ ´ hut pzq σ D ´ Θt pZ; ωq ´ ht pZ; ωq σ D ,
G|m
G|m
where ht pz; ωq ” ω i ht
pGq ` ω n ht
pφq is the weighted sum of the informational hedge of the informed and the
u
corresponding hedge of the noise trader. The aggregate hedging demand depends on hm
t pzq ” ht pzq ` ht pz; ωq.
To proceed, let us conjecture an affine aggregate hedging demand and a time-dependent equilibrium volatility,
D
hm
” phut pzq ` ht pz; ωqq σ D ” αh ptq z ` β h ptq Dt ` γ h ptq
t pzq σ
42
and
σtS non-stochastic.
(29)
Under this double conjecture,
¯
´
¯
´
¯
σS
σtS ´
p ptq Dt ´ γ
´ α ptq ` αh ptq z ´ β ptq ` β h ptq Dt ´ γ ptq ` γ h ptq ” t ´ α
p ptq Z ´ β
p ptq
Γ
Γ
θtm “
Z|D
θtm pzq ´ θt
pzq
“
”
¯
´
¯
¯
´
σtS ´
p ptq ` β D ptq Dt ´ γ
p ptq ` γ D ptq
´ α
p ptq ` αD ptq z ´ β
Γ
¯
¯
´
σtS ´
r ptq Dt ` γ
r ptq Dt ` γ ptq
´ α
r ptq z ` β
r ptq ” ´ α
r ptq z ` β
Γ
m
where γ ptq ” γ
r ptq ´ σtS {Γ. Substituting in the expression for hut pZq “ BDt Ht,T
gives,
hut pzq
“
“
“

„´
ż
żT
”
¯2 ˇˇ
ı
ˇ
1 T
p pvq Dv ` α
r z θvm pzq2 ˇ FtD dv “ ´ 1
rz
B Dt E
B Dt E
γ
p pvq ` β
p pvq z ˇˇ FtD dv
2 t
2 t
żT ˆ
” ˇ
”
ı
ı˙
p pvq2 BDt E
p pvq pp
r z Dv2 ˇ FtD dv
r z Dv | FtD ` 1 β
β
γ pvq ` α
p pvq zq BDt E
´
2
t
˙
żT ˆ
˘
`
1
p pvq2 b2 pt, vq ` c2 pt, vq z ` 2b1 pt, vq2 Dt dv.
p pvq pp
´
β
γ pvq ` α
p pvq zq b1 pt, vq ` β
2
t
´
Collecting terms and defining coefficients as indicated leads to the formulas stated.
D,Z
m
Lemma 32 Suppose that Fp¨q
“ Fp¨q
. If the conjectures in (29) are satisfied, the optimal hedging demand of the
G|m
G|m
G|m
G|m
h,i
h,u
informed is Nt “ Nt ` Γht
pG|Zq σ D {σtS , where ht
pG|Zq “ h1t pG|Zq ` h2t pG|Zq with,
G|m
h1t
i,G
i,Z
i
i
pG|Zq “ ψ11
ptq G ` ψ11
ptq Z ` ψ21
ptq Dt ` ψ31
ptq
i,G
i,Z
i
i
hG|m
pG|Zq “ ψ12
ptq G ` ψ12
ptq Z ` ψ22
ptq Dt ` ψ32
ptq
2t
ˆ
˙
żT
żT
1
i,G
i,Z
2
ψ11 ptq “ ´
α1 pvq β0 pvq b1 pt, vq dv, ψ11 ptq “ ´
α2 pvq β0 pvq b1 pt, vq ` β0 pvq c2 pt, vq dv
2
t
t
˙
ˆ
żT
żT
1
2
i
2
2
i
ψ21 ptq “ ´
β0 pvq b1 pt, vq dv, ψ31 ptq “ ´
γ0 pvq β0 pvq b1 pt, vq ` β0 pvq b2 pt, vq dv
2
t
t
żT
ż T ´´
¯
¯
i,G
p pvq α1 pvq b1 pt, vq dv, ψ i,Z ptq “
p pvq α2 pvq ` β0 pvq α
p pvq β0 pvq c2 pt, vq dv
ψ12
ptq “
β
β
p pvq b1 pt, vq ` β
12
t
i
ψ22
ptq “ 2
żT
t
t
p pvq b1 pt, vq2 dv,
β0 pvq β
i
ψ32
ptq “
ż T ´´
t
¯
¯
p pvq ` γ̌ pvq β0 pvq b1 pt, vq ` β0 pvq β
p pvq b2 pt, vq dv
γ0 pvq β
where γ̌ ptq ” γ
p ptq ´ σtS {Γ and b1 , b2 , c2 are defined in Lemma 28. The optimal hedging demand of the noise trader
has the same structure, but is evaluated at φ instead of G.
Proof of Lemma 32. The proof proceeds as for the uninformed. Optimal wealth is,
Xti
“
“
”
”
´
¯ı
´ ¯ ΓżT
m G|m
dv
Et ξt,v
θv px|zq θvG|m px|Zq ` 2θvm pzq
c0 pZ, Gq ` Xtu ´ Γ log ηtG ´
2 t
|x“G
´ ¯ ΓżT
”
´
¯ˇ
ı
r z θvG|m px|zq θvG|m px|zq ` 2θvm pzq ˇˇ FtD
c0 pZ, Gq ` Xtu ´ Γ log ηtG ´
E
dv
2 t
|z“Z,x“G
´ ¯
G|m
G|m
c0 pZ, Gq ` Xtu ´ Γ log ηtG ` ΓH1t,T pG, Zq ` ΓH2t,T pG|Zq
`
˘
where c0 pZ, Gq ” Γ log y u {y i and
G|m
H1t,T
px|zq “ ´
1
2
żT
t
ˇ
”
ı
r z θvG|m pxq2 ˇˇ FtD dv,
E
G|m
H2t,T
px|zq “ ´
żT
t
ˇ
”
ı
r z θvG|m pxq θvm pzqˇˇ FtD dv.
E
The hedging demand of the informed can be decomposed as hut ` hG|m
pG|Zq where hG|m
pG|Zq “ hG|m
pG|Zq `
t
t
1t
G|m
G|m
G|m
G|m
G|m
h2t pG|Zq with h1t pG|Zq ” BDt H1t,T pG|Zq and h2t pG|Zq ” BDt H2t,T pG|Zq. Given the affine structure of the
43
PIPR, θtG|m pxq“ α1 ptq x ` α2 ptq z ` β0 ptq Dt `γ0 ptq,
G|m
2h1t
px|zq
“
“
“
´
żT ´
t
żT
˘˘
`
2 pα1 pvq x ` α2 pvq z ` γ0 pvqq β0 pvq b1 pt, vq ` β0 pvq2 b2 pt, vq ` c2 pt, vq z ` 2b1 pt, vq2 Dt dv
ˆż T
˙
ˆż T
˙
`
˘
´2
α1 pvq β0 pvq b1 pt, vq dv x `
2α2 pvq β0 pvq b1 pt, vq ` β0 pvq2 c2 pt, vq dv z
´
t
G|m
h2t
px|zq
“
σtS
Γ
t
t
t
2
˙
2
β0 pvq b1 pt, vq dv Dt `
żT
t
t
`
˘
2γ0 pvq β0 pvq b1 pt, vq ` β0 pvq2 b2 pt, vq dv
´
¯
p ptq Dt ´ γ
p ptq Dt ` γ̌ ptq where γ̌ ptq ” γ
´α
p ptq Z ´ β
p ptq ” ´ α
p ptq Z ` β
p ptq ´ σtS {Γ,
¯
”
ı
p pvq ` pγ̌ pvq ` α
r z Dv | FtD dv
pα1 pvq x ` α2 pvq z ` γ0 pvqq β
p pvq zq β0 pvq BDt E
żT
t
żT ´
`
“
ˆż T
żT ´
`
“
`
t
´2
and using θtm “
”
ı
” ˇ
ı¯
r z Dv | FtD ` β0 pvq2 BDt E
r z Dv2 ˇ FtD dv
2 pα1 pvq x ` α2 pvq z ` γ0 pvqq β0 pvq BDt E
¯
p pvq ` pγ̌ pvq ` α
pα1 pvq x ` α2 pvq z ` γ0 pvqq β
p pvq zq β0 pvq b1 pt, vq dv
żT
żT
t
`2
” ˇ
ı
p pvq BDt E
r z Dv2 ˇ FtD dv
β0 pvq β
t
˘
`
p pvq b2 pt, vq ` c2 pt, vq z ` 2b1 pt, vq2 Dt dv
β0 pvq β
p pvq α1 pvq b1 pt, vq dvx `
β
żT
t
ż T ´´
¯
¯
p pvq α2 pvq ` β0 pvq α
p pvq β0 pvq c2 pt, vq dvz
β
p pvq b1 pt, vq ` β
t
p pvq b1 pt, vq dvDt `
β0 pvq β
2
ż T ´´
¯
¯
p pvq ` γ̌ pvq β0 pvq b1 pt, vq ` β0 pvq β
p pvq b2 pt, vq dv.
γ0 pvq β
t
Defining the coefficients as indicated in the lemma establishes the result.
D,Z
m
Lemma 33 Suppose that Fp¨q
“ Fp¨q
. If the conjectures in (29) are satisfied, the information-related component of
the residual hedging demand depends on,
G|m
G|m
ht pZ; ωq ” ω i ht
pG|Zq ` ω n ht
pφ|Zq “ h1t pZ; ωq ` h2t pZ; ωq
´
¯
i,G
i,Z
i
i
h1t pZ; ωq “ ψ11
ptq ` ωψ11
ptq Z ` ωψ21
ptq Dt ` ωψ31
ptq
´
¯
i,G
i,Z
i
i
h2t pZ; ωq “ ψ12
ptq ` ωψ12
ptq Z ` ωψ22
ptq Dt ` ωψ32
ptq .
The aggregate hedging demand depends on,
m
ψ2t
u
m
m
m
hm
t pZq ” ht pZq ` ht pZ; ωq “ ψ1t Z ` ψ2t Dt ` ψ3t
´
¯
i,Z
i,Z
m
u
i,G
i,G
ψ1t
“ ψ1t
` ψ11
ptq ` ψ12
ptq ` ω ψ11
ptq ` ψ12
ptq
´
¯
´
¯
u
i
i
m
u
i
i
“ ψ2t
` ω ψ21
ptq ` ψ22
ptq , ψ3t
“ ψ3t
` ω ψ31
ptq ` ψ32
ptq .
The aggregate hedging demand is an affine function of Z.
pG|Zq `
Proof of Lemma 33.
Substituting the formulas from Lemmas 30 and 31 in hG|m
pZ; ωq ” ω i hG|m
t
t
m
u
n G|m
pφ|Zq and ht pZq ” ht pZq ` ht pZ; ωq leads to the expressions stated.
ω ht
The coefficients of the aggregate hedging demand depend on the coefficients of the conjectures in (29). That is,
m
m
m
m
ht pZq “ αh ptq Z` β h ptq Dt ` γ h ptq “ ψ1t
Z ` ψ2t
Dt ` ψ3t
. For consistency, the following integral equations must
h
m D
h
m D
h
m D
m
m
m
be satisfied, α ptq “ ψ1t σ , β ptq “ ψ2t σ , γ ptq “ ψ3t σ . Simplifying the expressions for pψ1t
, ψ2t
, ψ3t
q gives,
m
ψ1t
“
´
żT ˆ
t
α
p pvq b1 pt, vq `
˙
żT
żT
1p
p pvq α1 pvq b1 pt, vq dv
β pvq2 c2 pt, vq dv ´
α1 pvq β0 pvq b1 pt, vq dv `
β
2
t
t
44
´ω
`ω
“
“
żT ˆ
t
żT
t
˙
1
β0 pvq2 c2 pt, vq dv
2
¯
¯
´´
p pvq α2 pvq ` β0 pvq α
p pvq β0 pvq c2 pt, vq dv
β
p pvq b1 pt, vq ` β
α2 pvq β0 pvq b1 pt, vq `
żT ´
¯¯¯
´
´
p pvq α2 pvq ` β0 pvq α
p pvq α1 pvq ` ω α2 pvq β0 pvq ´ β
´
p pvq
b1 pt, vq dv
α
p pvq ` α1 pvq β0 pvq ´ β
t
ˆ
˙˙
ˆ
żT
1
1p
p pvq β0 pvq
β pvq2 ` ω
β0 pvq2 ´ β
c2 pt, vq dv
´
2
2
t
żT ´
´
¯¯
p pvq b1 pt, vq dv
´
p1 ´ β pvqq α
p pvq ` α pvq β0 pvq ´ β
t
1
´
2
żT ´
m
ψ2t
t
´
¯¯
p pvq2 ` ωβ0 pvq β0 pvq ´ 2β
p pvq c2 pt, vq dv
β
“
´
“
m
ψ3t
“
´
´
´
t
ˆ żT
˙
żT
p pvq2 b1 pt, vq2 dv ` ω ´
p pvq b1 pt, vq2 dv
β
β0 pvq2 b1 pt, vq2 dv ` 2
β0 pvq β
żT ˆ
t
żT ˆ
t
`ω
“
´
żT
żT
t
żT ´
t
żT
t
t
ˆ
p pvq ´ 1 β0 pvq
p pvq2 ´ 2ωβ0 pvq β
β
2
˙˙
t
b1 pt, vq2 dv
˙
˙
żT ˆ
1p
1
p pvq γ
β
p pvq b1 pt, vq ` β
pvq2 b2 pt, vq dv ´ ω
γ0 pvq β0 pvq b1 pt, vq ` β0 pvq2 b2 pt, vq dv
2
2
t
´´
¯
¯
p pvq ` γ̌ pvq β0 pvq b1 pt, vq ` β0 pvq β
p pvq b2 pt, vq dv
γ0 pvq β
´
¯
¯
p pvq γ
p pvq ´ β0 pvq γ0 pvq ´ ωγ̌ pvq β0 pvq b1 pt, vq dv
β
p pvq ´ ω β
ˆ
ˆ
˙˙
1p
p pvq ´ 1 β0 pvq
β pvq2 ´ ωβ0 pvq β
b2 pt, vq dv
2
2
and substituting in the equations for αh , β h , γ h gives,
αh ptq
σD
“
´
żT ´
´
¯¯
p pvq b1 pt, vq dv
p1 ´ β pvqq α
p pvq ` α pvq β0 pvq ´ β
t
1
´
2
β h ptq
“´
σD
γ h ptq
σD
“
żT ´
t
żTˆ
t
´
¯¯
p pvq2 ` ωβ0 pvq β0 pvq ´ 2β
p pvq c2 pt, vq dv
β
ˆ
˙˙
p pvq2 ´ 2ωβ0 pvq β
p pvq ´ 1 β0 pvq
β
b1 pt, vq2 dv
2
żT ´
¯
¯
´
p pvq ´ β0 pvq γ0 pvq ´ ωγ̌ pvq β0 pvq b1 pt, vq dv
p pvq γ
β
p pvq ´ ω β
t
ˆ
˙˙
żT ˆ
1p
p pvq ´ 1 β0 pvq
´
β pvq2 ´ ωβ0 pvq β
b2 pt, vq dv.
2
2
t
(30)
(31)
´
(32)
where α pvq “ α1 pvq ` ωα2 pvq. The system (30)-(32) can be solved sequentially. As will be shown next, σtS depends
`
˘
only on β h . Hence, (31) is autonomous and determines β h . The coefficients αh , γ h follow from (30) and (32).
D,Z
m
Lemma 34 Suppose that Fp¨q
“ Fp¨q
. If the conjectures in (29) are satisfied, the equilibrium stock price is St “
p
p
p
A ptq Z` B ptq Dt `F ptq where,
p ptq “ B ptq B h ptq ,
B
B ptq “
ˆ
H pT q
H ptq
p “ λ pt, T q ` σ D
Aptq
ˆż T
t
˙ω ˆ
M pT q
M ptq
˙1´ω
,
B h ptq “ eσ
´
¯ ˙
p
p psq α
B
p psq ` βpsqλpt,
sq ds
45
D şT h
t β pvqdv
` D ˘2 ż T
żT
p psq γ
p ptq µD pT ´ tq ´ σ
p psq2 ds ` σ D
B
p psq ds ´ ω n Ip ptq µφ
Fp ptq “ B
B
Γ
t
t
` ˘2
żT
ω i σ D ps ´ tq
D
p
p
p
.
I ptq “ λ pt, T q ` σ
B psq β psq λ pt, sq ds,
λ pt, sq “
M ptq
t
p ptqσ D , a function of time.
The stock price volatility equals σtS “ B
Proof of Lemma 34. Straightforward calculations give,
ˇ

„
żT
ˇ
”
ı
ˇ m
D
D
D
D ˇ D,Z
ˇ
St “ E DT ´
F
µS
ds
s
ˇ t “ Dt ` µ pT ´ tq ` σ E WT ´ Wt ˇ Ft
t
ż
żT ”
´
¯ˇ
ı
1 T ´ S ¯2
ˇ
p
σs ds `
´
E σsS α
p psqZ ` βpsqD
ppsq ˇ FtD,Z
s `γ
Γ t
t
ˆż T
˙
ż
żT
1 T ´ S ¯2
D
Dt ` µ pT ´ tq ´
σs ds `
σsS γ
ppsqds `
σsS α
p psqds Z
Γ t
t
t
ˇ
”
ı żT
”
ı
Z,D
D
D
Dˇ
Sp
D,Z
`σ E WT ´ Wt ˇ Ft
`
σs βpsqE Ds | Ft
ds
“
t
”
p 0 pt, T q “
where G
E
ˇ
´
¯
”
ı żT
”
ı
p psq E Ds |FtD,Z ds
p 0 pt, T q Z ` F σ S , t ` σ D E WTD ´ WtD ˇˇ FtD,Z `
σsS β
Dt ` G
t
şT
t
„ż T
`
˘
şT
şT ` ˘2
p psq ds. Moreover,
σsS α
p psq ds and F σ S , t “ µD pT ´ tq ´ p1{Γq t σsS ds ` t σsS γ
σ
D
t
ˇ
ˇ
dWsD ˇˇ FtD,Z
”
ı
E Ds |FtD,Z
“
“
2
where λ pt, sq “
ω i pσ D q ps´tq
,
M ptq
żT
t
St
“
“
where,
“
“
´
´
¯
¯
λ pt, T q Z ´ ω i Dt ` µD pT ´ tq ´ ω n µφ
´
¯
λ pt, T q Z ´ ω i λ pt, T q Dt ´ λ pt, T q ω i µD pT ´ tq ` ω n µφ
´
´
¯
¯
Dt ` µD pT ´ tq ` λ pt, sq Z ´ ω i Dt ` µD pT ´ tq ´ ω n µφ
´
¯´
¯
Dt ` µD pT ´ tq 1 ´ ω i λ pt, sq ` λ pt, sq Z ´ ω n λ pt, sq µφ
so that,
”
ı
´
¯
´
¯
p
p 1 pt, T q Dt ` µD pT ´ tq ` G
p 2 pt, T q Z ´ ω n µφ
σsS βpsqE
Ds |FtD,Z ds “ G
p 1 pt, T q “
G
Hence,

żT
t
´
¯
p
σsS βpsq
1 ´ ω i λpt, sq ds,
G2 pt, T q “
żT
t
p
σsS βpsqλpt,
sqds.
