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Edited by: Yong Hyun Shin, Sookmyung Women's University, South Korea

Reviewed by: Kyoung Jin Choi, University of Calgary, Canada; Byung Hwa Lim, The University of Suwon, South Korea

*Correspondence: Fangfei Dong, Department of Applied Mathematics and Statistics, Stony Brook University, B148 Math Tower, Stony Brook, NY 11794, USA

This article was submitted to Mathematical Finance, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We propose a general equilibrium model for asset pricing that incorporates asymmetric information as the key element determining security prices. In our setting, the concepts of completeness, arbitrage, state price and equivalent martingale measure are extended to the case of asymmetric information. Our model shows that in a so-called quasi-complete market, agents with differential information can reach an agreement on an universal equilibrium price. The corresponding state price and martingale measure are derived. The key intuition is that agents evaluate consumption choices conditioned on their private information and the public information generated by the price. As a consequence, information asymmetry can lead to mispricing as well.

General equilibrium theory deals with an economy consisting of multiple agents in a market endowed with initial resources and willing to exchange commodities with others. It considers the behavior of the economy as a closed and inter-related system. In a general equilibrium perspective, each agent in the market optimizes his/her behavior to achieve maximum consumption utility. Agents' optimal behavior represents the behavior of the economy. Equilibrium prices are determined endogenously. The existence of such an equilibrium is based on the assumption of perfect competition among agents. In other words, the model assumes all individuals are price-takers, i.e., that they have zero price impact. This type of model is often called Walrasian equilibrium, from the Walras [

Demarzo and Skiadas [

Our work is in line with the formalization of the Consumption-based Capital Pricing Models in Duffie [

In Section 2, we describe the setup of our model. In Section 3, we re-introduce the concepts of arbitrage, state price and martingale measure in the context of asymmetric information. In Section 4, we define the equilibrium and discuss the characteristics of the equilibrium price. Section 5 provides several examples to illustrate the differences in equilibrium prices between the case with information asymmetry and the one without. Mispricing as a consequence of asymmetric information is also discussed.

In this section, We formulate a general equilibrium model with information asymmetry in discrete time. In our setting, there are finite number of agents in economy. As in other general equilibrium models, the agents are assumed to be price takers. Agents make their consumption plans based on not only their own private information but also the information generated by the price process. An agent prices a consumption process by her own pricing function, which is obtained from maximizing her utility. That means, for some consumption process, the prices as seen by different agents are allowed to be different. However, since agents are price takers and the market must clear, it is shown that there exists a so-called quasi pricing kernel such that the pricing function of each agent is the optimal projection of the quasi pricing kernel onto her own information filtration. Equivalently, there exists a probability measure, a so-called quasi equivalent martingale measure, under which a security price is viewed as a martingale for each agent (i.e., conditioned on individual information filtration).

There are _{0}, _{1}, …, _{T}}. _{t} denotes the information up to time _{T} =

In our economy, there are ^{(i)}), utility function ^{(i)}, endowment ^{(i)}, and information flow, represented by the filtration

Each individual filtration satisfies ^{(i)} is 𝔽^{(i)}-adapted for each

There exist _{j} = (δ_{j}(_{1}(_{2}(_{m}(_{u}:_{t} for all

For simplicity, we also assume all agents share the same (physical) probability measure, i.e., ^{(i)} =

The security price at time _{1}(_{m}(^{(i)}. Denote by _{t} = σ(_{u}:

Agents in the economy will choose their trading strategies to optimize their satisfaction (which will be represented by utility later). However, the trading strategies each agent can access are restricted by the information she can access. More information will give more choice of trading strategies. This means that an agent's trading strategies are adapted to the information available to her.

^{(i)}(_{++} which is given by the budget constraint:

The budget constraint makes sure that the consumption cannot exceed the sum of endowment, dividends and trading profit. Additionally, for convenience, we can define a strategy-generated dividend process as follows.

The dividend generated by trading strategy θ can be interpreted as the trading profit generated from price changes and dividends. Then the feasible consumption set for agent

^{θ} > 0.

