The state price density (SPD) is a central concept in modern asset pricing theory.1 Given the SPD and its probability distribution, we can price every asset. Thus, it is essential to derive the SPD for asset pricing. A lot of studies have attempted to derive the SPD from both a theoretical and an empirical viewpoint.
Most asset pricing models suppose that an agent can assign a unique probability distribution over a state space. However, it is commonly observed that such a unique probability is not available in the economy. It is known that expected utility, which is a dominant tool in asset pricing theory, cannot describe choice under ambiguity.2 Many ambiguity models, such as those of  , have been proposed to capture these choices. Following  , ambiguity is represented by a second-order probability over the set of first-order probabilities over the state space in this paper.
This paper unites the above two lines of research. The purpose of this paper is to characterize the SPD in the presence of ambiguity. More precisely, we examine equilibrium in a single-period pure exchange economy under ambiguity and derive the SPD by using the dual theory of the smooth ambiguity model  . We also derive the equilibrium excess return using the SPD, which can be viewed as an extension of the classical capital asset pricing model (CAPM)   .
The original theory of the smooth ambiguity model by  takes the “double” expected utility form as the second-order expectation of the transformed first-order expected utility. The tractability of this model is a distinct advantage compared with other existing ambiguity models. The ambiguity preference is reflected by the shape of the transformation. Unlike the original theory, the dual theory of the smooth ambiguity model by  captures ambiguity preference by a distortion of the second-order probability distribution. As shown in , an equivalent representation of the dual theory is the “single” expected utility with respect to a mixture of the first-order probability distributions with the distorted second-order probability distribution. This form reinforces the advantage of the original theory that the existing results in the expected utility are applicable to decision problems under ambiguity, while maintaining descriptive validity for ambiguity. This advantage makes it easy to characterize equilibria under ambiguity compared with the original theory as shown later in this paper. Thus the dual theory might be a powerful tool for these kinds of analyses.
A series of studies on equilibrium analysis in securities markets precede this paper. This paper derives the SPD in an economy with a representative agent. For the construction of the representative agent, we use the idea of assigning proper weights to each agent, which goes back to  .  gave the standard approach for an optimal/production model of a representative agent leading to equilibrium. The SPD can be viewed as a generalization of Bühlmann’s economic premium principle  in an economy under ambiguity. This paper also shows the existence and the uniqueness of the equilibrium in an economy.  pioneered establishing the existence of equilibria in a complete market. We adapt the dual method of   to show the existence and the uniqueness of the equilibrium in an economy.
Using the SPD, we extend the classical CAPM to an economy under ambiguity. Thus, this paper is related to the recent literature on the CAPM under ambiguity. In particular,  and  considered the CAPM under ambiguity using the smooth ambiguity model. Even though these papers derive a similar form of the excess return in the CAPM, it should be noted that we do not require any approximations to derive the CAPM, whereas  and  used a quadratic approximation. Furthermore, the optimal portfolio for each agent can be shown to consist of his specific portfolio in addition to a safe security and the market portfolio. As a special case, the optimal portfolio degenerates to the classical separation by , that is, an agent’s specific portfolio disappears.
Because we adopt the dual theory of the smooth ambiguity model by , ambiguity preference is embedded in the expected utility representation with a distorted probability. In other words, ambiguity preference does not explicitly appear for most of the main analysis. While this tractability is a distinct advantage compared with existing models, ambiguity preferences nonetheless have an effect on equilibria in securities markets. We clarify these effects through comparative statics of the SPD. Using the results, we also determine the effects of ambiguity preference on the excess returns of ambiguous securities based on the CAPM derived in the paper.
The organization of this paper is as follows. The next section introduces some settings and notions about the economy and agents, and provides some preliminary analysis. In Section 3, we derive the SPD and show its existence and uniqueness in the economy. In Section 4, we apply the SPD to obtain an equilibrium asset return which can be rewritten as the CAPM in the presence of ambiguity. Furthermore, we show how an agent’s portfolio is decomposed and obtain the classical two-fund separation from  as a special case. In Section 5, we perform some comparative statics to examine the effect of ambiguity on the SPD and the equilibrium excess return. The final section contains concluding remarks. All proofs of the propositions are collected in the Appendix.
2. The Model
We consider a single-period pure exchange economy. All of the uncertainty is described by a finite discrete state space . Agents in the economy can trade a safe security and S ambiguous securities. The rate of return of the safe security is a constant . The rate of return of the i-th ambiguous security is given by a random variable . The rate of return , , can be decomposed into a constant term and a random term such that
where if occurs for .