´
¯
p 0 pt, T q Z ` F σ S , t ` λ pt, T q Z ´ ω i λ pt, T q Dt
Dt ` G
´
¯
´
¯
´
¯
´λ pt, T q ω i µD pT ´ tq ` ω n µφ ` G1 pt, T q Dt ` µD pT ´ tq ` G2 pt, T q Z ´ ω n µφ
´
¯
¯
´
¯
´
p 0 pt, T q ` λ pt, T q ` G
p 2 pt, T q Z ` F σ S , t
p 1 pt, T q Dt ` G
1 ´ ω i λ pt, T q ` G
´
¯
´
¯
p 1 pt, T q µD pT ´ tq ´ λ pt, T q ` G
p 2 pt, T q ω n µφ ” AptqZ
p
p
p
` ´ω i λ pt, T q ` G
` BptqD
t ` F ptq
p “G
p 0 pt, T q ` λ pt, T q ` G
p 2 pt, T q “ λ pt, T q `
Aptq
żT
p
p 1 pt, T q “ 1 ´ ω i λ pt, T q `
Bptq
“ 1 ´ ω i λ pt, T q ` G
t
´
¯
p
σsS α
p psq ` βpsqλpt,
sq ds
żT
t
´
¯
p
σsS βpsq
1 ´ ω i λpt, sq ds
ˆ
˙
żT
¯
´
¯ ´
p ptq ´
p ptq ´ 1 µD pT ´ tq ´ B
σsS α
p psq ds ω n µφ .
Fp ptq “ F σ S , t ` B
t
46
p ptq σ D . The volatility coefficient is deterministic as conjectured. This
An application of Ito’s lemma gives σtS “ B
validates the construction of the equilibrium stock price to this stage. Substituting in the coefficients above yields,
p “ λ pt, T q ` σ D
Aptq
żT
t
´
¯
p
p psq α
B
p psq ` βpsqλpt,
sq ds,
p
Bptq
“ 1 ´ ω i λ pt, T q ` σ D
´
¯ ´
¯
p ptq σ D , t ` B
p ptq ´ 1 µD pT ´ tq ´ Ip ptq ω n µφ ,
Fp ptq “ F B
żT
t
´
¯
p
p psq βpsq
B
1 ´ ω i λpt, sq ds
p ptq ´ σ D
Ip ptq “ B
żT
t
p psq α
B
p psq ds.
`
˘
` ˘2 şT
ş
p psq2 ds ` σ D T B
p psq γ
Inserting F B ptq σ D , t “ µD pT ´ tq ´ Γ1 σ D
B
p psq ds in the last coefficient and collecting
t
t
terms leads to,
` D ˘2 ż T
żT
σ
D
2
D
p
p
p psq γ
p
F ptq “ B ptq µ pT ´ tq ´
B
p psq ds ´ Ip ptq ω n µφ
B psq ds ` σ
Γ
t
t
Ip ptq “ λ pt, T q ` σ D
In these expressions, with ω “ ω i ` ω n ,
p ptq “ β ptq ` β h ptq ,
β
α
p ptq “ α ptq ` αh ptq ,
γ ptq “ ´ω
żT
t
p psq λ pt, sq ds.
p psq β
B
γ
p ptq “ γ ptq ` γ h ptq ,
`
˘
1 ´ κt ω i µD pT ´ tq ´ ω n κt µφ D
σ ,
H ptq
α ptq “
λ pt, sq “
Equilibrium exists if the backward Volterra equation,
p
Bptq
“ 1 ´ ω i λ pt, T q ` σ D
ˆż T
t
1 ´ κt ω D
σ ,
H ptq
β ptq “ ´ω
` ˘2
ω i σ D ps ´ tq
.
M ptq
´
¯ ˙
p
p psq βpsq
B
1 ´ ω i λ pt, sq ds ,
p p¨q has a solution. This issue is addressed in the next lemma.
for the coefficient B
1 ´ κt ω i D
σ
H ptq
p pT q “ 1
B
(33)
´
¯ω ´
¯1´ω
M pT q
p ptq “ B ptq B h ptq where B ptq “ HpT q
Lemma 35 The unique solution of (33) is B
with M ptq “
Hptq
M ptq
´
¯
` i ˘2
` φ ˘2
ş
n 2
i
n
h
D T h
p
ω H ptq ` pω q σ
, ω “ ω ` ω and B ptq “ exp σ t β pvq dv . Moreover, B ptq ą 0 for t P r0, T s.
` ˘2
` ˘2
Proof of Lemma 35. With M ptq “ ω i H ptq ` pω n q2 σ φ , note that,
` i ˘2 ` D ˘2
` i ˘2 ` ζ ˘2
` ˘2
ω
σ
pT ´ tq
ω
σ
` pω n q2 σ φ
M pT q
1 ´ ω λ pt, T q “ 1 ´
“
”
M ptq
M ptq
M ptq
´
¯
` ˘2
` i ˘2 ` D ˘2
` ˘2
σ
pT ´ sq ` σ ζ
ω
` pω n q2 σ φ
M psq
i
“
.
1 ´ ω λpt, sq “
M ptq
M ptq
i
p
Substituting in (33) and using the change of variables Cptq “ BptqM
ptq leads to,
p
Bptq
“
ðñ
ˆż T
ˆż T
˙
´
¯ ˙ M pT q
M psq
i
D
p
p
p
p
1 ´ ω λ pt, T q ` σ
B psq βpsq 1 ´ ω λpt, sq ds “
`σ
ds
B psq βpsq
M ptq
M ptq
t
t
ˆż T
˙
˙
ˆż T
p
p
p psq βpsqM
p
C psq βpsqds
B
psq ds ðñ Cptq “ M pT q ` σ D
BptqM
ptq “ M pT q ` σ D
i
D
t
t
p
subject to ´the boundary¯ condition CpT qM pT q. Equivalently, dC ptq “ ´σ D C ptq βptqdt.
The solution is C ptq “
şT
p
p ptq “ β ptq ` β h ptq,
M pT q exp σ D t βpsqds
. Substituting, β
¯
ω
ω ´
1 ´ κt ω i σ D “ ´
β ptq “ ´
H ptq
H ptq
˜
¸
` i ˘2 ¸
` i ˘2
ω
ω H ptq
1
D
D
´
1´
σ “ ´ωσ
M ptq
H ptq
M ptq
˜
47
and performing the integration,
˙
ˆ
˙˙˙
ˆ
˙ω ˆ
˙´ω
ˆ ˆ ˆ
M pT q
H pT q
M pT q
H pT q
B h ptq .
´ log
B h ptq “ M pT q
C ptq “ M pT q exp ω log
H ptq
M ptq
H ptq
M ptq
Substituting Cptq “ BptqM ptq and rearranging leads to the formula stated.
p To validate the construction of hedging demands and equilibrium,
p depends on β.
Under the conjectures in (29), B
p This is done next.
conditional on the information structure, it remains to show the existence of β.
Lemma 36 The coefficient β h solves the Riccati equation,
h
β9 ptq “ κ0 ptq ` κ1 ptq β h ptq ´ σ D β h ptq2 ;
β h pT q “ 0
ˆ
˙
´
¯
1´ω
κ0 ptq ”
β ptq2 σ D ,
κ1 ptq ” ´2σ D β ptq ` β D ptq
ω
(34)
(35)
where β ptq is defined in (5). A unique solution exists.
r ptq yields,
Proof of Lemma 36. Differentiating (31) relative to time and using Bt b1 pt, vq2 “ ´2σ D b1 pt, vq2 β
ˆ
˙
h
h
β9 ptq
p ptq2 ´ 2ωβ0 ptq β
p ptq ´ 1 β0 ptq ´ 2σ D β ptq β
r ptq
“
β
D
σ
2
σD
r “ β ` βh ` βD , β
p “ β ` β h and β ptq “ ωβ0 ptq gives,
with boundary condition β h pT q “ 0. Using the definitions β
` 1´ω ˘
`
˘
h
after simplifications, β9 ptq “
β ptq2 σ D ´2σ D β ptq ` β D ptq β h ptq ´σ D β h ptq2 . Defining κ0 , κ1 as in (35)
ω
proves (34). Existence of a unique solution follows from continuity of the ODE and the fact (see Technical Appendix)
‰
` şs
˘
`
şT
that β h ptq P β ptq , 0 where β h ptq ” ´ t exp ´ t k1 puq du k0 psq ds ă 0.
` h h˘
The next lemma gives closed form expressions for α , γ .
`
˘
Lemma 37 The coefficients αh , γ h are given by,
αh ptq “ ´
where, with K pt, T q ”
żT
t
ˆ żs
˙
exp ´
κ2 pvq dv κ3 psq ds,
t
şT ´
t
γ h ptq “ ´
β h pvq2 `
κ0 pvq
σD
¯
żT
t
ˆ żs
˙
exp ´
κ4 pvq dv κ5 psq ds
b1 pt, vq2 dv,
´
¯
´ ¯2
r ptq σ D ` σ D K pt, T q
κ2 ptq ” 1 ´ β ptq ´ β
ˆ
ˆ
˙
˙
´ ¯2
´
¯
1´ω
p ptq σ D ` σ D K pt, T q α ptq ` αD ptq
κ3 ptq ” α ptq 1 ` β ptq
´β
ω
´
¯
κ4 ptq ” ´ β ptq ` β D ptq σ D ` σ D K pt, T q
ˆ
ˆ
˙
ˆ
˙˙
1´ω
σS
σS
κ5 ptq ” β ptq
γ ptq ` t σ D ` σ D K pt, T q µD ` σ D γ ptq ` γ D ptq ´ t
.
ω
Γ
Γ
Proof of Lemma 37.
(36)
t
(37)
(38)
(39)
(40)
r ptq and
As α ptq “ α1 ptq ` ωα2 ptq, b1 pt, tq “ 1, c2 pt, tq “ 0, Bt b1 pt, vq “ ´σ D b1 pt, vq β
48
r ptq, it follows from (30) that,
Bt c2 pt, vq “ ´2σ D b1 pt, vq2 α
r ptq ´σ D c2 pt, vq β
α9 h ptq
σD
“
“
´
¯
h
p ptq ´ σ D α ptq β
r ptq
p1 ´ β ptqq α
p ptq ` α ptq β0 ptq ´ β
σD
ˆ
˙˙
żT ˆ
1
1p
p pvq
β pvq2 ` ωβ0 pvq
β0 pvq ´ β
b1 pt, vq2 dvr
α ptq
`2σ D
2
2
t
ˆ
˙˙
˙
ˆ
żT ˆ
1p
1
p pvq
r ptq ` 2σ D
β pvq2 ` ωβ0 pvq
β0 pvq ´ β
b1 pt, vq2 dv αh ptq
1 ´ β ptq ´ β
2
2
t
´
¯
p ptq
`α ptq 1 ´ β ptq ` β0 ptq ´ β
ˆ
˙˙
żT ˆ
´
¯
1p
1
p pvq
`2σ D
β pvq2 ` ωβ0 pvq
β0 pvq ´ β
b1 pt, vq2 dv α ptq ` αD ptq
2
2
t
with boundary condition αh pT q “ 0. Expression (36) follows because,
ˆ
ˆ
˙˙
ˆ
´
¯2
´
¯˙
1p
1
β pvq
p pvq
2
β pvq2 ` ωβ0 pvq
β0 pvq ´ β
“
β pvq ` β h pvq ` β pvq
´ 2 β pvq ` β h pvq
2
2
ω
˙
ˆ
κ0 pvq
1
´ 1 “ β h pvq2 `
“ β h pvq2 ` β pvq2
ω
σD
where κ0 pvq is given in (35).
`
`
˘˘
Using Bt b2 pt, vq “ ´σ D b2 pt, vq β̃ ptq´2b1 pt, vq2 µD ` σ D γ ptq ` γ D ptq ´ σtS {Γ ´2σ D b1 pt, vq2 γ h ptq and b1 pt, tq “
b2 pt, tq “ 0, γ̌ ptq “ γ
p ptq ´ σtS {Γ, ωβ0 ptq “ β ptq, ωγ0 ptq “ γ ptq and differentiating the integral equation gives,
γ9 h ptq
σD
“
“
ˆ
˙
¯
´
S
h
p ptq ´ β0 ptq γ0 ptq ` σt β ptq ´ σ D γ ptq β
r ptq
β h ptq γ
p ptq ´ ω β
Γ
σD
ˆ
˙˙
żT ˆ
´
´
¯¯
1p
p pvq ´ 1 β0 pvq
β pvq2 ´ ωβ0 pvq β
b1 pt, vq2 dv µD ` σ D γ ptq ` γ D ptq ´ σtS {Γ ` γ h ptq
`2
2
2
t
ˆ
˙˙
˙
ˆ
żT ˆ
1
1
2
2
h
p
p
r
β pvq ´ ωβ0 pvq β pvq ´ β0 pvq
b1 pt, vq dv γ h ptq
β ptq ´ β ptq ` 2
2
2
t
ˆ
˙
¯
´
S
p ptq ´ β0 ptq γ0 ptq ` σt β ptq
` β h ptq γ ptq ´ ω β
Γ
ˆ
˙˙
żT ˆ
´
´
¯¯
1
1p
p pvq ´ β0 pvq
β pvq2 ´ ωβ0 pvq β
b1 pt, vq2 dv µD ` σ D γ ptq ` γ D ptq ´ σtS {Γ .
`2
2
2
t
The expressions for the coefficients in the Lemma and the representation of γ h in (36) follow.
D,Z
m
Proof of Proposition 4. Lemmas 29-37 establish that an equilibrium exists under the assumption Fp¨q
“ Fp¨q
.
D,Z
m
To complete the proof of existence, it remains to verify that the endogenous public filtration is indeed Fp¨q “ Fp¨q .
To this end, use the relations,
m
m
m
hm
t “ ψ1 ptq Z ` ψ2 ptq Dt ` ψ3 ptq ,
θtm “
Θt pZ; ωq “ α ptq Z ` β ptq Dt ` γ ptq ,
u
hm
t “ ht ` ht pZ; ωq
σtS
D
´ Θt pZ; ωq ´ hm
t pZ; ωq σ
Γ
to write the residual demand as,
Nta σtS
Γ
“
“
“
“
´
¯
´
¯
G|m
G|m
ω i θtm ` θt
pGq ` phut ` ht pG; ωqq σ D ` ω n θtm ` θt
pφq ` phut ` ht pφ; ωqq σ D
ˆ S
˙
σt
D
ω
` Θt pZ; ωq ` pωhut ` ht pZ; ωqq σ D
´ Θt pZ; ωq ´ hm
t pZ; ωq σ
Γ
´
¯
σS
ω t ` p1 ´ ωq Θt pZ; ωq ` ht pZ; ωq σ D
Γ
´´
¯
´
¯
¯
σtS
` p1 ´ ωq α ptq ` ψ1i ptq σ D Z ` β ptq ` ψ2i ptq σ D Dt ` γ ptq ` ψ3i ptq σ D .
ω
Γ
49
`
˘
As Nta , σtS and Dt are observed, the product α ptq ` ψ1i ptq σ D Z is known. If α p0q ` ψ1i p0q σ D ‰ 0, the endogenous
D,Z
m
signal Z is revealed. If α p0q ` ψ1i p0q σ D “ 0, then α ptq ` ψ1i ptq σ D ‰ 0 for t “ 0` . In both cases Fp¨q
“ Fp¨q
and the
D,Z
m
m
i
candidate equilibrium constructed is a rational expectations equilibrium. Moreover, as Fp¨q “ Fp¨q Ă Fp¨q
“ Fp¨q
“
D,G
Fp¨q , the equilibrium is a NREE.
Proof of Remark 5. Fix ω u and let ω i Ñ 0 and ω n Ñ 1 ´ ω u “ ω. Then,
αsi ptq “
σD
,
H ptq
β si ptq “ ´ω
σD
,
H ptq
κsi
t “ 0,
γ si ptq “ ´ω
´ ¯2
´ ¯2
H ptq “ σ D pT ´ tq ` σ ζ ,
µD pT ´ tq D
σ ,
H ptq
´ ¯2
M si ptq “ ω 2 σ φ
αD,si ptq “ β D,si ptq “ γ D,si ptq “ 0 ùñ α
r si ptq “ α
p si ptq ,
α
p si ptq “ αsi ptq ` αh,si ptq ,
β9
αh,si ptq “ ´
κsi
0 ptq “
ˆ
1´ω
ω
˙
żT
t
h,si
κsi
0
si
r si ptq “ β
p si ptq ,
β
p ptq “ β si ptq ` β h,si ptq ,
β
κsi
1
h,si
λsi pt, sq “ 0, s P rt, T s
D
h,si
(42)
γ
rsi ptq “ γ
psi ptq
γ
psi ptq “ γ si ptq ` γ h,si ptq
2
h,si
ptq “
ptq `
ptq β
ptq ´ σ β
ptq ;
β
pT q “ 0
ˆ żs
˙
ˆ żs
˙
żT
si
si
exp ´
κsi
γ h,si ptq “ ´
exp ´
κsi
2 pvq dv κ3 psq ds,
4 pvq dv κ5 psq ds
t
β si ptq2 σ D ,
t
(41)
(43)
(44)
(45)
(46)
t
´
¯
´ ¯2
si
D
D
p si
κsi
K si pt, T q (47)
2 ptq “ 1 ´ β ptq ´ β ptq σ ` σ
D si
κsi
1 ptq “ ´2σ β ptq ,
ˆ
ˆ
˙
˙
´ ¯2
si
1´ω
si
si
p
κsi
ptq
“
α
ptq
1
`
β
ptq
´
β
ptq
σ D ` σ D K si pt, T q αsi ptq
3
ω
(48)
si
D
κsi
` σ D K si pt, T q
4 ptq “ ´β ptq σ
ˆ
˙
ˆ
˙˙
σtS,si
σtS,si
1 ´ ω si
si
D
D
si
D
D
si
si
γ ptq `
γ ptq ´
σ ` σ K pt, T q µ ` σ
(49)
κ5 ptq “ β ptq
ω
Γ
Γ
¯
´
şT
κsi pvq
2
si
r si
r
bsi
where K si pt, T q ” t β h,si pvq2 ` 0σ D
1 pt, vq dv with b1 pt, vq as in Lemma 28, but with β ptq instead of β ptq.
ˆ
The stock price becomes Stsi “ Âsi ptq Z si ` B̂ si ptq Dt ` F̂ si ptq where Z si “ ωφ and,
˙ω
ˆ
D şT
H pT q
, B h,si ptq “ eσ t
B̂ si ptq “ B si ptq B h,si ptq , B si ptq “
H ptq
Âsi ptq “ σ D
ˆż T
t
F̂ si ptq “ B̂ si ptq µD pT ´ tq ´
`
˙
B̂ si psq α̂si psq ds ,
σD
Γ
˘2 ż T
t
Iˆsi ptq “ 0
B̂ si psq2 ds ` σ D
`
˘
The pair S si , D in the limit economy is uninformative.
If in addition ω n Ñ 0 (i.e., ω u Ñ 1), then M si ptq “ Z si “ 0 and,
αsi ptq “
σD
,
H ptq
si
β si ptq “ γ si ptq “ κsi
t “ λ pt, sq “ 0,
β h,si pvqdv
żT
t
(51)
B̂ si psq γ̂ si psq ds.
β si ptq
σD
“´
,
ω
H ptq
γ si ptq
µD pT ´ tq D
“´
σ
ω
H ptq
´
¯
´ ¯2
si
D
p si
κsi
κsi
` σ D K si pt, T q
0 ptq “ κ1 ptq “ 0,
2 ptq “ 1 ´ β ptq σ
ˆ
˙
´ ¯2
β si ptq p si
si
κsi
ptq
“
α
ptq
1
`
´
β
ptq
σ D ` σ D K si pt, T q αsi ptq
3
ω
ˆ
S,si ˙
si
D
si
si
D
si
D
D σt
κ4 ptq “ σ K pt, T q ,
κ5 ptq “ σ K pt, T q µ ´ σ
Γ
αD,si ptq “ β D,si ptq “ γ D,si ptq “ 0 ùñ α
r si ptq “ α
p si ptq ,
r si ptq “ β
p si ptq ,
β
γ
rsi ptq “ γ
psi ptq
p si ptq “ β h,si ptq ,
α
p si ptq “ αsi ptq ` αh,si ptq ,
β
γ
psi ptq “ γ h,si ptq
ˆ żs
˙
ˆ żs
˙
żT
żT
si
si
αh,si ptq “ ´
exp ´
κsi
γ h,si ptq “ ´
exp ´
κsi
2 pvq dv κ3 psq ds,
4 pvq dv κ5 psq ds.
t
t
t
50
(50)
t
(52)
h,si
r si ptq “ β
p si ptq “
Thus, β9
ptq “ ´σ D β h,si ptq2 , β h,si pT q “ 0, which has solution β h,si “ 0. It follows ´that β
¯
si
β ptq
si
h,si
D
si
si
β h,si ptq “ K si pt, T q “ κsi
ptq “ 0 and κsi
σ D . Finally,
4 ptq “ κ5 ptq “ γ
2 ptq “ σ , κ3 ptq “ α ptq 1 `
ω
˙
ˆ
2
D
pσ q
pT ´ tq and
B̂ si ptq “ B si ptq “ B h,si ptq “ 1 and Z si “ 0 so that Stsi,0 “ Dt ` F̂ si,0 ptq with F̂ si,0 ptq “ µD ´ Γ
σtS,si,0 “ σ D for all t P r0, T s.
Note that 0 ă B ptq ă B si ptq ă 1 for t ă T and B pT q “ B si pT q “ 1. Moreover,
ˆ
˙
´
¯
h
1´ω
β9 ptq “
β ptq2 σ D ´ 2σ D β ptq ` β D ptq β h ptq ´ σ D β h ptq2 ;
β h pT q “ 0
ω
ˆ
˙
h,si
1´ω
β9
ptq “
β si ptq2 σ D ´ 2σ D β si ptq β h,si ptq ´ σ D β h,si ptq2 ;
β h,si pT q “ 0
ω
˜
` ˘2 ¸ D
´ ¯2 σ D
pω n q2 σ φ
σ
1 ´ κt ω i D ´ i ¯2 1
σ ´ ω
σ D “ ´ω
´ ωi
β ptq ` β D ptq “ ´ω
H ptq
M ptq
H ptq
M ptq
M ptq
´
¯
´ ¯2 σ D
´ ¯2 σ D
´ ¯2 σ D
β si ptq ´ β ptq ` β D ptq “ ´ω ω i
` ωi
“ ωi
p1 ´ ωq ą 0
M ptq
M ptq
M ptq
˜
` ˘2 ¸2 ˆ D ˙2
˙2
˙
ˆ
˙ˆ
ˆ
pω n q2 σ φ
σ
1 ´ κt ω i D
1´ω
1´ω
2
σ
β ptq “
´ω
“ ω p1 ´ ωq
U”
ω
ω
H ptq
M ptq
H ptq
ˆ
˙
ˆ
˙ˆ
˙2
ˆ D ˙2
1´ω
σD
1´ω
σ
U si ”
β si ptq2 “
´ω
“ ω p1 ´ ωq
ω
ω
H ptq
H ptq
h,si
h
so that ∆ ” U si ´ U ą 0. Thus, β9
pT q ´ β9 pT q “ ∆ ą 0. Moreover, if β h,si ptq “ β h ptq at any t ă T , then
´
´
¯¯
h,si
h
β9
ptq ´ β9 ptq “ ∆σ D ´ 2σ D β si ptq ´ β ptq ` β D ptq β h,si ptq ą 0.
Thus, β h,si ptq ă β h ptq ă 0 for all t P r0, T s and β h,si pT q ´ β h pT q “ 0. Therefore B h,si ptq ă B h ptq. As
´ 0 ă B ptq
¯ ă
B si ptq ă 1, the overall effect on price volatility is ambiguous: σtS,si ż σtS for t ă T . However, max σtS,si , σtS ă
σtS,si,0 “ σ D for t ă T and limtÑT σtS “ limtÑT σtS,si “ limtÑT σtS,si,0 “ σ D .
Proof of Remark 6. Fix ω n and let ω u Ñ 0 and ω i Ñ 1 ´ ω n . Then, ω Ñ 1 and,
αsu ptq “
γ
su
1 ´ κsu
t
σD ,
H ptq
i
1 ´ κsu
t ω
σD ,
H ptq
ω i H ptq
M su ptq
κsu
t “
`
˘ D
` ˘2
i
φ
1 ´ κsu
µ pT ´ tq ´ ω n κsu
ω i σ D ps ´ tq
t ω
t µ
D
su
ptq “ ´
σ ,
λ pt, sq “
, s P rt, T s
H ptq
M su ptq
´ ¯2
´ ¯2
´ ¯2
´ ¯2
H ptq “ σ D pT ´ tq ` σ ζ ,
M su ptq “ ω i H ptq ` pω n q2 σ φ
α
r su ptq “ α
p su ptq ` αD,su ptq ,
α
p su ptq “ αsu ptq ` αh,su ptq ,
αD,su ptq ”
αh,su ptq “ ´
κsu
0 ptq “ 0,
β su ptq “ ´
ωiσD
,
M su ptq
żT
t
r su ptq “ β
p su ptq ` β D,su ptq ,
β
p su ptq “ β su ptq ` β h,su ptq ,
β
β D,su ptq ” ´ω i αD,su ptq ,
γ
psu ptq “ γ su ptq ` γ h,su ptq
´
¯
γ D,su ptq ” ´αD,su ptq ω i µD pT ´ tq ` ω n µφ
t
(54)
(55)
γ
rsu ptq “ γ
psu ptq ` γ D,su ptq
h,su
h,su
β9
ptq “ κsu
ptq ´ σ D β h,su ptq2 ;
β h,su pT q “ 0
1 ptq β
ˆ żs
˙
ˆ żs
˙
żT
su
su
exp ´
κsu
γ h,su ptq “ ´
exp ´
κsu
2 pvq dv κ3 psq ds,
4 pvq dv κ5 psq ds
t
(53)
t
(56)
(57)
(58)
´
¯
´
¯
´ ¯2
D
su
r su ptq σ D ` σ D K su pt, T q
κsu
β su ptq ` β D,su ptq ,
κsu
ptq ´ β
1 ptq “ ´2σ
2 ptq “ 1 ´ β
´
¯
´ ¯2
´
¯
su
p su ptq σ D ` σ D K su pt, T q αsu ptq ` αD,su ptq
κsu
ptq 1 ´ β
3 ptq “ α
´
¯
su
κsu
ptq ` β D,su ptq σ D ` σ D K su pt, T q
4 ptq “ ´ β
51
su
κsu
ptq
5 ptq “ β
ˆ
ˆ
˙˙
σ S,su
σtS,su D
σ ` σ D K su pt, T q µD ` σ D γ su ptq ` γ D,su ptq ´ t
Γ
Γ
şT
2
su
r su ptq instead of β
r ptq. As the
where K su pt, T q ” t β h,su pvq2 bsu
1 pt, vq dv with b1 pt, vq as in Lemma 28, but with β
h,su
su
solution of (57) is β
ptq “ 0, it follows that K pt, T q “ 0 and,
´
¯
´
¯
D
su
r su ptq σ D
κsu
κsu
β su ptq ` β D,su ptq ,
κsu
ptq ´ β
(59)
0 ptq “ 0,
1 ptq “ ´2σ
2 ptq “ 1 ´ β
´
¯
su
p su ptq σ D ,
κsu
ptq 1 ´ β
3 ptq “ α
α
r su ptq “ α
p su ptq ` αD,su ptq ,
su
α
p
su
ptq “ α
´
¯
su
κsu
ptq ` β D,su ptq σ D ,
4 ptq “ ´ β
h,su
ptq ` α
r
β
su
ptq ,
p
ptq “ β
p
β
B̂ su ptq “ B su ptq ,
Âsu ptq “ λsu pt, T q ` σ D
F̂ su ptq “ B su ptq µD pT ´ tq ´
`
σD
Γ
su
ptq “ β
su
ptq ,
γ
p
su
B su psq2 ds ` σ D
Iˆsu ptq “ λsu pt, T q ` σ D
żT
t
B su psq β̂
żT
su
t
σtS,su D
σ
Γ
γ
rsu ptq “ γ
psu ptq ` γ D,su ptq
ptq “ γ
su
ptq ` γ
h,su
H pT q
, B h,su ptq “ 1
H ptq
´
¯ ˙
su
B su psq α̂su psq ` β̂ psqλsu pt, sq ds
t
t
ptq ` β D,su ptq ,
B su ptq “
ˆż T
˘2 ż T
su
su
κsu
ptq
5 ptq “ β
ptq
B su psq γ̂ su psq ds ´ ω n Iˆsu ptq µφ
psq λsu pt, sq ds.
(60)
(61)
(62)
(63)
(64)
(65)
(66)
If, in addition, ω n Ñ 0, then ω i Ñ 1, M su ptq “ H ptq, κsu
t “ 1 and,
` D ˘2
σ
ps ´ tq
, s P rt, T s
(67)
α ptq “ β ptq “ γ ptq “ 0,
λ pt, sq “
H ptq
´
¯
su
su
su
D D,su
D,su
r su ptq σ D , κsu
κsu
ptq , κsu
ptq σ D (68)
0 ptq “ κ3 ptq “ κ5 ptq “ 0, κ1 ptq “ ´2σ β
2 ptq “ 1 ´ β
4 ptq “ ´β
su
su
su
su
αh,su ptq “ γ h,su ptq “ 0,
α
r su ptq “ αD,su ptq ,
r
β
su
p
α
p su ptq “ β
su
ptq “ β D,su ptq ,
ptq “ γ
psu ptq “ 0
(69)
γ
rsu ptq “ γ D,su ptq
(70)
H pT q
, B h,su ptq “ 1,
Âsu ptq “ Iˆsu ptq “ λsu pt, T q
(71)
B̂ su ptq “ B su ptq , B su ptq “
H ptq
` D ˘2 ż T
żT
σ
F̂ su ptq “ B su ptq µD pT ´ tq ´
B su psq2 ds ` σ D
B su psq γ̂ su psq ds.
(72)
Γ
t
t
`
˘
As Z su,0 “ G, the pair D, S su,0 is fully revealing. Price volatilities in the different economies are σtS,su,0 “ σtS,su “
B̂ su ptq σ D “ B su ptq σ D ă B ptq σ D “ σtS ă σ D for t ă T . As t Ñ T, σtS,su,0 “ σtS,su Ñ σ D and σtS Ñ σ D .
Proof of Remark 7. Fix ω u and let ω i Ñ 1 ´ ω u and ω n Ñ 0. Then, ω Ñ ω i Ñ 1 ´ ω u and,
´ ¯2
M sn ptq “ ω i H ptq ,
sn
α
ptq “ β
sn
α
r sn ptq “ α
p sn ptq ` αD,sn ptq ,
sn
αD,sn ptq ”
αh,sn ptq “ ´
żT
t
α
p
i
h,sn
ptq “ α
D
ωσ
,
M sn ptq
ptq “ γ
sn
κsn
t “
ptq “ 0,
λ
ω i H ptq
,
M sn ptq
sn
i
κsn
t ω “ 1
` ˘2
ω i σ D ps ´ tq
pt, sq “
M sn ptq
r sn ptq “ β
p sn ptq ` β D,sn ptq ,
β
ptq ,
p sn ptq “ β h,sn ptq ,
β
β D,sn ptq ” ´ω i αD,sn ptq ,
(73)
γ
p
sn
γ
rsn ptq “ γ
psn ptq ` γ D,sn ptq
ptq “ γ
h,sn
ptq
γ D,sn ptq ” ´αD,sn ptq ω i µD pT ´ tq
h,sn
h,sn
β9
ptq “ κsn
ptq ´ σ D β h,sn ptq2 ;
β h,sn pT q “ 0
1 ptq β
ˆ żs
˙
ˆ żs
˙
żT
sn
sn
exp ´
κsn
γ h,sn ptq “ ´
exp ´
κsn
2 pvq dv κ3 psq ds,
4 pvq dv κ5 psq ds
t
t
52
t
(74)
(75)
(76)
(77)
(78)
(79)
´
¯
´ ¯2
D D,sn
r sn ptq σ D ` σ D K sn pt, T q
κsn
ptq ,
κsn
1 ptq ” ´2σ β
2 ptq ” 1 ´ β
´ ¯2
D
D,sn
κsn
K sn pt, T q αD ptq ,
κsn
ptq σ D ` σ D K sn pt, T q
3 ptq ” σ
4 ptq ” ´β
ˆ
ˆ
˙˙
σ S,sn
D
sn
κsn
pt, T q µD ` σ D γ ptq ` γ D ptq ´ t
5 ptq ” σ K
Γ
ş
T
2
2
sn
r sn ptq instead of β
r ptq. As the
where K sn pt, T q ” t β h,sn pvq bsn
1 pt, vq dv with b1 pt, vq as in Lemma 28, but with β
sn
sn
h,sn
D,sn
sn
p
r
solution of (78) is β
ptq “ 0, it follows that β ptq “ 0, β ptq “ β
ptq and K pt, T q “ 0. Then,
´
¯
sn
sn
D D,sn
D,sn
D,sn
κsn
κsn
ptq , κsn
ptq σ D , κsn
ptq σ D
0 ptq “ κ3 ptq “ κ5 ptq “ 0,
1 ptq “ ´2σ β
2 ptq “ 1 ´ β
4 ptq “ ´β
(80)
p sn ptq “ γ
αh,sn ptq “ γ h,sn ptq “ 0,
α
p sn ptq “ β
psn ptq “ 0
(81)
κsn
0 ptq “ 0,
B̂ sn ptq “ B sn ptq ,
 ptq “ Iˆsn ptq “ λsn pt, T q ,
sn
F̂
B sn ptq “
sn
ptq “ B
sn
H pT q
,
H ptq
B h,sn ptq “ 1
` D ˘2 ż T
σ
B sn psq2 ds.
ptq µ pT ´ tq ´
Γ
t
D
(82)
(83)
The pair pD, S sn q, in the limit economy, is fully revealing because Z “ ω i G. The stock price volatility is σtS,sn “
B̂ sn ptq σ D “ B sn ptq σ D where B sn ptq ă B̂ ptq. Volatilities rank as σtS,sn ă σtS ă σtD for t ă T .
The proofs of Corollaries 8-10 follows from Lemma 38, given next. The proofs for Lemma 38 are straightforward,
but long and tedious. They are in a companion Technical Appendix.
Lemma 38 The following holds,
´ ¯2
´ ¯2 BH ptq
BM ptq
BH ptq
“ ´ σ D ă 0,
“ ωi
ă0
Bt
Bt
Bt
` ˘2
´ ¯2 M psq
ω i pω n q2 σ φ BH ptq
Bλ pt, sq
Bκt
“
ă 0,
“ ´ω i σ D
ă0
2
Bt
Bt
Bt
M ptq
M ptq2
˜
` ˘2 ¸
p1 ´ ωq ω i
ω
BH ptq
BB ptq
“ ´B ptq
`
ą0
Bt
H ptq
M ptq
Bt
` ˘2
pω n q2 σ φ ωκt ` p1 ´ κt ωq M ptq BH ptq D
Bα ptq
1
“´
σ ż 0 ðñ κ2t ž i
Bt
Bt
ωω
M ptq H ptq2
` ˘4
` ˘2
` ˘2
pω n q4 σ φ ` 2 ω i H ptq pω n q2 σ φ BH ptq D
Bβ ptq
“ω
σ ă0
Bt
Bt
M ptq2 H ptq2
ˆ
¯
´ ¯2 ´
¯
´ ¯2 ˙
Bκt ´ i D
Bγ ptq
ωσ D
n φ
n
φ
D
i
D
“
ω
µ
pT
´
tq
´
ω
µ
H
ptq
´
ω
κ
µ
σ
`
1
´
κ
ω
µ
σζ
t
t
Bt
Bt
H ptq2
#
?
Bγ ptq {Bt ą 0 ðñ 0 ď H ptq ă H ptq`
´b ` b2 ´ 4ac
`
,
H
“
2a
Bγ ptq {Bt ă 0 ðñ H ` ă H ptq
˙
ˆ ´ ¯
´
´
¯
¯
´
´
¯
¯
´
¯2
´ ¯2
2
2
2
2
2
ωi
a “ s2 s σ D µφ ` σ φ µD , b “ ´2s2 σ φ µD σ ζ , c “ ´ σ φ
σ φ µD σ ζ , s “ n
ω
(84)
(85)
Proof of Proposition 11. Marginal utility is exp p´XTu {Γq “ y u ξTm where the shadow price of initial wealth y u
u
u
m
m
satisfies X0u “ ´Γ log py u q ´ ΓE0 rξTm log ξTm s, and therefore,
´ y u“ exp p´X0 {Γ ´ ¯E0 rξT log ξT sq. Interim expected
X0
u
u
m
m
utility of the public investor is U ” E0 ru pXT qs “ ´Γ exp ´ Γ ´ E0 rξT log ξT s . In the presence of private inforşT
şT ` ˘2
mation, θvm “ σvS {Γ ´ ϑv pZ|Dv q. It follows that ´E0 rξTm log ξTm s “ 2Γ1 2 0 σvS dv´ Γ1 0 σvS ϑv pZ|E0 rξvm Dv sq ` I u .
¯
´
p p0q Z ` B
p p0q D0 ` Fp p0q . In the absence of private information, θvm,ni “ σ D {Γ
Initial wealth is X0u “ N0u S0 “ N0u A
`
˘
“ ` ˘ˇ ‰
2
such that ´E r ξTm log ξTm | F0m s “ ´ 2Γ1 2 σ D T . The corresponding expression for the insider is E u XTi ˇ F0i “ ´Γ
´
¯
ˇ ‰
ˇ ‰
“
“ m
‰
“
Xi
exp ´ Γ0 ´ E ξTG log ξTG ˇ F0i . As E ξTG log ξTG ˇ G0 “ E0 ξt,T
log pξTm ηTx q |x“G “ E0 rξTm log ξTm s`E0 rξTm log ηTx s|x“G
53
“ `
˘ˇ ‰
`
˘
and N0u “ N0j “ 1, we have X0i “ N0i S0 “ N0u S0 “ X0u and E u XTi ˇ F0i “ U u exp I i pG|Zq . The informed is
always better off, i.e. U u ă 0 and I i pG|Zq ď 0. This inequality follows as by optimality of the informed investor’s
`
`
˘
˘
m
i
choice and Fp¨q
Ă Fp¨q
, necessarily, U i ě U u exp I i pG|Zq ´ 1 . This establishes the results announced.
´ ´ pu
¯¯
Proof of Proposition 12. Proposition 11 shows that U u “ U u,ni exp ´ ∆P Γ pZq ` ∆Tpu pZq
with
∆Pp u pZq ” S0 ´ S0ni “
∆Tpu pZq “
Hence,
1
∆Vp
´
2Γ2
Γ
żT
´
¯´
¯
p
p psq γ
p p0q ´ 1 D0 ` µD T ` A
p p0q Z ´ ∆V ` σ D
B
p psq ds ´ ω n Ip p0q µφ
B
Γ
0
żT
0
´ ¯2 ż T ´
¯
p pvq2 ´ 1 dv.
σtS ϑ pZ|E0 rξtm Dt sq dt ` I u pZq and ∆Vp ” σ D
B
0
∆U ”U u ´ U u,ni “
˜
˜ ˜
exp ´
∆Pp u pZq
` ∆Tpu pZq
Γ
¸¸
¸
´ 1 U u,ni
and, as U u,ni ď 0, ∆U ě 0 if and only if ∆Pp u pZq {Γ ` ∆Tpu pZq ě 0.
To study the limit behavior, note that limΓÑ8 γ
p psq “ γ ptq ` γ h pt; 8q ” γ
p pt; 8q where
γ h ps; 8q ” ´
κ5 pt; 8q ”
ˆ
β ptq
h
ˆ
1´ω
ω
˙
γ ptq `
h
żT ˆ
t
żT
s
ˆ żv
˙
exp ´
κ4 puq du κ5 pv; 8q dv
s
β h pvq2 `
κ0 pvq
σD
˙
´
´
¯¯˙
σD .
b1 pt, vq2 dv µD ` σ D γ ptq ` γ D ptq
Moreover, as γ psq ď γ ps; 8q uniformly in Γ for all s P r0, T s, Lebesgue’s dominated convergence theorem gives,
żT
¯´
¯
´
p p0q ´ 1 D0 ` µD T ` A
p p0q Z ` σ D
p psq γ
lim ∆Pp u pZq “ B
B
p ps; `8q ds ´ ω n Ip p0q µφ .
ΓÑ8
0
´
¯
pu
Hence, limΓÑ8 ´ ∆PΓ pzq “ 0. As θtm “ σtS {Γ ´ ϑt pZ|Dt q ,
«
ff
ˇ

„
˙2 ˇˇ
żT
żT ˆ
ˇ u
1
σtS
1
ˇ u
m
m
2
m
ˇ
θt ´
ϑt pZ|Dt q dtˇ F0 “ E ξT
dtˇ F0
I pZq “ E ξT
ˇ
2
2
Γ
0
0
u
ˇ
„
ˆ
ˆ
˙
˙
„
żT
żT
¯
ˇ u
1
u
u
m
m 2
m
m 2
p
ˇ
lim ´∆T pZq “ lim p´I pZqq “ lim ´ E ξT
pθv q dv
ď 0.
“ E0 lim ´ξT
pθt q dtˇ F0
ΓÑ8
ΓÑ8
ΓÑ8
ΓÑ8
2
0
0
şT
The third equality above follows from the Lebesgue dominated convergence theorem and ´ξTm 0 pθvm q2 dv ď 0 uni´ ş m
¯
¨
m
m
p pvq Dv ´ γ pvq ´ γ h pv; `8q ” θm
”
formly in Γ. But limΓÑ8 θvm “ ´p
α pvq Z ´ β
v and limΓÑ8 ξv “ E ´ 0 θ s dWs
´
v
m
ξ v (indeed, as θtm has finite variance uniformly in Γ for all t P r0, T s, the sequence θtm indexed by Γ is tight, so
´ m ¯2
şT
şT
şT
m
m şT
that limΓÑ8 0 θtm dWtm “ 0 θt dWtm when limΓÑ8 θtm “ θt ). Thus, limΓÑ8 ξTm 0 pθvm q2 dv “ ξ T 0 θv
dv and
´
¯
u
u,ni
p
p
limΓÑ8 ´ ∆T pZq ` ∆P pZq {Γ ď 0. Because limΓÑ8 U
“ ´8, it follows that limΓÑ8 ∆U “ `8.
For small risk tolerance,
γ pvq “ lim Γp
γ pvq “ lim Γγ h pvq “ ´
dγ h pvq ” lim Γr
ΓÑ0
ΓÑ0
dκ5 psq ” lim Γκ5 psq “
ΓÑ0
The price effect is,
ˆ
ΓÑ0
β psq ´ σ D
żT ˆ
s
β h pvq2 `
żT
v
ˆ żs
˙
exp ´
κ4 puq du dκ5 psq ds
κ0 pvq
σD
v
˙
˙
b1 pt, vq2 dv σsS σ D ă 0.
żT
¯´
¯
´
p p0q ´ 1 D0 ` µD T ` ΓA
p p0q Z ´ ∆Vp ` Γσ D
p psq γ
Γ∆P pZq “ Γ B
B
p psq ds ´ Γω n Ip p0q µφ
0
lim Γ∆P pZq “ ´∆Vp ` σ D
ΓÑ0
żT
0
´
¯
p pvq dγ h pv; 0q dv ” ´ ∆Vp ` ∆H
pS
B
54
where dγ h pv; 0q “ limΓÑ0 Γp
γ psq “ limΓÑ0 Γγ h psq ą 0 and
p S ” ´σ D
∆H
żT
0
p pvq dγ h pv; 0q dv ă 0
B
(86)
measures the impact of hedging on the price effect. To find the trading impact for small risk tolerance, note that
˜
` D ˘2 ¸
ż
¯
´
σ
T
1 T S
∆Vp
m
m,ni
m
u
m
u
p
´
” ´ H0,T
´ H0,T
σt ϑ pZ|E0 rξt Dt sq dt ` I pZq “ ´ H0,T ´
∆T pZq “
2
2
2Γ
Γ 0
2Γ
m
where H0,T
” ´ 12
şT
0
“
‰
r z θvm pzq2
E
“
‰
r z θvm pzq2 dv from Lemma 31 . Using (27)-(28),
E
“
“
“
“
«ˆ
˙2 ff
σvS
p
Ẽ
´γ
p pvq ´ α
p pvq z ´ β pvq Dv
Γ
˙2
˙
ˆ S
ˆ S
“ ‰
σv
σv
p pvq E
r z rDv s ` β
p pvq2 E
r z Dv2
´γ
p pvq ´ α
p pvq z ´ 2
´γ
p pvq ´ α
p pvq z β
Γ
Γ
ˆ S
˙2
˙
ˆ S
σv
σv
´γ
p pvq ´ 2
´γ
p pvq α
p pvq z ` α
p pvq2 z 2
Γ
Γ
ˆ S
˙
σv
p pvq pa1 p0, vq z ` b1 p0, vq D0 ` g1 p0, vqq
´2
´γ
p pvq ´ α
p pvq z β
Γ
`
˘
p pvq2 a2 p0, vq z ` b2 p0, vq D0 ` c2 p0, vq D0 z ` d2 p0, vq z 2 ` b1 p0, vq2 D02 ` g2 p0, vq
`β
˙2
˙
ˆ S
ˆ S
σv
σv
p pvq g1 p0, vq ` β
p pvq2 g2 p0, vq
´γ
p pvq ´ 2
´γ
p pvq β
Γ
Γ
˙´

„ ˆ S
¯
σv
p pvq a1 p0, vq ´ 2p
p pvq g1 p0, vq ´ β
p pvq2 a2 p0, vq z
´γ
p pvq
α
p pvq ` β
α pvq β
´ 2
Γ
´
¯
p pvq a1 p0, vq ` β
p pvq2 d2 p0, vq z 2
` α
p pvq2 ` 2p
α pvq β
„ ˆ S
˙

σv
2
p
p
´ 2
´γ
p pvq β pvq b1 p0, vq ´ β pvq b2 p0, vq D0
Γ
”
ı
p pvq b1 p0, vq ` β
p pvq2 c2 p0, vq D0 z ` β
p pvq2 b1 p0, vq2 D02 .
` 2p
α pvq β
z
`
˘
Using Γk psq ” σ D Γγ psq “ σ D Γr
γ psq ´ σsS and γ
r psq “ γ D psq ` γ
p psq gives,
dγ h psq
“
dg1 psq
”
dg2 psq
”
db2 psq
”
da2 psq
”
it follows that,
´
¯
´
¯
lim Γp
γ psq “ lim Γγ h psq ,
lim Γk psq “ σ D lim Γp
γ psq ´ σsS “ σ D dγ h psq ´ σsS
ΓÑ0
ΓÑ0
ΓÑ0
ΓÑ0
˙´
ˆ
żs
żs
¯
D
D
r puq du
β
dγ h pvq ` σvS dv
lim Γg1 p0, sq “ ´σ
exp σ
ΓÑ0
v
ż0s
´
¯
lim Γ2 g2 p0, sq “ 2σ D
b1 pv, sq dγ h pvq ´ σvS dg1 pvq dv
ΓÑ0
ż s0
´
¯
D
lim Γb2 p0, sq “ 2σ
b1 pv, sq dγ h pvq ´ σvS b1 p0, vq dv
ΓÑ0
ż0s
´´
¯
¯
D
lim Γa2 p0, sq “ 2σ
b1 pv, sq2 dγ h pvq ´ σvS a1 p0, vq ` α
r pvq dg1 pvq dv
ΓÑ0
0
¯
´
¯2
“
‰ ´
p pvq2 dg pvq
p pvq dg pvq ` β
r z θvm pzq2 “ σvS ´ dγ h pvq ´ 2 σvS ´ dγ h pvq β
lim Γ2 E
2
1
ΓÑ0
m
lim Γ2 H0,T
ΓÑ0
1
2
“
´
“
1
´
2
żT
0
żT
0
“
‰
r z θvm pzq2 dv
lim Γ2 E
ΓÑ0
ˆ´ ¯
˙
´
¯
2
p pvq ` β
p pvq2 dg2 pvq dv
σvS ´ 2σvS dγ h pvq ` dγ h pvq2 ´ 2 σvS ´ dγ h pvq dg1 pvq β
55
¯
´
¯
1´ p
m
m,ni
pT
“
lim Γ2 ∆Tpu pZq “ ´ lim Γ2 H0,T
´ H0,T
∆V ` ∆H
ΓÑ0
ΓÑ0
2
p T measures the effect of hedging on the trading impact,
where ∆H
Thus,
lim Γ
ΓÑ0
2
żT ´
¯
´
¯
p pvq ` β
p pvq2 dg2 pvq dv.
´2σvS dγ h pvq ` dγ h pvq2 ´ 2 σvS ´ dγ h pvq dg1 pvq β
pT ”
∆H
˜
∆Pp pZq
` ∆Tpu pZq
Γ
1 p
´ ∆H
2
(87)
0
¸
´
´
´
¯
¯
¯
p S ` 1 ∆Vp ` ∆H
p T “ ´ 1 ∆Vp ´ ∆H
p S ` 1 ∆H
p T ” ´ 1 ∆Vp ` ∆H
p
“ ´ ∆Vp ` ∆H
2
2
2
2
pS `
´∆H
”
1 pT
∆H ” σ D
2
żT
0
p pvq dγ h pvq dv
B
ż
¯
¯
´
1 T´
p pvq ` β
p pvq2 dg pvq dv.
´2σvS dγ h pvq ` dγ h pvq2 ´ 2 σvS ´ dγ h pvq dg1 pvq β
2
2 0
¯¯
¯
´
´
´
¯¯
¯
´
´ ´ p
p
´ 1 U u,ni and
Note that ∆U “ exp ´ ∆PΓpZq ` ∆Tpu pZq ´ 1 U u,ni “ exp 2Γ1 2 ∆Vp ` ∆H
`
U
u,ni
˜
` D ˘2 ¸
` D ˘2 ¸
σ
T
T
E rDT s
1 σ
X0u,ni
“ ´Γ exp ´
´
`
” ´Γ exp ´
Γ
2Γ2
Γ
2
Γ2
˜
` ˘2
because θvm,ni “ σ D ´
{Γ, N0u,ni “¯1 and X0u,ni “ S0ni “ E rDT s ´ σ D T {Γ. It follows that limΓÑ0 U u,ni “ ´8 and
p ˆ8. This completes the proof.
limΓÑ0 ∆U “ ´sgn ∆Vp ` ∆H
Proof of Corollary 13. As,
κ5 ptq
”
˙
„
żT ˆ
´
´
¯¯
κ0 pvq
1´ω
2
D
D
D
σD
b
pt,
vq
dv
µ
`
σ
γ ptq `
γ
ptq
`
γ
ptq
β h pvq2 `
β ptq
1
ω
σD
t
˙

„
żT ˆ
κ0 pvq
1
1
h
2
2
D
β pvq `
b1 pt, vq dv σtS σ D ” κ5,0 ptq ` κ5,1 ptq
β ptq ´ σ
`
D
Γ
σ
Γ
t
we can write, using (36),
γ h ptq “ γ0h ptq ` γ1h ptq
1
,
Γ
γih ptq “ ´
żT
t
ˆ żs
˙
exp ´
κ4 pvq dv κ5,i psq ds, i “ 0, 1.
t
`
˘
r ptq ´ σtS {Γ “ γ ptq ` γ h ptq ` γ D ptq ´ σtS {Γ and
Likewise, using γ ptq ” γ
k ptq
ˆ
˙
σS
µD ` σ D γ ptq “ µD ` σ D γ ptq ` γ h ptq ` γ D ptq ´ t
Γ
˙
ˆ
h
S
k1 ptq
γ ptq ´ σt
” k0 ptq `
µD ` σ D γ ptq ` γ0h ptq ` γ D ptq ` 1
Γ
Γ
”
“
we have,
g1 pt, sq ”
a2 pt, sq
żs
t
”
“
”
b1 pv, sq k pvq dv “
ˆż s
żs
t
˙
ˆ
g1,1 pt, sq
k1 pvq
dv ” g1,0 pt, sq `
b1 pv, sq k0 pvq `
Γ
Γ
2
żs
2
˙
b1 pv, sq α
r pvq g1 pt, vq dv
t
˙
żs
2
b1 pv, sq2 k0 pvq a1 pt, vq dv ` σ D
b1 pv, sq2 α
r pvq g1,0 pt, vq dv
t
t
ˆż s
˙
żs
1
2
D
`2
b1 pv, sq k1 pvq a1 pt, vq dv ` σ
b1 pv, sq2 α
r pvq g1,1 pt, vq dv
Γ
t
t
ˆ ˙
1
a2,0 pt, sq ` a2,1 pt, sq
Γ
2
t
ˆż s
b1 pv, sq k pvq a1 pt, vq dv ` σ
D
56
ˆż s
˙
żs
1
b1 pv, sq2 k pvq b1 pt, vq dv “ 2
b1 pv, sq2 k0 ptq b1 pt, vq dv ` 2
b1 pv, sq2 k1 ptq b1 pt, vq dv
Γ
t
t
t
ˆ ˙
1
” b2,0 pt, sq ` b2,1 pt, sq
Γ
żs
żs
c2 pt, sq ” 2σ D
b1 pv, sq2 α
r pvq b1 pt, vq dv,
d2 pt, sq ” 2σ D
b1 pv, sq2 α
r pvq a1 pt, vq dv
b2 pt, sq
”
2
żs
t
g2 pt, sq
”
”
t
¸
¸
˜ ` ˘2
˜ ` ˘2
żs
σD
σD
2
` k pvq g1 pt, vq dv “ 2
` k0 pvq g1,0 pt, vq dv
b1 pv, sq
2
b1 pv, sq
2
2
t
t
ˆż s
˙
ˆż s
˙ ˆ ˙2
1
1
`2
b1 pv, sq2 pk1 pvq g1,0 pt, vq ` k0 pvq g1,1 pt, vqq dv
`2
b1 pv, sq2 k1 pvq g1,1 pt, vq dv
Γ
Γ
t
t
ˆ ˙
ˆ ˙2
1
1
g2,0 pt, sq ` g2,1 pt, sq
` g2,2 pt, sq
.
Γ
Γ
żs
2
and
“
‰
r z θvm pzq2
E
“
ˆ
˙2
˙
ˆ S
σvS
σv
p pvq g1 p0, vq ` β
p pvq2 g2 p0, vq
´γ
p pvq ´ 2
´γ
p pvq β
Γ
Γ
˙´

„ ˆ S
¯
σv
p pvq a1 p0, vq ´ 2p
p pvq g1 p0, vq ´ β
p pvq2 a2 p0, vq z
´γ
p pvq
α
p pvq ` β
α pvq β
´ 2
Γ
´
¯
p pvq a1 p0, vq ` β
p pvq2 d2 p0, vq z 2
` α
p pvq2 ` 2p
α pvq β
„ ˆ S

˙
σv
p pvq b1 p0, vq ´ β
p pvq2 b2 p0, vq D0
´ 2
´γ
p pvq β
Γ
”
ı
p pvq b1 p0, vq ` β
p pvq2 c2 p0, vq D0 z ` β
p pvq2 b1 p0, vq2 D02 .
` 2p
α pvq β
Substituting the expressions for γ
p pvq “ γ pvq ` γ h pvq , g1 p0, vq , g2 p0, vq , a2 p0, vq , b2 p0, vq,
“
‰
r z θvm pzq2
E
where,
m
k0,v
“
“
ˆ
˙˙2
γ h pvq
σvS
´ γ pvq ` γ0h pvq ` 1
Γ
Γ
ˆ S ˆ
˙˙
ˆ
˙
σv
γ h pvq
p pvq g1,0 p0, vq ` g1,1 p0, vq
´2
´ γ pvq ` γ0h pvq ` 1
β
Γ
Γ
Γ
˜
ˆ ˙
ˆ ˙2 ¸
p pvq2 g2,0 p0, vq ` g2,1 p0, vq 1 ` g2,2 p0, vq 1
`β
Γ
Γ
„ˆ S ˆ
˙˙ ´
˙
ˆ
¯
h
σv
γ1 pvq
g1,1 p0, vq
h
p
p
´2
´ γ pvq ` γ0 pvq `
z
α
p pvq ` β pvq a1 p0, vq ´ α
p pvq β pvq g1,0 p0, vq `
Γ
Γ
Γ
˙
ˆ
„
´
¯
p pvq a1 p0, vq ` β
p pvq2 d2 p0, vq z 2
p pvq2 a2,0 pt, sq ` a2,1 pt, sq
z` α
p pvq2 ` 2p
α pvq β
` β
Γ
ˆ
˙˙
˙
„ ˆ S ˆ
γ1h pvq
b2,1 pt, sq
σv
h
2
p
p
´ γ pvq ` γ0 pvq `
β pvq b1 p0, vq ´ β pvq b2,0 pt, sq `
D0
´ 2
Γ
Γ
Γ
”
ı
m
m 1
m 1
p pvq b1 p0, vq ` β
p pvq2 c2 p0, vq D0 z ` β
p pvq2 b1 p0, vq2 D02 “ k0,v
` 2p
α pvq β
` k1,v
` k2,v
Γ
Γ2
ˆ
´
¯
¯2
p pvq g1,0 p0, vq ` β
p pvq2 g2,0 p0, vq
γ pvq ` γ0h pvq ` 2 γ pvq ` γ0h pvq β
” ´´
¯´
¯
¯
ı
p pvq a1 p0, vq ` α
p pvq g1,0 p0, vq ` β
p pvq2 a2,0 pt, sq z
` 2 γ pvq ` γ0h pvq α
p pvq ` β
p pvq β
ı
´
¯
” ´
¯
p pvq b1 p0, vq ` β
p pvq2 b2,0 pt, sq D0
p pvq a1 p0, vq ` β
p pvq2 d2 p0, vq z 2 ` 2 γ pvq ` γ0h pvq β
` α
p pvq2 ` 2p
α pvq β
”
ı
p pvq b1 p0, vq ` β
p pvq2 c2 p0, vq D0 z ` β
p pvq2 b1 p0, vq2 D02
` 2p
α pvq β
(88)
´
57
m
k1,v
“
´
¯´
¯
”´
¯
´
¯
ı
p pvq
´2 σvS ´ γ1h pvq γ pvq ` γ0h pvq ´ 2 σvS ´ γ1h pvq g1,0 p0, vq ´ γ pvq ` γ0h pvq g1,1 p0, vq β
´
¯ ”´
¯
ı
p pvq2 g2,1 p0, vq ´ 2 σvS ´ γ1h pvq
p pvq a1 p0, vq z ` β
p pvq b1 p0, vq D0
`β
α
p pvq ` β
p pvq g1,1 p0, vq z ` β
p pvq2 pa2,1 pt, sq z ` b2,1 pt, sq D0 q
`2p
α pvq β
´
¯
´
¯2
m
p pvq .
p pvq2 g2,2 p0, vq ` σvS ´ γ1h pvq ´ 2 σvS ´ γ1h pvq g1,1 p0, vq β
k2,v
“β
Thus,
m
m
m
´H0,T
“ K0,T
` K1,T
Using,
(89)
1
m 1
` K2,T
,
Γ
Γ2
m
´
¯
K1,T
m
m,ni
m
∆Tpu pZq “ ´ H0,T
´ H0,T
“ K0,T
`
`
Γ
p
∆P
K1,T
leads to,
˜
m
Ki,T
“
m
K2,T
´
1
2
żT
m
ki,v
dv,
0
i “ 0, 1, 2
(90)
(91)
` D ˘2 ¸
∆Tp
∆Tp
σ
T
K1,T
K2,T
1
∆Tp
”
K
`
`
.
0,T
2
Γ2
Γ
Γ2
żT
∆Pp
∆Pp
K1,T
K2,T
∆Pp u pZq
∆Pp
p psqγ1h psq ds
p ` σD
B
“
`
,
K
“
´∆
V
2,T
Γ
Γ
Γ2
0
żT
´
¯
´
¯´
¯
p psq γ psq ` γ1h psq ds ´ ω n Ip p0q µφ
p p0q ´ 1 D0 ` µD T ` A
p p0q Z ` σ D
B
“ B
0
p
p
p
p
∆T
∆P
∆T
∆P
∆U
∆U
K1,T
` K1,T
K2,T
` K2,T
K1,T
K2,T
∆Pp u pZq
∆Tp
∆U
` ∆Tpu pZq “ K0,T
`
`
”
K
`
`
.
0,T
Γ
Γ
Γ2
Γ
Γ2
∆U
∆U
∆U 2
As ∆U “ U u,ni pexp p´l p1{Γqq ´ 1q where l pxq ” K0,T
` K1,T
x ` K2,T
x , it follows that ∆U ě 0 iff l p1{Γq ě 0.
∆U
From the proof of Proposition 12, K0,T ě 0. Four cases can be distinguished.
` ∆U ˘
∆U
∆U
If K2,T
ą 0, the function l p¨q is a convex parabola. It has a unique minimum at 1{Γ˚ “ ´K1,T
{ 2K2,T
at which
` ∆U ˘2 ` ∆U ˘
` ∆U ˘
` ∆U ˘2
∆U
∆U
∆U
l p1{Γq “´K0,T ´ K1,T¯ { 4K2,T “ ´J{ 4K2,T where J ” K1,T ´ 4K0,T K2,T . If J ě 0, there are two roots
?
` ∆U ˘
∆U
∆U
. If K1,T
ă 0, the roots are positive, implying ∆U ě 0 (ă 0) for 1{Γ ď 1{Γ´ and
1{Γ˘ “ ´K1,T
˘ J { 2K2,T
∆U
1{Γ` ď 1{Γ (1{Γ´ ď 1{Γ ď 1{Γ` ). If K1,T
ě 0, the roots are negative and l p1{Γq ě 0 for all 1{Γ ě 0. In this case
∆U ě 0. If J ă 0, there are no roots and ∆U ě 0 for all 1{Γ ě 0.
` ∆U ˘
∆U
˚
∆U
If K2,T
ă 0, the function l p¨q is a concave parabola. The
¯is `at 1{Γ ˘ “ ´K1,T { 2K2,T at which
´ maximum
?
` ∆U ˘
∆U
∆U
l p1{Γq “ ´J{ 4K2,T As J ě 0, there are two roots 1{Γ˘ “ ´K1,T
, such that 1{Γ´ ă 0 ă 1{Γ` .
˘ J { 2K2,T
Thus, ∆U ě 0 (ă 0) for 1{Γ´ _ 0 ď 1{Γ ď 1{Γ` (1{Γ` ď 1{Γ).
∆U
∆U
∆U
∆U
∆U
If K2,T
“ 0, the function l pxq ” K0,T
` K1,T
x is affine with minimum (maximum) K0,T
ě 0 if K1,T
ą 0
∆U
∆U
∆U
(K1,T ă 0) for x ě 0. If K1,T ą 0, then l p1{Γq ě 0 for all 1{Γ ě 0 and ∆U ě 0. If K1,T ď 0, then ∆U ě 0 for
∆U
∆U
1{Γ ď 1{Γ0 ” ´K0,T
{K1,T
.
∆U
∆U
∆U
If K1,T “ K2,T “ 0, then l p1{Γq “ K0,T
ě 0, so that ∆U ě 0 for all Γ ě 0.
p
Proposition
12
shows
that
the
behavior
of
welfare gains for small risk tolerance depends on the sign of ∆Vp ` ∆H.
´
¯
` ∆U ˘
p {2 “ ´ limΓÑ0 Γ2 l p1{Γq. The sign of the right hand side is governed by ´sgn K2,T
In fact, ∆Vp ` ∆H
.
Proof of Corollary 14. The noise trader maximizes CARA utility under the conditional beliefs dP x {dP “ pηTx q´1
`
˘
evaluated at x “ φ. The ex-ante value is U n “ u pX0n q exp p´E0 rξTm ηTx log pξTm ηTx qsq “ U u exp I i pφ|Zq . Hence,
`
˘
`
˘
ş
ş
f
pxq
E0 rU n s “ U u R exp I i px|Zq f φ|Z pxq dx “ U u R exp I i px|Zq Lφ,G px|Zq fG|Z pxq dx. The formula in the Corollary
G|Z
“
`
˘‰
follows. As, U u ď 0, the interim utility is larger (less negative) if and only if E0 Lφ,G pG|Zq exp I i pG|Zq ď 1. The
“
‰
conditions ω i “ ω n , V AR rGs “ V AR rφs, E rφs “ E rGs imply Lφ,G pG|Zq “ 1 so that E0 rU n s “ E0 U i . To prove
the last statement, note that Lφ,G p x| zq “ h1 exp p´h2 q where h1 “ ω n {ω i and
h2 “
$
’
’
’
’
’
’
&
’
’
’
’
’
’
%
2
2
i
n 2
q Hp0qpσ φ q
1 pω q pω
´
¯
2 M p0q pω n q2 ´pω i q2
¨
ˆ
˚
M p0q˝µφ ´ErGs`
ω̄
¨
2
2
˝x ´
pσφ q2 ´Hp0q
M p0q
˙
pω n q2 µφ ´pω i q ErGs`
pω n q2 ´pω
˛¨
‹˚
pz´ErZsq‚˝µφ `ErGs`
2
2 ω̄σ φ Hp0q
p
q
58
3
pωn q3 pσφ q ´pωi q Hp0q
ω̄
ˆ
M p0q
i 2
q
pσφ q2 `Hp0q
M p0q
˙
pz´ErZsq
˛
‹
pz´ErZsq´2x‚
˛2
‚ if ω i ‰ ω n
if ω i “ ω n ” ω̄.
` ˘2
If ω i “ ω n , µφ “ E rGs and σ φ “ V AR rGs, then Lφ,G p x| zq “ 1. This establishes the result.
Proof of Corollary 15. Note that,
¸¸
¸
˜
˜ ˜
”
´
¯ı
∆Pp pZq
u
n
u,ni
u,ni
i
p
` ∆T pZq
E0 exp I pG|Zq Lφ,G pG|Zq ´ 1
E0 rU s ´ U
“U
exp ´
Γ
As U u,ni ă 0, welfare improves only if,
¸¸
˜ ˜
”
´
¯ı
∆Pp pZq
u
p
` ∆T pZq
E0 exp I i pG|Zq Lφ,G pG|Zq ă 1.
exp ´
Γ
This condition can be expressed in terms of the certainty equivalent gain for the noise trader. The fact that
Lφ,G pG|φq “ 1 under unbiasedness, equal weights and equal variances of the noise and private signals, and the
fact that the noise trader attains the interim utility of the informed are proved in Corollary 14.
“
`
˘‰
Proof of Corollary 16. The condition ∆Pp pZq ` Γ∆Tpu pZq ą Γ log E0 exp I i pG|Zq Lφ,G pG|Zq is equivalent to
E0 rU n s ´ U u,ni ą 0. It follows that the noise trader is better off.
Proof of Corollary 18. Let ω n “ 0 and ω u “ 1´ω i . Consider first the case where the informed acquires information
G
D Ž
and suppose that a fully revealing equilibrium prevails. The common information filtration is then Fp¨q
” Fp¨q
σ pGq.
Market clearing implies,
θtm,wa “
σtS,wa
´ hut ,
Γ
„

żT
1
m
pθvm,wa q2 dv
hut “ ´ BDt Et ξt,T
2
t
Stwa “ E r DT | Dt , Gs ´
σtS,wa “
ˆ
BDt E r DT | Dt , Gs ´
żT
t
żT
t
ˇ
ı
”
ˇ
E σvS,wa θvm,wa ˇ Dt , G
ˇ
ı˙
”
ˇ
BDt E σvS,wa θvm,wa ˇ Dt , G σ D
(the superscript stands for “with acquisition”). A solution to this system of equations is,
θtm,wa
1´
σ S,wa
BDt E r DT | Dt , Gs S,wa
“ t
“
σt
“
Γ
Γ
V ARr DT |Dt s
V ARr G|Dt s
Γ
σ
D
` ζ ˘2
1 σ
“
σD
Γ H ptq
` ζ ˘2
˙
σ
V AR r DT | Dt s
D
“ 1´
σ “
σD
V AR r G| Dt s
H ptq
` ζ ˘2 ´
` D ˘2
ż
¯ 1 żT ´
¯2
σ
σ
pT ´ tq
1 T ´ S,wa ¯2
G`
Dt ` µD pT ´ tq ´
σv
σtS,wa dv
dv “
Stwa “ E r DT | Dt , Gs ´
Γ t
H ptq
H ptq
Γ t
` D ˘2
` ζ ˘2
where we recall that H ptq “ σ
pT ´ tq ` σ . The price is indeed fully revealing, thus validating the initial
conjecture. The wealth CE of the informed (including the information acquisition costs C ą 0) is,
¨
˛
ż ˜ ` ζ ˘2 ¸2
´ ¯2
σ
1 ˝ T
i,wa
wa
CE0
“ S0 `
dv ‚ σ D ´ C.
2Γ
H pvq
0
σtS,wa
ˆ
Full revelation implies that CE0u,wa “ CE0i,wa ` C ą CE0i,wa .
“
‰
The ex-ante utility function V i,wa ” E U i,wa is,
V
i,wa
“
“
“
˙
ˆ
„
ˆ
żT ´
¯2 ˙
1
S0wa ´ C
S,wa
exp ´ 2
σv
dv
´ΓE exp ´
Γ
2Γ 0
«
˜ ` ˘2
¸ff
˜
˜ ` ˘2
¸
¸
ż
¯
σζ ´
σD T
1
1 T ´ S,wa ¯2
C
D
´ΓE exp ´
G
exp ´
D0 ` µ T ´
σv
dv `
H p0q Γ
Γ H p0q
2Γ 0
Γ
¨ ` ˘
˜
¸
˜
¸˛
` ζ ˘2 ´
żT ´
2
¯ 1 `σ D ˘2 T 2
¯
¯2
σ
σD T ´
1
1
D
D
S,wa
D0 ` µ T `
D0 ` µ T ´
H p0q ´
σv
dv ´ C ‚
´Γ exp ˝´
H p0q Γ
2 H p0q Γ
Γ H p0q
2Γ 0
59
“
¨
¨
´` ˘ ¯2
2
σD T
1
˚ 1˚
´Γ exp ˝´ ˝D0 ` µD T ´
Γ
2Γ
H p0q
1
´
2Γ
żT ´
σvS,wa
0
¯2
The ex-ante certainty equivalent is,
CE
i,wa
1
“ D0 ` µ T ´
2Γ
D
´` ˘ ¯2
2
σD T
H p0q
1
´
2Γ
˛˛
‹‹
dv ‚‚
żT ´
¯2
σvS,wa dv ´ C.
0
Next, consider the case where the informed does not acquire information. The common information filtration is then
D
Fp¨q
. The fully revealing equilibrium is,
θtm,na “
σtS,na
BDT E r DT | Dt s σ D
σD
“
“
,
Γ
Γ
Γ
σtS,na “ σ D ,
Stna “ Dt ` µD pT ´ tq ´
1 ´ D ¯2
σ
pT ´ tq
Γ
(the superscript stands for “no acquisition”). Utility functions are the same as in a no information equilibrium,
U
j,na
j,ni
“U
X0j “ N0j
“u
˜
D0 `
The CE wealth becomes CE i,na “ S0na `
´
X0j
¯
ˆ
1
exp ´ 2
2Γ
żT ´
σtS,ni
0
¯2
˙
j P tu, iu
dt ,
˘2 ¸ ¸
σD
T ,
σtS,ni “ σ D ,
N0j “ 1.
Γ
¯` ˘
´ş
` D ˘2
2
T
1
1dv σ D “ D0 ` µD T ´ 2Γ
σ
T . Information production is
0
˜
µD ´
1
2Γ
`
suboptimal if and only if CE0i,wa ă CE0i,na , i.e.,
1
´
2Γ
´`
σD
˘2 ¯2
T
1
´
2Γ
H p0q
żT ´
σvS,wa
0
¯2
dv ´ C ă ´
1 ´ D ¯2
σ
T.
2Γ
˚
This condition is equivalent to C ą C where,
` D ˘2 2 ff
` D ˘2 «
` ζ ˘2
` D ˘2 2 ff
` ζ ˘4 ¸
` D ˘2 «ż T ˜
σ
T
σ
σ
T
σ
T
σ
σ
˚
dv ´
“
T´
´
“ 0.
1´
C “
2Γ
H p0q
2Γ
H p0q
H p0q
H pvq2
0
The first equality above uses σvS,wa “
ż T ` D ˘2
σ
0
H pvq
´`
σζ
dv “
2
˘2
¯
{H pvq σ D . The second relies on,
` D ˘2
σ
T
1
H p0q ´ H pT q
1
´
“
“
.
H pT q
H p0q
H pT q H p0q
pσ ζ q2 H p0q
Hence, information acquisition is always suboptimal if acquisition is costly. In the absence of speculation, the noinformation equilibrium prevails.
To evaluate the welfare implications of speculation, it remains to compare interim utilities with and without
speculation. But, the equilibrium without speculation corresponds to the equilibrium without information. Thus,
welfare improves under the conditions of Corollary 16.
Proof of Proposition 19. Consider the information infrastructure ℑ pnq with n signals equally spaced in time
n
ttn
0 , . . . , tn u. The interim utility is as in the one period model,
Utinn´1
˜
“ ´Γ exp ´
Xtinn´1
Γ
´E
”
ξtGnn ,tnn
n´1
ˇ
ˇ
log ξtGnn ,tnn ˇ Gtnn´1
n´1
¸
ı
.
Choosing Xtnn´1 optimally at time tn
n´2 leads to the first order condition and optimal wealth,
˙
ˆ
ˇ
ı¯
´ ”
Xtn
G
ˇ
exp ´ n´1 “ ytnn´2 ξtnn´1,tn exp E ξtGnn ,tnn log ξtGnn ,tnn ˇ Gtnn´1
n´1
n´1
n´2 n´1
Γ
60
´
Xtnn´2
ˇ
ı
ıˇ
”
”
G
G
G
ˇ
ˇ
log ytnn´2 ` E ξtnn´1,tn log ξtnn´1,tn ` ξtnn´1,tn E ξtGnn ,tnn log ξtGnn ,tnn ˇ Gtnn´1 ˇ Gtnn´1
n´1
n´1
n´2 n´1
n´2 n´1
n´2 n´1
ˇ
ff
«
n´1
k
ˇ
ź
ÿ
Gi`1
Gk`1 ˇ
n
n
ξtn ,tn log ξtn ,tn ˇ Gtn´2 .
log ytn´2 ` E
i i`1
k k`1 ˇ
k“n´2 i“n´2
“
Γ
“
Interim utility at tn
n´2 is,
˜
Utinn´2
“ ´Γ exp ´
Xtinn´2
Γ
´E
«
k
ź
n´1
ÿ
k“n´2 i“n´2
G
ξtni`1
n
i ,ti`1
Iterating this argument gives,
˜
Uti0 “ ´Γ exp ´
As Xtin0 “ S0m,n where S0m,n “ E
”ś
n
i“0
«
Xtin0
Γ
´E
«
n´1
k
ÿ ź
ˇ
ff¸
.
ˇ
ff¸
ˇ
G
ˇ n
log ξtnk`1
G
.
,tn ˇ t0
i`1
k k`1 ˇ
G
ξtni`1
,tn
k“0 i“0
i
ˇ
ı
ˇ
ξtmni ,tni`1 DT ˇ F0m,n , the ex-ante utility associated with ℑ pnq is,
˜
S m,n
´E
V pnq “ ´ΓE exp ´ 0
Γ
i
ˇ
ˇ
G
ˇ n
log ξtnk`1
,tn ˇ Gtn´2
k k`1
«
n´1
k
ÿ ź
k“0 i“0
G
ξtni`1
,tn
i i`1
Using,
ˇ
ˇ
G
ˇ
log ξtnk`1
,tn ˇ G0
k k`1
ˇ
ff¸ff
ˇ
ˇ
ˇ
”
ı
”
ı
”
ı
Gk`1 ˇ
G
ˇ
ˇ
m
n
E ξtkk`1
log ξtmnk ,tnk`1 ˇ Ftm,n
` E ξtmnk ,tnk`1 log ηtxnk ,tnk`1 ˇ Ftm,n
n
n
,tn
,tk`1 log ξtn ,tn ˇ Gtk “ E ξtn
k k`1
k
and defining K0n ” E
«
k`1
k
ˇ
ı
ˇ m,n
n´1 śk
m
m
n
n ,tn
F
log
ξ
gives,
ˇ
,t
t
0
k“0
i“0 ξtn
i i`1
k k`1
”ř
˜
S m,n
´ K0n ´ E
V pnq “ ´ΓE exp ´ 0
Γ
i
P
As ηtxnk ,tnk`1 “
Φ“
ż `8
´8
˜
P
ˆ
m,n
Gk`1 Pdx|F n
t
k`1
m,n
Gk`1 Pdx|F n
t
k
¸
˙
«
n´1
k
ÿ ź
ξtmni ,tni`1
k“0 i“0
it follows that Φ ”
ş `8
´8
ż `8
´8
k
log ηtxnk ,tnk`1 P
p Gk`1 P
ˇ
ˇ
x“Gk`1
ˇ
dx| F0m,n qˇ F0m,n
ˇ
ff¸ff
.
log ηtxnk ,tnk`1 P p Gk`1 P dx| F0m,n q becomes,
˛
m,n
m,n
P
p
G
P
dx|
F
q
P
p
G
P
dx|
F
q
0
0
˝log ´ k`1
¯ ´ log ´ k`1
¯ ‚P p Gk`1 P dx| F0m,n q ” DtKL
n
n
n
´ DtKL
” ∆DtKL
k`1
k
k
P Gk`1 P dx| Ftm,n
P Gk`1 P dx| Ftm,n
n
n
¨
k`1
k
where
is the Kullback-Leibler divergence between the signal density conditional on initial information F0m,n and
information Ftm,n . Ex-ante utility is,
ˇ
«
«
˜
ff¸ff
n´1
k
ˇ
ÿ ź
S0m,n
m
KL ˇ m,n
n
i
ξtni ,tni`1 ∆Dtnk ˇ F0
´ K0 ´ E
V pnq “ ´ΓE exp ´
.
ˇ
Γ
k“0 i“0
DtKL
As densities are Gaussian,
DtKL
n
∆DtKL
k
1
“
2
ˆ
“
”
ı
˛
¨
V AR Gk`1 | Ftm,n
n
1
1
1
1
k`1
”
”
”
ı ` ˝
ı´
ı ‚V AR r Gk`1 | F0m,n s
log
m,n
2
2 V AR Gk`1 | F m,n
V AR Gk`1 | Ftm,n
V
AR
G
|
F
n
n
n
k`1
t
t
V AR r Gk`1 | F0m,n s ` pE r Gk`1 | F0m,n s ´ E r Gk`1 | Ftm,n sq2
V AR r Gk`1 | Ftm,n s
log
m,n ´ 1 `
V AR r Gk`1 | F0 s
V AR r Gk`1 | Ftm,n s
k
k`1
61
k
˙
¨¨
˛2 ¨ ”
˛2 ˛
”
ı
ı
m,n
m,n
m,n
m,n
E
G
|
F
´
E
r
G
|
F
s
E
G
|
F
´
E
r
G
|
F
s
k`1
k`1
k`1
k`1
‹
‹ ‹
˚
0
0
tn
tn
1 ˚˚
k`1
k
˚
‹ ´˚
‹ ‹.
c
c
` ˚
”
ı
”
ı
‚
‚‚
˝
2 ˝˝
m,n
m,n
V AR Gk`1 | Ftn
V AR Gk`1 | Ftn
k`1
k
The corresponding ex-ante certainty equivalent is,
ˇ ff¸ff
«
˜
«
n´1
k
ˇ
ÿ ź
S0m,n
Gi`1
Gk`1 ˇ
i
CE pnq “ ´Γ log E exp ´
ξtn ,tn log ξtn ,tn ˇ G0
´E
i i`1
k k`1 ˇ
Γ
k“0 i“0
ˇ
«
˜
«
ff¸ff
n´1
k
ˇ
ÿ ź m
S0m,n
KL ˇ
m,n
n
ξtni ,tni`1 ∆Dtnk ˇ F0
“ ´Γ log E exp ´
´ K0 ´ E
.
ˇ
Γ
k“0 i“0
The informed is a price-taker and takes S0m,n and ξtmni ,tni`1 as given when making the information acquisition decision.
n
Only the impacts of log ηtxn ,tn and therefore ∆DtKL
on ex-ante utility will be taken into account when choosing the
k k`1
k
optimal infrastructure. Thus, the informed only considers the value of all Kullback-Leibler divergence gains of the
infrastructure for a given state price density. The optimal choice is the maximal solution such that,
ˇ
ˆ
„
˙ ˛
¨ „
m,n˚
ˇ m,n˚
řn˚ śk
S0
n˚
m
KL
ˇ
k“0
i“0 ξtn˚ ,tn˚ ∆Dtn˚ `1 ˇ F0
˚ E exp ´ Γ ´ K0 ´ E
‹
`
˘
` ˘
i
i`1
k
ˇ
„
ˆ
„
˙ ‹
´Γ log ˚
ď k n˚ ` 1 ´ k n˚ .
˚
˝
‚
m,n
ˇ
ř
ś
˚
˚
S0
˚
n ´1
k
m,n
m
KL ˇ
E exp ´ Γ ´ K0n ´ E
k“0
i“0 ξ n˚ n˚ ∆D n˚ ˇ F0
ti
,ti`1
tk
It remains to show that the optimal number of signals n˚ (i.e., the frequency) is finite. As limnÑ8 kpnq “ `8 by
assumption, it is shown that given the market price of risk and the state price density, the certainty equivalent gain
of the informed is bounded. For this, note that ´ log is a convex function so that, by Jensen’s inequality,
ff
«
k
n´1
ÿ ź
m
KL
i
m,n
n
ξtni ,tni`1 ∆Dtnk .
CE pnq ď E rS0 s ` ΓE rK0 s ` ΓE
k“0 i“0
By Proposition 22, the information structure is isomorph to one with signal process Gt “ DT ` r
ζ where r
ζ “
” ı
” ı t ` ˘ t
´1
1
˘
` 1
ζN
V ARrζNt s
1Nt
´1
ζ 2
t
r
r
and V AR ζ t “ 1{ 1Nt V AR rζNt s 1Nt . The assumption limNt Ñ8 V AR ζ t “ σ
from
´1
11N V ARrζNt s
1Nt
t
above, implies,
ff
«
„ż T

ż `8 ´
k
n´1
¯
ÿ ź
m
m 2
1
m
KL
n
r
ξtni ,tni`1 ∆Dtnk
“
E rK0 s ` E
ξ 0,v
E
θvG|m pxq ` r
θv
P p Gv P dx| F0m q dv
2
´8
0
k“0 i“0
„ż T

ż `8 ´
¯
m
m 2
1
G|m
r
ξ 0,v
E
θv pxq ` r
θv
P p Gv P dx| F0m q dv
ď
2
´8
0
m
x´E r DT |Fv
s
.
pσ D q2 pT ´vq`pσ ζ q2
Proof of Proposition 22. The private information process is Gt “ DT 1Nt ` ζNt where 1Nt is an Nt ˆ 1 vector and
ζNt is an Nt ˆ 1 vector of independent noises. The PIPR is,
G|m
where θv
pxq “ σ D
G|m
θt
pGt q
“
“
“
`
˘´1
pGt ´ Et rDT s 1Nt q1 V ARt rDT s 1Nt 11Nt ` V AR rζNt s
1Nt BDt Et rDT s
`
˘´1
1
1
ppDT ´ Et rDT sq 1Nt ` ζNt q V ARt rDT s 1Nt 1Nt ` V AR rζNt s
1Nt BDt Et rDT s
˙1
ˆ
`
˘
pDT ´ Et rDT sq 1Nt ` ζNt
´1
1Nt 11Nt ` Ψt
1Nt BDt Et rDT s
V ARt rDT s
V ARrζNt s
where Et r¨s ” Et r ¨| Ftm s , V ARt rDT s ” V AR r ¨| Ftm s and Ψt ” V ARt rD
. The Sherman-Morrison formula (Golub
s
˘´1
˘´1 T´1
`
`
´1
1
1
Ψt 1Nt , so that,
and van Loan (1996)) gives 1Nt 1Nt ` Ψt
1Nt “ 1 ` 1Nt Ψt 1Nt
θtG|m
pGt q
“
DT ´ Et rDT s
V ARt rDT s
˜
1
1
1 ` 1Nt Ψ´1
t 1Nt
62
¸
11Nt Ψ´1
t 1Nt BDt Et rDT s
`
“
1
V ARt rDT s
˜
1
1 ` 11Nt Ψ´1
t 1Nt
¸
1
ζN
Ψ´1
t 1Nt BDt Et rDT s
t
” ı
r t ´ Et G
rt
G
r
DT ´ Et rDT s ` ζ t
” ı BDt Et rDT s ”
” ı BDt Et rDT s
rt
ζt
V ARt rDT s ` V AR r
V ARt G
11N V AR rζNt s´1 ζNt
r t “ DT ` r
r
and G
ζt.
ζt “ 1 T
1NT V AR rζNt s´1 1Nt
´ ¯
r ´Et rG
rts
φ
G|m r
The noise trader’s demand depends on θt
φt “ VtAR Gr BDt Et rDT s. The WAPR is therefore based on the
tr ts
r.
rt ` ωnφ
public signal Zrt “ ω i G
t
´ ¯2
” ı
Proof of Corollary 23. To simplify notation, let vtζ ” σtζ “V AR r
ζ t and note that vtζ is constant for t ‰ tj
and jumps down for t “ tj ,
∆vtζ
where
ζ
k2,t
j´1
“
$
’
’
&
¨
0
’
´vtζj´1 ˝ ˆ
’
%
ζ
k
˙2
ˆ
ζ
k1,t
j´1
˙2
ζ
ζ
`vt
k2,t
1,tj´1
j´1
j´1
for t ‰ tj
˛
‚ for t “ tj .
ı´1
”
ı
”
ζ
1
COV
ζ
,
ζ
k1,t
”
1
´
1
V
AR
ζ
N
j
N
N
t
t
tj´1
j´1
j´1
j´1
”
”
”
ı´1
ı1
ı
” V AR rζj s2 ´ COV ζNtj´1 , ζj V AR ζNtj´1
COV ζNtj´1 , ζj
ζNtj´1 is an Ntj´1 ˆ 1 vector of error noises of the private signals received before time tj and ζj is the error noise of
”
ı´1
ζ
the private signal received at time tj . As e1Ntj V AR ζNtj
eNtj “ 1{k2,t
where e1Ntj is the Ntth
unit vector and
j
j´1
ı´1
ı
”
”
ζ
ζ
are positive definite, it follows that k2,tj´1 ą 0 and ∆vt ă 0.
as V AR ζNtj and therefore V AR ζNtj
In between information arrival times, as vtζ is constant, the time-behavior of the stock price volatility is determined
by the same factors as in the benchmark model with a single private signal. Price volatility therefore increases. At
information arrival times, the jump in volatility is determined by the jump in vtζ and the impact of vtζ on volatility.
To find the latter, note that, in the absence of hedging,
˜
` ˘2
` ˘2 ¸
p1 ´ ωq ω i
p1 ´ ωq ω i
ω
ω
Bvζ B ptq “ B ptq
´
`
´
ą0
t
H pT q
H ptq
M pT q
M ptq
implying that price volatility jumps down. The effect due to hedging is Bvζ B h ptq “ σ D B h ptq
t
Bvζ β h pvq “
t
żT
t
şT
t
Bvζ β h pvq dv where,
t
ˆż v ´
´
¯
¯ ˙´
¯
exp
κ1 psq ` 2 1 ` 2σ D β h psq ds
Bvζ κ0 pvq ` Bvζ κ1 pvq β h pvq
t
t
ˆ
t
˙
´
¯
1´ω
β ptq Bvζ β ptq ,
Bvζ κ1 ptq “ 2σ D Bvζ β ptq ` Bvζ β D ptq
t
t
t
t
t
ω
˜
` i ˘4 D
` φ ˘2 ¸
ˆ´ ¯
˙
n 2
2
ω σ
pω q σ
i
D
M ptq
ω
H ptq ` 1 ą 0,
Bvζ β ptq “
ą 0.
Bvζ β ptq “ ω
2
t
t
M ptq
pM ptq H ptqq
Bvζ κ0 ptq “ 2
Thus Bvζ B h ptq ą 0 implying that the hedging factor jumps down as well. Hence, Bvζ σtS ą 0 and price volatility
t
decreases at information arrival times tj , j “ 1, ..., J.
Proof of Proposition 24. Assume heterogeneous risk tolerances. The market clearing condition gives,
´
¯
´
¯
´
¯
ÿ
G|m
G|m
G|m
G|m
ω j Γj θtm ` hut σ D ` ω i Γi θt
pGq ` ht
pG|Zq σ D ` ω n Γn θt
pφq ` ht
pφ|Zq σ D “ σtS .
jPtu,i,nu
63
Rearranging terms and using the definitions of ω
ri, ω
r n leads to,
´
¯
´
¯
σS
G|m
G|m
G|m
G|m
θtm ` hut σ D ` ω
r i θt
pGq ` ht
pG|Zq σ D ` ω
r n θt
pφq ` ht
pφ|Zq σ D “ ta .
Γ
The WAPR and weighted average of the informational hedging components follow. The equilibrium market price of
risk is identical to the one under homogeneous information modulo the parameter substitution announced.
Proof of Corollary 25.
Under the conditions of the corollary BΓa ” ω i BΓi ` ω n BΓn “ 0 ðñ BΓn {BΓi “
` i n˘
`
˘
i
´ ω {ω ă 0 and B ω
r “ Bω
r ` Bω
r n “ ω i BΓi ` ω n BΓn {Γa “ 0. Suppose BΓn ą 0. Simple calculations show,
B
ω i Γi
ω i BΓi Γn ´ Γi BΓn
ω
ri
1 ω n Γn ` ω i Γi n
“´ n
BΓ ă 0
n “ B n n “
2
n
ω
r
ω Γ
ω
ω
pΓn q
pΓn q2
` ˘2
` φ ˘2
pr
ω n q2 σ φ
σ
M pT q
“ ` i ˘2
“
´ ¯2
M ptq
ω
ri
H ptq ` pr
ω n q2 pσ φ q2
ω
r
H ptq ` pσ φ q2
n
B
BB ptq “ B
ˆ
M pT q
“ ´ ˆ´ ¯
2
M ptq
ω
ri
H pT q
H ptq
˙ωr ˆ
B h ptq “ eσ
ω
rn
M pT q
M ptq
ω
r
` φ ˘2
σ
H ptq ` pσ φ q2
˙1´ωr
D şT h
t β pvqdv
,
“ p1 ´ ω
rq
˙2 2
ˆ
ˆ
ω
ri
ω
rn
H pT q
H ptq
˙
H ptq B
˙ωr ˆ
BB h ptq “ B h ptq σ D
ω
ri
ą0
ω
rn
M pT q
M ptq
żT
t
˙´ωr
B
M pT q
ą0
M ptq
Bβ h pvq dv
` ˘2
pr
ω n q2 σ φ
1 ´ κt ω
ri D
M pT q
σ “ ´r
ω
σ D “ ´r
ω
σD
H ptq
M ptq H ptq
M ptq H ptq
` i ˘2 D
σ
ω
r
ωiσD
D
D
i D
α ptq ”
,
β ptq ” ´r
ω α ptq “ ´
M ptq
M ptq
»
´ n ¯2 ` ˘2 fi
` φ ˘2 ` i ˘2
ω
r
n 2
D
H
ptq
`
ω
r
σφ ffi σD
H ptq σ
ω
r pr
ω q σ
` ω
r
ω
ri
—
D
“ ´–
β ptq ` β ptq “ ´
fl
` n ˘2
M ptq
H ptq
H ptq
H ptq ` ωrωr i pσ φ q2
β ptq “ ´r
ω
´ n¯` ˘ ´ n¯
2
2H ptq ωrωr i
σ φ B ωrωr i
σD
D
B β ptq ` β ptq “ p1 ´ ω
rq ”
ą0
ı2
` ωr n ˘2
H ptq
H ptq ` ωr i pσ φ q2
ˆ
˙
ˆ
˙
1´ω
r
1´ω
r
κ0 ptq ”
β ptq2 σ D ,
Bκ0 ptq ” 2
β ptq Bβ ptq σ D ą 0
ω
r
ω
r
´
¯
´
¯
κ1 ptq ” ´2σ D β ptq ` β D ptq ,
Bκ1 ptq ” ´2σ D B β ptq ` β D ptq ă 0.
´
h
¯
As β h solves β9 ptq “ κ0 ptq`κ1 ptq β h ptq´σ D β h ptq2 with terminal condition β h pT q “ 0, the derivative Bβ h relative to
“
‰
h
the risk tolerance perturbation described above satisfies, B β9 ptq “ Bκ0 ptq ` Bκ1 ptq β h ptq ` κ1 ptq ´ 2σ D β h ptq Bβ h ptq
with boundary condition Bβ h pT q “ 0. The solution is,
Bβ h ptq “ ´
żT
t
´
¯
şv
D h
e´ t pκ1 psq´2σ β psqqds Bκ0 pvq ` Bκ1 pvq β h pvq dv
which is strictly negative if Bκ0 pvq ` Bκ1 pvq β h pvq ą 0 for all v P r0, T s. As Bκ0 pvq ą 0, Bκ1 pvq ă 0, β h pvq ă 0 for
all v P r0, T s, the result stated follows.
Proof of Proposition 26. It is first shown that in equilibrium the informed can infer Z “ ω i G ` ω n φ from the
residual demand, where G “ DT ` ζ. For this note that for arbitrary ι P r0, 1s, µ pdιq, a.e.,
Gι |m
θt
Gι |m
pGι q`ht
pGι q “
σ̌S
m
´θ̌t ´ω i
Γ
ż
r0,1s{tιu
´
Gι |m
θt
Gι |m
pGι q ` ht
64
ż
¯
pGι q µ pdιq´ω n
r0,1s
´ ι
¯
G |m
Gι |m
θt
pφι q ` ht
pφι q µ pdιq .
ι
ι
ι
The Gaussian structure of Z, Dt implies that θtG |m pGι q ` htG |m pGι q is an affine function of Gι , Z, Dt and θtG |m pφι q `
Gι |m
pφι q an affine function
of φ ι , Z, Dt . With
each individual
ht
´ ι
¯ a continuum
´of informed,
¯ is negligible µ pdιq a.e. on
ş
ş
G |m
Gι |m
Gι |m
Gι |m
r0, 1s. Thus, r0,1s{tιu θt
pGι q ` ht
pGι q µ pdιq “ r0,1s θt
pGι q ` ht
pGι q µ pdιq and there exist coeffiι
ι
m
cients ᾰ ptq, β̆ ptq, and γ̆ ptq such that µ pdιq a.e., θtG |m pGι q ` htG |m pGι q “ σ̌S {Γ ´ θ̌t ´ ᾰ ptq Z ´ β̆ ptq Dt ´ γ̆ ptq.
It follows that the residual demand of each privately informed individual reveals Z “ ω i G ` ω n φ. Defining,
ARt rGs
, individual PIPRs are,
̺ιt ” V ARtVrGs`V
ARrζ ι s
ˆ
˙
x ´ Et rDT s
x ´ Et rDT s
D
ι
B
E
rD
s
σ
“
̺
BDt Et rDT s σ D ” ̺ιt θtG|m pxq
D
t
T
t
V ARt rGs ` V AR rζ ι s t
V ARt rGs
˙
ˆ
x ´ Et rDT s
BDt Et rDT s σ D “ α1 pvq x ` α2 pvq z ` β0 pvq Dv ` γ0 pvq .
θvG|m pxq “
V ARt rGs
ş1
Define the population moments ̺rns
” 0 p̺ιt qn µ pdιq for n “ 1, 2, and the transformed variables,
t
ι
θtG
|m
pxq “
r1s
x̌ ptq ” ̺t x ptq and x ptq ” x̌ ptq ` x̌h ptq
for x P tα1 , α2 , α, β0 , β, γ0 , γu .
Under the conjecture that volatility σ̌S
t is deterministic and that the market price of risk is an affine function of Z, Dt
m
S
given by, θ̌t “ σ̌ t {Γ ´ α ptq Z ´ β ptq Dt ´ γ ptq, it holds that,
”
ı
“ m
‰
m
BDt Et ξ̌ t,v θvG|m pxq2
“ BDt Et ξ̌ t,v ppα1 pvq x ` α2 pvq Z ` β0 pvq Dv ` γ0 pvqqq2
“ m
‰
‰
“ m
“ 2 pα1 pvq x ` α2 pvq Z ` γ0 pvqq β0 pvq BDt Et ξ̌ t,v Dv ` β0 pvq2 BDt Et ξ̌ t,v Dv2
ı
”
m m
BDt Et ξ̌ t,v θ̌ v θvG|m pxq
“
“
˙

ˆ S
„
σ̌v
m
´ α pvq Z ´ β pvq Dv ´ γ pvq pα1 pvq x ` α2 pvq Z ` β0 pvq Dv ` γ0 pvqq
BDt Et ξ̌ t,v
Γ
ˆˆ S
˙
˙
‰
“ m
σ̌ v
´ α pvq Z ´ γ pvq β0 pvq ´ pα1 pvq x ` α2 pvq Z ` γ0 pvqq β pvq BDt Et ξ̌ t,v Dv
Γ
“ m
‰
´β pvq β0 pvq BDt Et ξ̌ t,v Dv2
”
ı
r z Ds | FtD “ b̌1 pt, sq ,
B Dt E
” ˇ
ı
r z Dv2 ˇ FtD “ b̌2 pt, vq ` č2 pt, vq Z ` 2b̌1 pt, vq2 Dt .
B Dt E
ş1
As 1 ą ̺ιt ą 0, it follows from Kolmogorov’s strong law of large number that 0 p̺ιv qn ζ ι µ pdιq “ 0 for n “ 1, 2. Then,
also using Gι “ G ` ζ ι and the expressions above leads to,
k1,t,v
”
ż1
“
2 pα1 pvq G ` α2 pvq Z ` γ0 pvqq β0 pvq b̌1 pt, vq
0
”
ı
ι
m
BDt Et ξ̌ t,v θvG |m pxq2
`2α1 pvq β0 pvq b̌1 pt, vq
“
“
k2,t,v
”
“
“
|x“Gι
ż1
0
µ pdιq “
ż1
0
”
ı
m
p̺ιv q2 BDt Et ξ̌ t,v θvG|m pxq2
ż1
0
|x“Gι
µ pdιq
p̺ιv q2 µ pdιq
˘
`
p̺ιv q2 ζ ι µ pdιq ` β0 pvq2 b̌2 pt, vq ` č2 pt, vq Z ` 2b̌1 pt, vq2 Dt
ż1
0
2
p̺ιv q2 µ pdιq
˘‰
`
2 pα1 pvq G ` α2 pvq Z ` γ0 pvqq β0 pvq b̌1 pt, sq ` β0 pvq2 b̌2 pt, vq ` č2 pt, vq Z ` 2b̌1 pt, vq Dt ̺r2s
v
ż1
ż1
ı
ı
”
”
ι
m m
m m
µ pdιq “
µ pdιq
BDt Et ξ̌ t,v θ̌v θvG |m pxq
̺ιv BDt Et ξ̌ t,v θ̌v θvG|m pxq
|x“Gι
|x“Gι
0
0
˙
˙
ˆˆ S
ż1
σ̌ v
̺ιv µ pdιq
´ α pvq Z ´ γ pvq β0 pvq ´ pα1 pvq G ` α2 pvq Z ` γ0 pvqq β pvq b̌1 pt, vq
Γ
0
ż
ż1
˘ 1 ι
`
̺v µ pdιq
̺ιv ζ ι µ pdιq ´ β pvq β0 pvq b̌2 pt, vq ` č2 pt, vq Z ` 2b̌1 pt, vq2 Dt
´α1 pvq β pvq b̌1 pt, vq
0
0
ˆˆ S
˙
˙
σ̌ v
´ α pvq Z ´ γ pvq β0 pvq ´ pα1 pvq G ` α2 pvq Z ` γ0 pvqq β pvq b̌1 pt, vq ̺r1s
v
Γ
˘
`
´β pvq β0 pvq b̌2 pt, vq ` č2 pt, vq Z ` 2b̌1 pt, vq2 Dt ̺r1s
v
65
and the informed hedging components related to the PIPR become,
ż1
Gι |m
h1t
0
ż1
Gι |m
i,G
ψ̌ 11
ptq “ ´
i
żT
i,G
ψ̌ 12
ptq “
żT
t
t
i,G
żT
t
i,G
i,Z
ψ̌11
i
ψ̌22
ż T ˆˆ
t
ptq “ ´
ψ̌ 31 ptq “ ´
i,Z
β pvq α1 pvq b̌1 pt, vq ̺r1s
v dv, ψ12 ptq “
i
i
i
i
i
β 0 pvq2 b̌1 pt, vq2 ̺r2s
v dv,
ψ̌ 32 ptq “
i,Z
k1,t,v dv “ ψ̌ 11 ptq G ` ψ̌ 11 ptq Z ` ψ̌ 21 ptq Dt ` ψ̌ 31 ptq
α1 pvq β 0 pvq b̌1 pt, vq ̺r2s
v dv,
t
ψ̌ 21 ptq “ ´
żT
i
i,Z
pGι |Zq µ pdιq “ ψ̌ 12 ptq G ` ψ12
ptq Z ` ψ̌ 22 ptq Dt ` ψ̌ 32 ptq
h2t
0
where,
1
2
pGι |Zq µ pdιq “ ´
ptq “ 2
żT
t
żT
t
``
żT ˆ
t
żT ˆ
t
˙
1
2
α2 pvq β 0 pvq b̌1 pt, vq ` β 0 pvq č2 pt, vq ̺r2s
v dv
2
γ 0 pvq β 0 pvq b̌1 pt, vq `
˙
1
β 0 pvq2 b̌2 pt, vq ̺r2s
v dv
2
˘
˘
β pvq α2 pvq ` β 0 pvq α pvq b̌1 pt, vq ` β pvq β 0 pvq č2 pt, vq ̺r1s
v dv
β 0 pvq β pvq b̌1 pt, vq2 ̺r1s
v dv
ˆ
˙
˙
˙
σ̌ S
γ 0 pvq β pvq ` γ pvq ´ v β 0 pvq b̌1 pt, vq ` β 0 pvq β pvq b̌2 pt, vq ̺r1s
v dv.
Γ
u
u
u
The hedging components associated with the MPR depend on hut pZq “ ψ̌ 1t Z ` ψ̌2t Dt ` ψ̌ 3t where,
u
ψ̌ 1t “ ´
żT ˆ
t
α pvq b̌1 pt, vq `
u
ψ̌ 3t “ ´
żT ˆ
t
˙
1
β pvq2 č2 pt, vq dv,
2
β pvq γ pvq b̌1 pt, vq `
u
ψ̌ 2t “ ´
żT
t
β pvq2 b̌1 pt, vq2 dv
˙
1
β pvq2 b̌2 pt, vq dv.
2
The aggregate hedging demand is the the same as in Lemma 31 with coefficients,
´ i,Z
¯
m
u
i,G
i,G
i,Z
ψ̌ 1t “ ψ̌ 1t ` ψ̌ 11 ptq ` ψ̌ 12 ptq ` ω ψ̌ 11 ptq ` ψ̌12 ptq
¯
´
i
i
m
u
ψ̌ 3t “ ψ̌3t ` ω ψ̌31 ptq ` ψ̌ 32 ptq
¯
´
i
i
m
u
ψ̌ 2t “ ψ̌2t ` ω ψ̌21 ptq ` ψ̌ 22 ptq ,
still to be identified. Given the conjecture about the aggregate structure of hedging demands, it must be that
m
h
m
m
α̌h ptq “ ψ̌ 1t σ D , β̌ ptq “ ψ̌ 2t σ D and γˇh ptq “ ψ̌ 3t σ D . Therefore,
α̌h ptq
σD
“
´
ż T ´´
t
1
´
2
h
β̌ ptq
“´
σD
żT
t
¯
´
¯¯
r1s
1 ´ β pvq ̺r1s
α pvq ` α pvq β 0 pvq ̺r2s
b̌1 pt, vq dv
v
v ´ β pvq ̺v
ˆ
˙˙
ˆ
1
β pvq2 ´ 2ωβ 0 pvq β pvq ̺r1s
č2 pt, vq dv
β 0 pvq ̺r2s
v ´
v
2
żT ˆ
t
2
ˆ
β pvq ´ 2ωβ 0 pvq β
pvq ̺r1s
v
1
´ β 0 pvq ̺r2s
v
2
˙˙
b̌1 pt, vq2 dv
˙
˙˙
ˆˆ
˙
ˆ
σ̌S
r2s
β 0 pvq
b̌1 pt, vq dv
β pvq γ pvq ´ ω γ 0 pvq β pvq ̺r1s
γ pvq ´ v ̺r1s
v `
v ´ γ 0 pvq ̺v
Γ
t
ˆ
˙˙
ˆ
żT
1
1
β pvq2 ´ ωβ 0 pvq β pvq ̺r1s
b̌2 pt, vq dv.
´
β pvq ̺r2s
v
v ´
2
2 0
t
´
¯
h
Using Bt b̌1 pt, vq2 “ ´2σ D b̌1 pt, vq2 β ptq ` β̌ ptq ` β D ptq and β ptq “ ωβ0 ptq leads to,
γ̌ h ptq
σD
“
´
żT ˆ
h
Bt β̌ ptq
σD
“
˙
ˆ
1
r2s
β
ptq
̺
β ptq2 ´ 2ωβ 0 ptq β ptq ̺r1s
´
t
t
2 0
66
ˆ żT ˆ
ˆ
˙˙
˙
´
¯
1
h
´2 ´
β pvq2 ´ 2ωβ 0 pvq β pvq ̺r1s
b̌1 pt, vq2 dv σ D β ptq ` β̌ ptq ` β D ptq
β 0 pvq ̺r2s
t ´
t
2
t
˙
ˆ
´
¯
1
h
h
r2s
r1s
´ 2β̌ ptq β ptq ` β̌ ptq ` β D ptq
β ptq2 ´ 2ωβ 0 ptq β ptq ̺t ´ β 0 ptq ̺t
2
ˆ
˙
´
¯
1
h
h
β ptq2 ´ 2β ptq β ptq ̺r1s
´ 2β̌ ptq β ptq ` β D ptq ´ 2β̌ ptq2 .
β 0 ptq ̺r2s
t ´
t
2
“
“
h
Straightforward simplifications, with the definitions β ptq “ β ptq ` β̌ ptq and β ptq “ ωβ0 ptq, give,
˙
ˆ´
´
¯
¯2
¯
1
h
h
h
h
r2s
r1s
´ 2β̌ ptq β ptq ` β D ptq ´ 2β̌ ptq2
β ptq ` β̌ ptq ´ 2β ptq
β ptq ` β̌ ptq ̺t ´ β 0 ptq ̺t
2
´
¯
´
¯
h
h
r1s
D
“
1 ´ 2̺r1s
β ptq2 ` β ptq β 0 ptq ̺r2s
´ β̌ ptq2
t
t ´ 2β̌ ptq β ptq ` β ptq ̺t
˙
ˆ
´
¯
1 r2s
κ̌0 ptq
κ̌1 ptq h
h
h
2
β ptq2 ´ 2β̌ ptq β D ptq ` β ptq ̺r1s
“
1 ´ 2̺r1s
`
´ β̌ ptq2 ”
̺
`
β ptq ´ β h ptq(92)
t
t
t
ω
σD
σD
´
¯
¯
´
r1s
r2s
r1s
where κ̌0 ptq ” 1 ´ 2̺t ` ω1 ̺t β ptq2 σ D and κ̌1 ptq “ ´2σ D β ptq ̺t ` β D ptq .
´
¯
h
Using b̌1 pt, tq “ 1, č2 pt, tq “ 0, Bt b1 pt, vq “ ´σ D b1 pt, vq β ptq ` β̌ ptq ` β D ptq and
´
h
Bt β̌ ptq
σD
“
´
¯
´
¯
h
Bt č2 pt, vq “ ´2σ D b̌1 pt, vq2 α ptq ` α̌h ptq ` αD ptq ´ σ D č2 pt, vq β ptq ` β̌ ptq ` β D ptq
gives,
Bt α̌h ptq
σD
“
“
“
´
¯´
¯
´
¯
r1s
r1s
r2s
1 ´ β ptq ̺t
α ptq ` α̌h ptq ` α ptq β 0 ptq ̺t ´ β ptq ̺t
ż T ´´
´
¯¯ ´
¯ ´
¯
¯
h
r1s
α pvq ` α pvq β 0 pvq ̺r2s
´σ D b1 pt, vq dv β ptq ` β̌ ptq ` β D ptq
´
1 ´ β pvq ̺r1s
v ´ β pvq ̺v
v
t
˙˙ ´
ˆ
ż ˆ
¯ ´
¯
1 T
1
h
2
r1s
r2s
´
β pvq ´ 2ωβ 0 pvq β pvq ̺v ´ β 0 pvq ̺v
´σ D č2 pt, vq dv β ptq ` β̌ ptq ` β D ptq
2 t
2
ˆ
˙˙ ´
żT ˆ
¯ ´
¯
1
1
r2s
β pvq2 ´ 2ωβ 0 pvq β pvq ̺r1s
´
β
pvq
̺
´2σ D b̌1 pt, vq2 dv α ptq ` α̌h ptq ` αD ptq
´
v
v
0
2 t
2
ˆ h
˙
¯
¯
´
¯´
¯
´
α̌ ptq ´
h
r1s
r1s
r2s
D
´ σD
ptq
β
ptq
`
β̌
ptq
`
β
1 ´ β ptq ̺t
α ptq ` α̌h ptq ` α ptq β 0 ptq ̺t ´ β ptq ̺t
σD
˙˙
ˆ
żT ˆ
´
¯
1
β pvq r2s
`2σ D
b̌1 pt, vq2 dv α ptq ` α̌h ptq ` αD ptq
β pvq2 ` β pvq
̺v ´ β pvq ̺r1s
v
2
2ω
t
κ̌2 ptq h
κˇ3 ptq
α̌ ptq `
σD
σD
where,
´
¯
κ̌2 ptq
h
r1s
D
”
1´β
ptq
̺
´
β
ptq
`
β̌
ptq
`
β
ptq
`2σ D
t
σD
κ̌3 ptq
σD
”
żT ˆ
t
1
β pvq2 ` β pvq
2
ˆ
β pvq r2s
̺ ´ β pvq ̺r1s
v
2ω v
˙˙
b̌1 pt, vq2 dv (93)
˙
ˆ
β ptq r2s
r1s
β
ptq
̺
α ptq 1 ´ β ptq ̺r1s
`
̺
´
t
t
ω t
˙˙
ˆ
żT ˆ
´
¯
1
β pvq r2s
`2σ D
b̌1 pt, vq2 dv α ptq ` αD ptq .