Recall ^{(i), S} the space of process adapted to the filtration generated by ^{(i), S} = {δ^{θ}:θ ∈ ^{(i), S}} and ^{θ}:θ ∈

^{θ}) = 0 ^{(i), S} for some i

^{(i), S} for all i

_{D}(_{D}(0) = 0, and ϵ_{D}(_{f}(_{f}(0) = 0 and ϵ_{f}(

Then this example can be illustrated by the tree with (δ(_{D}(

Now suppose there two agents. One is uninformed and can only observe dividends, and the other is an insider who knows the true fundamental. Thus, the uninformed agent cannot distinguish the two nodes A,B as shown in the red boxes, and the insider can distinguish between the two nodes. Let price given by probability measure

We call a strictly positive process π, which is adapted to 𝔽, a state-price deflator if, for all

Suppose there exists a risk-free short-rate process

In the classic case of symmetric information, a security's price equals the expectation of aggregated future discounted dividend flow under the equivalent martingale measure, and also equals the expectation of aggregated discounted future dividend flows under the physical probability measure. To extend the analysis to asymmetric information, we allow each agent to have her ownstate-price deflator, which is the optimal projection of the universal state-price deflator to her own information.

Let us first investigate some process _{t}}_{t ∈ T}, where _{t} ⊆ _{t} for all ^{ℍ} the optimal projection of

Some simple facts:

^{ℍ} =

^{ℍ} is ℍ-martingale if

For an equivalent probability measure ^{ℍ} on some smaller filtration ℍ as the optimal projection of ξ onto ℍ, i.e.,

_{t}-measurable for all 0 ≤

If there exists no arbitrage in _{t} is the information set can be accessed at time _{t}-measurable, which means security prices are public information. Moreover, for any strategy in ^{ℍ}, the space of ℍ-adapted processes,

Now, bearing in mind that all agents are price takers, we can start to construct the martingale measure and state-price deflator for asymmetric information. Assume that the risk-free rate is known to all agents.

Denote the density processes of

Under a Quasi-EMM, a security's price equals the expectation of discounted future dividends conditioned on individual information for every agent. All agents in our economy agree on the same (fair) prices of securities, which are exactly consistent with the assumption that all agents are price takers. Correspondingly, we can define a quasi state-price deflator as follows.

^{(i)} is the optimal projection of π onto

^{(i)}^{(i), S}, i

The following theorem shows the relationship between a quasi-EMM and a quasi state-price deflator, which is similar to the classic case of symmetric information.

The following proposition gives a strong condition for the existence of a quasi state-price deflator, which is actually for the case of symmetric information.

^{(i), S} for all i

The use of quasi state-price deflator π is to price any consumption process, that is, the price of a consumption process _{+} is given by

If ^{(i), S} for some

But for some other agent

This argument can be extended to any consumption _{+}. The price of any consumption _{+} for an agent

And the information fee η is

Thus, η = 0 when ^{(i), S}.

The objective of each agent is to maximize her individual utility by choosing the optimal feasible trading strategy. Then the optimization problem for agent

The following claims show the relation between arbitrage and individual optimization problem.

^{(i), S}

^{(i), S} and U^{(i)} is continuous

If the individual optimization (1) has a strictly positive solution ^{*}, and ^{(i)} is continuously differentiable at ^{*}, then

Note that ∇^{(i)}(^{*}; ·):

^{*} and U has a strictly positive continuous derivatives at c^{*}. Then the Riesz representation π of ∇U^{(i)}(c^{*}; ·) satisfies that

If we restrict utility satisfying additive form, then we can have the following. Suppose ^{(i)}, for each

Then for any

^{(i)} solves individual optimization

If a SSE equilibrium exists, that means the equilibrium consumption
^{(i)}(^{(i)*}; ·) gives this agent ^{(i)}, which is adapted to individual information filtration, up to a positive multiplier. In order to construct a universal pricing kernel π, such that π^{(i)} equals to the optimal project of π on individual information. If this universal pricing kernel π exists, then π is a quasi state-price deflator, and Π(^{(i)}(^{(i)}(^{(i)}) for all ^{(i)}(·) = 〈π^{(i)}, ·〉.