We define the matrix by
where , . We assume that the rank of is equal to S, that is, the security market is assumed to be complete.3 Ambiguity is represented by a finite set of probability distributions on :
Given a unique probability distribution over the index set , a compound probability distribution on is defined by
Each agent is assumed to consider this to be the reference probability distribution in the economy.4 Without loss of generality, we assume that , .
We define the SPD as a non-negative random variable on satisfying
There are agents in the economy. Let , , denote agent ’s terminal income if the state , , occurs, let denote agent ’s initial wealth, and let denote the amount that agent k invests in the i-th ambiguous security. For a given state , the terminal wealth , , of agent is given by
where and , and where is an S-dimensional vector of 1s and is an -matrix of 1s.5 The portfolio is called admissible for initial wealth if
We assume that the axioms of the dual theory of the smooth ambiguity model  hold in the economy. Under this setting, agent is assumed to evaluate their terminal wealth using
Here, is a probability distribution on defined by
where the probability distribution is a distortion of
that reflects agent ’s attitude towards ambiguity (see Corollary 1 of  ). To avoid technical difficulties, we set the utility function to be a map from to that is strictly increasing, strictly concave and
continuously differentiable, with and
For a given initial wealth , agent chooses an admissible portfolio so as to maximize his welfare represented by (4) over the class of portfolios
In other words, each agent computes the value function
To solve this problem, we define a function by
where is the likelihood ratio defined by , and is the inverse function of the marginal utility . We note that is a map from onto itself with and .
Under the settings above, the agent’s optimal wealth and the optimal portfolio are given by the following proposition.
Proposition 1. Suppose that
Then agent ’s optimal wealth and optimal portfolio are given by
where is a solution to the equation of the budget constraint:
and is the inverse of .8
In this section, we show the existence and the uniqueness of the equilibrium and derive the SPD. Before proceeding with the analysis, let us start by defining equilibria in the economy.
Definition 1. An equilibrium is defined as a set of pairs , , of an optimal portfolio and an optimal terminal wealth satisfying the following equations:
where and are the aggregate initial wealth and the aggregate terminal income defined by and , , respectively.
From (6), it follows that (10) is equivalent to
For arbitrary and for each , let the function be defined by
Then (11) is equivalent to
with . If the inverse function of is defined by
then the SPD in equilibrium is given by
The budget constraint (8) in equilibrium can be rewritten as
This means that the equilibrium can be characterized by to satisfy (13).
For each , and , we define the utility function of the representative agent by
where denotes , .
The SPD in equilibrium is characterized in the following proposition.
We note that the utility function of the representative agent has positive homogeneity with respect to by the definition, that is, for any positive constant , . Hence, this proposition implies that also has this property with respect to . That is, for each , and each positive constant c, the following equation holds:
From this fact, we can also confirm that the budget constraint (13) does not change for any positive homogeneity with respect to . That is, we can write
for every positive constant .
Let be the index of absolute risk aversion for the representative agent, defined by
We immediately obtain the following corollary to the above proposition.
We note that Corollary 1 is an extension of the general economic premium principle of Bühlmann under risk  to that under ambiguity.
The following proposition states the existence and the uniqueness of the equilibrium.
Proposition 3. There exists a satisfying (13). Furthermore, suppose that the marginal utilities , , satisfy the condition
is increasing with respect to . (17)
Then is unique up to positive constant multiples.
4. Asset Pricing
The following proposition characterizes the excess return in equilibrium.
Let be a portfolio satisfying
We note that there exists a unique because the market is complete. We refer to as the market portfolio. The terminology of the market portfolio is used as the common portfolio that every agent in the economy holds. At the end of this section, we show that the market portfolio degenerates to the classical market portfolio under certain conditions. The following corollary is a natural extension of CAPM   under ambiguity.
Corollary 2. Let be the rate of return of the market portfolio and let be its expected return.
and Cov and Var denote the covariance and the variance under , respectively.
Next, we show that the classical two-fund separation theorem  holds in a special case of my model. By this result, our analysis can be seen as a natural extension of CAPM under risk to that under ambiguity. Before proceeding, we note that each agent’s optimal portfolio can be decomposed as a sum of the market portfolio and his specific portfolio. From (7) and (19), the following proposition holds.
Proposition 5. For agent , the optimal portfolio can be decomposed as a sum of the market portfolio and his specific portfolio as follows:
The following is the two-fund separation theorem under ambiguity.
Proposition 6. Assume that all agents have quadratic utility functions, that they are all ambiguity neutral, and that all their terminal incomes are proportional to the aggregate income . Then the optimal portfolio for each agent consists of the market portfolio and the safe security.
5. Some Comparative Statics
We examine how ambiguity preference influences the SPD in equilibrium. We also determine the effects of ambiguity preference on equilibrium excess returns based on the CAPM derived in the previous section.
To keep the analysis simple, we consider a specific case with , , and .9 We refer to this economy as a two-state economy. In the two-state economy, the probability distributions over the index set are given as follows:
Then the reference probability distribution , and agent ’s probability distribution , , are given by
By Definition 1, the SPDs , , in equilibrium are given by a solution of the following simultaneous equations:
We impose the following assumption on the economy to get explicit results.
Assumption 1. a) Each agent gains strictly higher expected utility under the first-order probabilities with than with ; that is, for ,
b) Each agent has log utility; that is, , .
Remark 1. We note that if Assumption 1 (a) holds and agent k, , is strictly ambiguity averse (loving) in the two-state economy, then holds from Corollary 1 and Proposition 1 of  .
First, we consider the effects of ambiguity aversion and loving on the SPD in equilibrium.
Proposition 7. In the two-state economy, suppose that
is satisfied under Assumption 1. Then the following statements hold.
1) If each agent is strictly ambiguity averse, then the SPD with ambiguity is strictly lower (higher) than that without ambiguity, and the SPD with ambiguity is strictly higher (lower) than that without ambiguity.
2) If each agent is strictly ambiguity loving, then the SPD with ambiguity is strictly higher (lower) than that without ambiguity, and the SPD with ambiguity is strictly lower (higher) than that without ambiguity.
As stated in Remark 1, if all agents are strictly ambiguity averse, then they uniformly increase the weights , , of the index as under Assumption 1 (a). This leads to for both under the condition . As a result, ambiguity aversion increases (decreases) and decreases (increases) . The same reasoning can be applied to the case of ambiguity loving.
Next, we consider the effect of more ambiguity aversion on the excess returns of the ambiguous securities in equilibrium. We compare two economies consisting of the same two-state economy except for the ambiguity preferences of each agent. To distinguish between the two economies, we call them Economy A and Economy B. We assume that all of the agents in Economy A are more ambiguity averse than those in Economy B in the sense of  . Under this assumption, we obtain comparative static predictions for how more ambiguity aversion influences the equilibrium excess returns as a corollary of Proposition 7.
Corollary 3. Assume that the conditions of Proposition 7 hold. If the random terms of the rate of return for the i-th ambiguous security are arranged in the order
then the excess return in equilibrium in Economy A is lower (higher) than that in Economy B.
If the order of the random terms is reversed, that is, , then the excess return in equilibrium in Economy A is higher (lower) than that in Economy B.
We note that, from Theorem 2 of  and Proposition 7, the SPD in Economy A is lower (higher) than that in Economy B if . Thus, the excess return in Economy A is lower (higher) than that in Economy B in the case where (22) holds. The last statement of the corollary also holds by the same reasoning.
6. Concluding Remarks
This paper studies an equilibrium asset pricing model for a static pure exchange economy with ambiguity. The preference of an agent in the economy is represented by the dual theory of the smooth ambiguity model from  . An equilibrium is fully characterized by the SPD, so we derive the SPD and show its existence and uniqueness in the economy. Applying the SPD, the equilibrium excess return is derived. Equilibrium excess returns can be rewritten in an extended version of CAPM under ambiguity. The optimal portfolio consists of the agent’s specific portfolio in addition to the safe asset and the market portfolio. The classical separation theorem is obtained as a special case. We also conduct comparative statics analysis on a specific case of the two-state economy and show how ambiguity preferences influence the SPD and returns of ambiguous securities in the equilibrium.
This work was supported by JSPS KAKENHI Grant Number JP17K03825.
Appendix A. Proofs
A.1. Proof of Proposition 1
is trivially an admissible portfolio because from (6). We first show that . It is obvious that holds by (8).
Following   and , the convex dual of is defined by
and is a decreasing, convex and continuously differentiable function on , satisfying
Using the convex dual, we have
Because , we have , and so
From (23), we have
Next, we show that is optimal. For all and ,
The first inequality is due to the budget constraint, . The second inequality and the equality are due to the definition of . From (6) and (8), the expression (24) holds with equality if and only if and . This means that is optimal from (7).
A.2. Proof of Proposition 2
Let be Defined by
by the definitions of and . This means that satisfies (10).
Since is the inverse of , (25) can be rewritten as
Since is strictly concave,
for all . This holds with equality if and only if . Hence, we have
Differentiating the above equation completes the proof.10
A.3. Proof of Corollary 1
From (16), there is a constant for which
The result then follows from Proposition 2 and the fact that .
A.4. Proof of Proposition 3
We first show the existence. Let be an index set of agents and let be the -dimensional unit coordinate vectors. For any , we denote the convex hull of by
Let be the set
For each , we define a function by
To prove that the proposition holds, we have to show that there exists a satisfying for each .
Because the function is continuous, the set
is closed. On the other hand, from (11) and (12),
Now, suppose that there exists a such that . Then for all , which contradicts (26). Therefore,
Furthermore, suppose that there exists a such that . Then for all . In this case, let for . Then and , which again contradicts (26). Therefore,
From (27) and the Knaster-Kratowski-Mazurkiewicz Theorem (cf. p. 26 of  ), the set is nonempty. For any , we have
Otherwise we would have , which contradicts (26). We also have , because if there exists a such that , then , which contradicts (28). Therefore, we can conclude that there exists a that belongs to .
Next, we show the uniqueness. For any pair of vectors , we consider the usual order:
Let and be vectors in , both of which satisfy (13). We define another vector by for a positive constant . Then, from (15), also satisfies (13), and . If , then is a positive constant multiple of . Hence, we have only to show that does not hold.
Suppose that . Then, from the definition of ,
holds. Hence, for such that , we have
On the other hand, because is the inverse of , (17) is equivalent to decreasing with respect to . Hence, noting that , , , we have
This inequality leads to
Combining (29) and (30), we have
However, this contradicts (28). That is, never holds.
A.5. Proof of Proposition 4
From (2), we have
where we use by (1).
A.6. Proof of Corollary 2
From (18) and (19),
where we have used the fact: in the second equality. From (31) and the definition of , we have
Canceling out from (31) and (32), we obtain
we obtain the result.
A.7. Proof of Proposition 6
Because all agents have quadratic utility functions, we can assume that the marginal utility for each agent k, , is given by
for some constant . Noting that ambiguity neutrality implies that , , (see,  ), it follows from (6) that
Because the terminal income is proportional to the aggregate income , there exists a constant satisfying and . Hence, from (8),
Substituting (34) into (33), we have from (3) that
Summing the above equations for , and applying (9), we have
where . Substituting this into (35), we have
Hence, from Proposition 2 and by the definition of the market portfolio , the optimal portfolio satisfies
where . This means that the optimal portfolio consists of the safe security and the market portfolio.
A.8. Proof of Proposition 7
Under the condition: , if each agent is strictly ambiguity averse, from the definition of the probability distributions , , and Remark 1,
Similarly, under the same condition, if each agent is strictly ambiguity loving, . Hence, to prove the proposition it is sufficient to show that
From Assumption 1 (b), (21) can be explicitly rewritten as
Solving the above equations with respect to , we obtain
where we put , and . From this, we obtain for :
where we put , . Hence we obtain (36).
A.9. Proof of Corollary 3
Let be a probability distribution over the states defined by
We first note that, from (18) and Proposition 2, the excess returns in equilibrium are given by
where denotes the expectation under . For each state , let and denote the SPD in equilibrium for Economies A and B, respectively. Similarly, let and denote the probability distribution s for Economies A and B, respectively. From Theorem 2 of  and Proposition 7, the following implication holds:
From (38), this implies that if , then and , while if , then and . Hence, from (39), we obtain the result.
1There are various terminologies for representing the same concept, such as stochastic discount factor, pricing kernel and others (see, e.g.,   ).
2Risk is defined as a situation in which the probabilities over the state space are uniquely assigned. Ambiguity is defined as a situation in which the probabilities over the state space are either not uniquely assigned or are unknown.
3While we assume a complete market for simplicity, we can relax this assumption by using, for example, the embedded market approach in , i.e., an augmented complete market with constraints on sales and purchases for some securities.
4Ambiguity neutral agents, who are equivalent to expected utility maximizers, evaluate ambiguity using this probability distribution.
5 denotes the transpose.
6See  .
7 denotes the negative part of the utility, that is,
8Because the market is assumed to be complete, this inverse matrix exists.
9Although a similar result is expected to obtain in the general case, we leave this for future research.
10From (12), we use the fact that
 Karatzas, I., Lehoczky, J.P. and Shreve, S.E. (1990) Existence and Uniqueness of Multi-Agent Equilibrium in a Stochastic, Dynamic Consumption/Investment Model. Mathematics of Operations Research, 15, 80-128.
 Lintner, J. (1965) The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economic Statistics, 47, 13-37.