β pvq2 ` β pvq
̺v ´ β pvq ̺r1s
v
2
2ω
t
(94)
h
Likewise, using b̌2 pt, tq “ 0, b ptq “ β ptq ` β̌ ptq, β ptq “ ωβ0 ptq, γ ptq “ ωγ0 ptq and
´
¯
h
Bt b1 pt, vq “ ´σ D b1 pt, vq β ptq ` β̌ ptq ` β D ptq
ˆ
ˆ
˙˙
´
¯
σ̌S
h
Bt b̌2 pt, vq “ ´σ D b̌2 pt, vq β ptq ` β̌ ptq ` β D ptq ´ 2b̌1 pt, vq2 µD ` σ D γ ptq ` γ D ptq ´ t
´ 2σ D b̌1 pt, vq2 γ̌ h ptq
Γ
67
leads to,
Bt γ̌ h ptq
σD
ˆ
ˆˆ
˙
˙
˙
σ̌S
r2s
β ptq γ ptq ´ ω γ 0 ptq β ptq ̺r1s
γ ptq ´ t ̺r1s
β 0 ptq
t `
t ´ γ 0 ptq ̺t
Γ
˙
˙˙
ˆ
ˆˆ
˙
żT ˆ
σ̌S
r1s
r2s
v
β
pvq
Bt b̌1 pt, vq dv
β pvq γ pvq ´ ω γ 0 pvq β pvq ̺r1s
`
γ
pvq
´
̺
´
γ
pvq
̺
´
v
v
v
0
0
Γ
t
ˆ
˙˙
żT ˆ
1
1
β pvq2 ´ ωβ 0 pvq β pvq ̺r1s
Bt b̌2 pt, vq dv
´
β 0 pvq ̺r2s
v
v ´
2
2
t
ˆ h
˙
˙
˙
˙
ˆˆ
ˆ
¯
γ̌ ptq ´
σ̌S
r2s
r1s
r1s
D
t
β 0 ptq ´ σ
̺t ´ γ 0 ptq ̺t
β ptq γ ptq ´ ω γ 0 ptq β ptq ̺t `
γ ptq ´
β ptq ` β D ptq
D
Γ
σ
ˆ
ˆ
ˆ
˙˙
˙˙
żT ˆ
1
β pvq r2s
σ̌S
h
D
t
2
D
D
`2
β pvq2 ´ β pvq β pvq ̺r1s
´
γ
ptq
`
γ̌
ptq
`
γ
ptq
´
b̌
pt,
vq
dv
µ
`
σ
̺
1
v
2
2ω v
Γ
t
κˇ5 ptq
κˇ4 ptq h
γ̌ ptq `
σD
σD
“
“
“
where,
κ̌4 ptq
r1s
” ´β D ptq ´ β ptq ̺t ` 2σ D
σD
κ̌5 ptq
σD
”
ˆ
żT ˆ
t
ˆˆ
ˆ
˙˙
β pvq r2s
1
β pvq2 ´ β pvq β pvq ̺r1s
´
b̌1 pt, vq2 dv
̺
v
v
2
2ω
(95)
˙
˙
γ ptq r2s
β ptq γ ptq ´ γ ptq β
`
β ptq
̺
´
ω t
ˆ
˙˙
ˆ
ˆ
˙˙
żT ˆ
σ̌ S
1
β pvq r2s
D
t
2
D
D
γ
ptq
`
γ
ptq
´
b̌
pt,
vq
dv
µ
`
σ
β pvq2 ´ β pvq β pvq ̺r1s
̺
(96)
`2
´
1
v
2
2ω v
Γ
t
ptq ̺r1s
t
σ̌S
γ ptq ´ t
Γ
˙
̺r1s
t
h
S
h
and where σ̌ S
t {Γ is as σt in (14) but with β replaced by β̌ . This completes the proof.
Proof of Proposition 27. Continuity of the fundamental process D implies DT ´ “ DT , at time T´ . At T´ , the
informational advantage of the informed has disappeared. The Gaussian
´ structure and the¯conditional independence
of dividend jumps, implies that optimal demands at T´ are NTι ´ “ Γ DT ´ ` E rǫs ´ SrT ´ {V AR rǫs for ι P tu, i, nu.
ř
Market clearing, ιPtu,i,nu NTι ´ “ 1, gives V AR rǫs {Γ “ DT ´ ` E rǫs ´ SrT ´ , or equivalently, SrT ´ “ DT ´ ` E rǫs ´
V AR rǫs {Γ. The value function of an informed is,
ˆ
„
ˆ
˙ „
ˆ
˙ˇ
ˇ ˙
Xι
∆XTι ˇˇ m ˇˇ ι
´ΓE exp ´ T ´ E exp ´
U ι “ sup
F
ˇ T ´ ˇ F0
ι ,N ι
Γ
Γ
XT
´
T´
´
¯
where ∆XTι “ NTι ´ DT ´ ` ǫ ´ SrT ´ “ NTι ´ pǫ ´ E rǫs ` V AR rǫs {Γq. At the equilibrium price, optimal holdings are
¯ı
”
´
˘
`
∆X ι
“ exp 2Γ1 2 V AR rǫs , it follows that,
NTι ´ “ 1, ∆XTι “ ∆S̃T “ ǫ ´ E rǫs ` V AR rǫs {Γ. As ET ´ exp ´ Γ T
U ι “ sup
ι
XT
´
ˆ
ˆ
˙
„
ˆ
˙ˇ ˙
ˇ
Xι
1
exp
V
AR
rǫs
.
´E Γ exp ´ T ´ ˇˇ F0ι
Γ
2Γ2
For t P r0, T q, the signal structure of the informed is the same as in Section 2. The optimal XTι ´ is the same as the
optimal XTι in the model without jumps. The same holds for the optimal portfolio demands, the WAPR and the market
r
price of risk. As SrT ´ “ DT ´ ` E rǫs ´ V AR rǫs {Γ, the stock price now solves the BSDE, dSrt “ σtS pθtm dt ` dWtm q
subject to SrT ´ “ DT ´ ` E rǫs ´ V AR rǫs {Γ. Therefore, for t ă T ,
żT
”
ı żT r
r
Srt “ Et SrT ´ ´
σvS Et rθvm s dv “ Et rDT ´ s ` E rǫs ´ V AR rǫs {Γ ´
σvS Et rθvm s dv.
t
r
´
Volatility solves the BSDE σtS “ BDt Et rDT ´ s ´
t
şT
t
¯
r
σvS BDt Et rθvm s dv σ D . As Et rDT ´ s “ Et rDT s, by continuity of
r
the fundamental process, this BSDE is the same as without dividend surprises. It follows that σtS “ σtS with σtS as
in Proposition 4. For t ă T , the only effect of the surprise is that the price shifts by the constant E rǫs ´ V AR rǫs {Γ.
Given the price jump ∆SrT “ SrT ´ SrT ´ “ ǫ ´ E rǫs ` V AR rǫs {Γ at T , the formulas stated follow.
68
References
[1] Albuquerque, R. and J. Miao, 2014. Advance Information and Asset Prices, Journal of Economic Theory 149: 236-275.
[2] Andrei, D. and M. Hasler, 2014. Optimal Asset and Attention Allocation, Working Paper,
UCLA.
[3] Banerjee, S. and B. Green, 2015. Signal or noise? Uncertainty and Learning about whether
Other Traders are Informed, forthcoming Journal of Financial Economics.
[4] Black, F., 1986. Noise, Journal of Finance 41: 529-543.
[5] Brennan, M. and H. Cao, 1996. Information, Trade and Derivative Securities, Review of Financial Studies 9: 163-208.
[6] Detemple, J. and R. Kihlstrom, 1987. Acquisition d’Information dans un Modèle Intertemporel
en Temps Continu, L’Actualité Économique 63: 118-137.
[7] Detemple, J. and M. Rindisbacher, 2013. A Structural Model of Dynamic Market Timing,
Review of Financial Studies 26: 2492-2547.
[8] Duffie, D., 2014. Challenges to a Policy Treatment of Speculative Trading Motivated by Differences in Beliefs, Working Paper, Stanford University.
[9] Dumas, B., Kurshev, A. and R. Uppal, 2009. Equilibrium Portfolio Strategies in the Presence
of Sentiment Risk and Excess Volatility, Journal of Finance 64: 579-629.
[10] Ferson, W. and R. Schadt, 1996. Measuring Fund Strategy and Performance in Changing
Economic Conditions, Journal of Finance 51: 425-62.
[11] Glosten, L.R. and P.R. Milgrom, 1985. Bid, Ask and Transaction Prices in a Specialist Market
with Heterogeneously Informed Traders, Journal of Financial Economics 14: 71-100.
[12] Golub, G.H. and C.F. Van Loan, 1996. Matrix Computations, 3rd ed. Baltimore, MD: Johns
Hopkins.
[13] Grossman, S.J., 1976. On the Efficiency of Competitive Stock Markets where Trades have
Diverse Information, Journal of Finance 31: 573-585.
[14] Grossman, S.J., 1978. Further Results on the lnformational Efficiency of Competitive Stock
Markets, Journal of Economic Theory 18: 81-101.
[15] Grossman, S.J., 1980. An Introduction to the Theory of Rational Expectation under Asymmetric Information, Review of Economic Studies 48: 541-585.
[16] Grossman, S.J. and J.E. Stiglitz, 1980. On the Impossibility of Informationally Efficient Markets, American Economic Review 70: 393–408.
[17] Hasler, M. 2012. Portfolio Choice and Costly Dynamic Information Acquisition. Working Paper,
Swiss Finance Institute.
[18] Huang, S, 2015. The Effect of Options on Information Acquisition and Asset Pricing, Working
Paper, University of Hong Kong.
69
[19] Huang, L. and H. Liu, 2007. Rational Inattention and Portfolio Selection, Journal of Finance
62: 1999-2040.
[20] He, H. and J. Wang, 1995. Differential Information and Dynamic Behavior of Stock Trading
Volume, Review of Financial Studies 8: 919-972.
[21] Hellwig, M.F., 1980. On the Aggregation of Information in Competitive Markets, Journal of
Economic Theory 22: 477-498.
[22] Hong, H., Torous, W. and R. Valkanov, 2007. Do Industries Lead Stock Markets. Journal of
Financial Economics 83: 367-396.
[23] Hirshleifer, J., 1975. Speculation and Equilibrium: Information, Risk, and Markets. Quarterly
Journal of Economics 89: 519-542.
[24] Jagannathan, R., Malakhov, A. and D. Novikov, 2010. Do Hot Hands Exist Among Hedge
Fund Managers? An Empirical Evaluation. Journal of Finance 65, 217-255.
[25] Jensen, M., 1968, The Performance of Mutual Funds in the Period 1945–1964, Journal of
Finance 23: 389-416.
[26] Kyle, A.S., 1985. Continuous Auctions and Insider Trading,
Econometrica 53: 1315-1335.
[27] Kyle, A.S., 1989. Informed Speculation and Imperfect Competition, Review of Economic Studies
56: 317-355.
[28] Leland, H.E., 1992. Insider Trading: Should it be Prohibited, Journal of Political Economy
100: 859-887.
[29] Merton, R.C., 1971. Optimum Consumption and Portfolio Rules in a Continuous-Time Model,
Journal of Economic Theory 3: 373-413.
[30] Posner, D. and G. Weyl, 2012. Not Worth the Gamble: Why it Should Be Illegal to Speculate
using Financial Derivatives, Slate, April.
[31] Townsend, R.M., 1983. Forecasting the Forecasts of Others, Journal of Political Economy 91:
546-588.
[32] Vayanos, D. and J. Wang, 2012. Liquidity and Asset Prices under Asymmetric Information and
Imperfect Competition, Review of Financial Studies 25: 1339-1365.
[33] Wang, J., 1993. A Model of Intertemporal Asset Prices under Asymmetric Information, Review
of Economic Studies 60: 249-282.
[34] Wang, J., 1994. A Model of Competitive Stock Trading Volume, Journal of Political Economy
102: 127-168.
[35] Working, H., 1962. New Concepts Concerning Futures Markets and Prices, American Economic
Review 52: 431-59.
70
Figure 1: Sensitivity of WAPR to private information: The figure presents the dynamic
behavior of α ptq for t P r0, T s. Parameter values are T “ 1, σ D “ 0.2, µD “ 0.05, µφ “ E rGs,
σ φ “ ST D rGs, σ ζ “ 0.1. The weight of the informed is ω i “ 1{10. The weight of the speculating
noise trader varies between ω n “ 1{125 (left panel), ω n “ 1{60 (middle panel) and ω n “ 1{4 (right
panel).
-0.45
-0.2
18
α(t)
α(t)
α(t)
-0.5
-0.3
16
-0.55
-0.5
-0.6
-0.7
WAPR sensitivity: α(t)
14
WAPR sensitivity: α(t)
WAPR sensitivity: α(t)
-0.4
-0.6
-0.65
-0.7
-0.75
-0.8
-0.8
12
10
8
6
-0.85
-0.9
4
-0.9
-1
-0.95
0
0.1
0.2
0.3
0.4
0.5
time: t
0.6
0.7
0.8
0.9
1
2
0
0.1
0.2
0.3
0.4
0.5
time: t
71
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
time: t
0.6
0.7
0.8
0.9
1
Figure 2: Stock volatility, and sensitivity of WAPR and stock price to fundamental
p ptq, β ptq and B
p ptq, B ptq for
information: The figure presents the dynamic behavior of σ
pSt , σtS , β
D
D
φ
t P r0, T s. Parameter values are Γ “ 1{4, T “ 1, σ “ 0.2, µ “ 0.05, µ “ E rGs, σ φ “ ST D rGs.
The weights of informed and the speculating noise traders are ω i “ 1{10 and ω n “ 1{5. Signal
noise volatility varies between σ ζ “ 0.25 (left panel), σ ζ “ 0.1 (middle panel), and σ ζ “ 0.05 (right
panel).
0.2
0.2
0.2
σbtS
σtS
0.195
σbtS
σtS
σbtS
σtS
0.18
0.18
0.19
0.16
0.18
0.175
0.17
0.14
volatility
0.16
volatility
volatility
0.185
0.14
0.12
0.12
0.1
0.08
0.165
0.06
0.1
0.16
0.04
0.155
0.08
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.02
0
0.1
0.2
0.3
0.4
time: t
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
time: t
0
-0.45
0.6
0.7
0.8
0.9
1
0
b
β(t)
β(t)
-0.5
0.5
time: t
b
β(t)
β(t)
b
β(t)
β(t)
-1
-5
-0.6
-0.65
-0.7
-2
WAPR sensitivity
WAPR sensitivity
WAPR sensitivity
-0.55
-3
-4
-10
-15
-0.75
-20
-5
-0.8
-0.85
-6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-25
0
0.1
0.2
0.3
0.4
time: t
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
time: t
1
1
1
b
B(t)
B(t)
b
B(t)
B(t)
b
B(t)
B(t)
0.9
0.9
0.95
0.8
0.9
0.85
0.8
price sensitivity
price sensitivity
price sensitivity
0.5
time: t
0.7
0.6
0.7
0.6
0.5
0.4
0.3
0.8
0.5
0.2
0.75
0.4
0
0.1
0.2
0.3
0.4
0.5
time: t
0.6
0.7
0.8
0.9
1
0.1
0
0.1
0.2
0.3
0.4
0.5
time: t
72
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
time: t
Figure 3: Pareto ranking of equilibria: This figure presents the different components of ∆P̂ {Γ `
∆T̂ u as a function of risk tolerance Γ. Parameter values are T “ 1{24, D0 “ 1, ω i “ 1{10, ω n “ 1{5,
σ D “ 1.18, µD “ 0.05, µφ “ E rGs, σ φ “ ST D rGs. The precision of the private information signal
varies between σ ζ “ 0.5 (left panels), σ ζ “ 0.1, (middle panels) and σ ζ “ 0.05 (right panels). The
top row shows welfare components based on optimal demands. The bottom row presents welfare
components based on demands excluding hedging components.
0.2
2.5
0.2
∆T̂ u + ∆ΓP̂
∆P̂
Γ
u
∆T̂ u + ∆ΓP̂
y
u
∆P̂ y
Γ
∆T̂ u
2
∆T̂ u + ∆ΓP̂
∆T̂ u
∆ P̂ y
Γ
0
-0.2
0
welfare component
welfare component
welfare component
1.5
-0.2
∆T̂
u
u
1
0.5
0
-0.5
-1
-0.4
-0.4
0
1
2
3
4
5
6
-1.5
0
1
2
risk tolerance: Γ
3
4
5
10
∆T u + ∆PΓ
8
∆P y
Γ
∆T u
4
4
2
0
-2
∆T
2
0
-2
2
0
-2
-6
-8
-8
-8
-10
-10
-10
6
0
1
2
3
risk tolerance: Γ
73
4
5
u
∆P y
Γ
4
-4
5
∆T u + ∆PΓ
6
-6
4
6
∆T u
-4
3
5
8
-6
risk tolerance: Γ
4
u
-4
2
3
u
welfare component
6
welfare component
6
1
2
10
u
∆P y
Γ
0
1
risk tolerance: Γ
10
∆T u + ∆PΓ
8
welfare component
0
6
risk tolerance: Γ
6
0
1
2
3
risk tolerance: Γ
4
5
6
1.5
2
1
1
2
0
0
-0.5
-1
-1.5
-1
-2
-3
-4
-5
Gains from acquisition: GA
CE no acquisition: CE i,na
CE with acquisition: CE i,wa
Acquisition costs: C
-2
certainty equivalents (CE)
0
0.5
certainty equivalents (CE)
certainty equivalents (CE)
Figure 4: Pareto ranking of equilibria without speculative noise trading: This figure
presents the gains from information acquisition in the absence of speculative noise traders as a
function of risk tolerance Γ (GA “ CE i,wa ´ CE i,na ) where CE i,wa and CE i,na are the certainty
equivalents of the informed with and without information production. Parameter values are T “ 1,
D0 “ 1, ω i “ 1{10, σ D “ 0.2, µD “ 0.05, µφ “ E rGs, σ φ “ ST D rGs. The precision of the private
information signal is σ ζ “ 0.25 (left panel), σ ζ “ 0.10 (middle panel) and σ ζ “ 0.05 (right panel).
Certainty equivalent acquisition costs are C “ 0.5
Gains from acquisition: GA
CE no acquisition: CE i,na
CE with acquisition: CE i,wa
Acquisition costs: C
-6
-2.5
1
2
3
risk tolerance: Γ
4
5
6
-4
-6
-8
Gains from acquisition: GA
CE no acquisition: CE i,na
CE with acquisition: CE i,wa
Acquisition costs: C
-10
-7
0
-2
-12
0
1
2
3
risk tolerance: Γ
74
4
5
6
0
1
2
3
risk tolerance: Γ
4
5
6