^{(1)}, …, π^{(n)}) obtained from Proposition 6 admits a quasi state-price deflator

When the market is quasi-complete, the existence of the universal π will reduce the individual optimization to the following form:

Since ^{(i)} is strictly increasing, there is a Lagrange multiplier λ^{(i)} such that the optimization above is equivalent to

Define the utility function _{λ}: _{+} → ℝ by

^{(i)} is of additive form

_{λ} is also of additive form

Denote by 𝔼_{i, t} the expectation conditioned on _{i, t} the variance conditioned on _{i, t} the covariance conditioned on

Now, let us define the capital returns generated by a trading strategy θ by

Denote by ^{(i), 0}(^{(i), 0} is the return of a strategy θ_{0}, such that,
_{i, t} is the correlation conditioned on

Each agent also believes in her own market portfolio. That is, for each agent ^{(i), S} by maximizing the the correlation with π conditioned on their own information. That is , at time

Let ^{(i), M}(^{(i), S}. Note that

Then we can obtain the beta form of the CAPM:

_{D}(_{D}(0) = 0, and ϵ_{f}(_{D}(1), ϵ_{D}(2), ϵ_{f}(1), ϵ_{f}(2) are jointly normally distributed.

Now suppose no agents has private information, i.e., the only information agents can access are generated by dividend and price.

Let (Quasi-)EMM be the probability measure under which ϵ_{D}(1), ϵ_{D}(2), ϵ_{f}(1), ϵ_{f}(2) are independent standard normally distributed

In this case, price reveals no additional information beyond dividends.

_{y}(

_{D}(1), ϵ_{D}(2), ϵ_{f}(1), ϵ_{f}(2), ϵ_{y}(1) are independent standard normally distributed

_{f}(1), ϵ_{D}(1), ϵ_{y}(1), ϵ_{f}(2), ϵ_{D}(2) are jointly normally distributed with mean

Then the price given by the following is valid,

In this example, we can see when there exists private information, either case can happen: (i) the signal is incorporated into price as factor, and (ii) price is very inefficient and reveals no private information.

_{1}(_{2}(

where ϵ_{y}(_{1}, and agent 2 can access signal _{2}.

Let Quasi-EMM be the probability measure under which ϵ_{D}(1), ϵ_{D}(2), ϵ_{f}(1), ϵ_{f}(2), ϵ_{1}(1), ϵ_{2}(1) are independent standard normally distributed

In this example, price reflects aggregated private information.

Suppose we have a pricing kernel π and that its optimal projections on individual information filtration are π^{(i)}'s. Consider a derivative written by agent ^{θ}, where θ is a trading strategy in ^{(w), S}. That means the writer, agent

If θ ∈ ^{(i), S}, agent ^{(i)} + δ^{θ} cannot increase the utility of this agent

If θ ∉ ^{(i), S}, agent ^{(i)} + δ^{θ} can increase her maximum utility, i.e., ^{(i)}(^{(i)}(^{(i)*}). The maximum price at which agent ^{(i)}(^{(i)}(^{(i)*}). In this case, agent

In this paper, we formulate a general equilibrium asset pricing model in discrete time. It is shown that with the existence of differential information, agents can still achieve agreement on a universal trading price. This equilibrium price reflects the private information through the individual choice of consumption. As an inter-related system, the equilibrium price is also viewed as a public information resource by each agent. The information asymmetry also influences individuals portfolio choices and the consumption beta. This is rooted in the fact that agents price the same consumption choices differently. Mispricing is also discussed as a consequence of these facts.

Both authors contributed significantly to all aspects of this work.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Many thanks to Professor Svetlozar Rachev, Professor Noah Smith, Professor Haipeng Xing, and Professor Aaron Kim for stimulating discussions. Without their help, this paper could not be finished.

The Supplementary Material for this article can be found online at: