Endogenous incompleteness of financial markets: The role of ambiguity and ambiguity aversion

Incompleteness of financial markets has been widely questioned in the literature, but traditional research has been mainly focused on the role of transaction costs and asymmetric information in determining such incompleteness. This paper, instead, focuses on agents’ preferences, showing that the introduction of ambiguity and ambiguity aversion may induce investors to restrict their trading to a simpler set of assets, relative to which they are less likely to make errors.     

A dual approach to ambiguity aversion

In the present paper, the assumption of monotonicity of Anscombe and Aumann (1963) is replaced by an assumption of monotonicity with respect to first-order stochastic dominance. This yields a representation result where ambiguous distributions of objective beliefs are first aggregated into “equivalent unambiguous beliefs” and then risk preferences are used to compute the utility of these equivalent unambiguous beliefs. Such an approach makes it possible to disentangle uncertainty aversion, related to the processing of information, from risk aversion, related to the evaluation of the equivalent unambiguous beliefs. An application to portfolio choice shows the tractability of the framework and its intuitive appeal.     

Optimal investment and reinsurance policies for an insurer with ambiguity aversion

In this paper, we study the optimal investment and reinsurance problem for an insurer based on the variance premium principle, in which three cases are considered. First, we assume that the financial market does not exist. The insurer only holds an insurance business, and the optimal reinsurance problem is studied. Subsequently, we assume that there exists a financial market with an accurately modeled risky asset. The optimal investment and reinsurance problem is investigated under these conditions. Finally, we consider the general case in which the insurer is concerned about the model ambiguity of both the insurance market and the financial market. In all three cases, the value function is set to maximize the expected utility of terminal wealth. By employing the dynamic programming principle, we derive the Hamilton–Jacobi–Bellman (HJB) equations, which are satisfied by the value functions and obtain closed-form solutions for optimal reinsurance and investment policies and the value functions in all three cases. Most interestingly, we elucidate how investment improves the insurer’s utility and find that the existence of ambiguity can significantly affect the optimal policies and value functions. We also compare the ambiguities in the two markets and find that ambiguity in the insurance market has much more significant impact on the value function than the ambiguity in the financial market. It implies that it is more valuable for insurer to precisely evaluate the insurance risk. We also provide some numerical examples and economic explanations to illustrate our results.     

Endowment-regarding preferences

We consider a general equilibrium model of pure exchange economies with endowment externalities. Consumers’ behaviors depend not only on their own consumption but also on the endowments of the other consumers. Applying the same method of analysis in Balasko (2015) about wealth concerns, we first show that almost all properties of equilibrium, including smooth equilibrium manifold and genericity of regular economies, can be directly extended to the economy where the demand function depends on the endowments of others and wealth of only one consumer. Next, we clarify the sufficient conditions under which those properties remain true in the economy with the most general form of endowment externalities. Finally, we generalize the above sufficient conditions to derive generic regularity results in the economy with both consumption and endowment externalities.     

Cross-sectional implications of dynamic asset pricing with stochastic volatility and ambiguity aversion

Building on recent research that highlights the importance of macroeconomic volatility and ambiguity aversion in explaining the dynamics of stock returns, in this paper we propose a dynamic asset pricing model that simultaneously accounts for stochastic macroeconomic volatility and ambiguity, assuming that investors deal with uncertainty about the mechanics of macroeconomic fluctuations using first-release consumption and revisions to aggregate consumption on vintage data. Our results show that the proposed model captures a large fraction of the cross-sectional variation of excess returns for a wide range of market anomaly portfolios. Furthermore, while the price of risk for ambiguity is positive and significant for the vast majority of assets under study, macroeconomic volatility yields ambiguous outcomes, although it significantly increases the explanatory power of the model for specific assets. Our results suggest that macroeconomic volatility and ambiguity complement each other in explaining the cross-sectional behavior of stock returns.     

Generic non-existence of general equilibrium with EUU preferences under extreme ambiguity

In the presence of three or more realisations of the aggregate endowment that are extremely ambiguous, in the sense that all relative probabilities are admissible, if agents have preferences that are representable by expected uncertain utility functions (Gul and Pesendorfer, 2014), general equilibrium does not generically exist in finite economies. It always exists, however, in continuum economies.     

A note on the equilibrium theory of economies with asymmetric information

In this note, we give an equilibrium existence theorem for exchange economies with asymmetric information and with an infinite dimensional commodity space. In our model, we assume that preferences are represented by well behaved utility functions, the positive cone has a non empty interior and the individual rational utility set is compact. Our result complements the corresponding one in Podczeck and Yannelis (2008), in the sense that is applicable to commodity spaces in which the order intervals are (possibly) not compact with respect to any Hausdorff linear topology.     

Existence of an equilibrium for infinite horizon economies with and without complete information

This work proves the existence of an equilibrium for an infinite horizon economy where trade takes place sequentially over time. There exist two types of agents: the first correctly anticipates all future contingent endogenous variables with complete information as in Radner [Radner, R. (1972). Existence of equilibrium of plans, prices and price expectations in a sequence of markets. Econometrica, 289–303] and the second has exogenous expectations about the future environment as in Grandmont [Grandmont, J. M. (1977). Temporary general equilibrium theory. Econometrica, 535–572] and information based on the current and past aggregate variables including those which are private knowledge. Agents with exogenous expectations may have inconsistent optimal plans but have predictive beliefs in the context of Blackwell and Dubbins [Blackwell, D., Dubins, L. (1962). Merging of opinions with increasing information. The Annals of Mathematical Statistics, 882–886] with probability transition rules based on all observed variables. We provide examples of this framework applied to models of differential information and environments exhibiting results of market selection and convergence of an equilibrium. The existence result can be used to conclude that, by adding the continuity assumption on the probability transition rules, we obtain the existence of an equilibrium for some models of differential information and incomplete markets.     

Existence of equilibria in economies with externalities and non-convexities in an infinite-dimensional commodity space

We prove an equilibrium existence theorem for economies with externalities, general types of non-convexities in the production sector, and infinitely many commodities. The consumption sets, the preferences of the consumers, and the production possibilities are represented by set-valued mappings to take into account the external effects. The firms set their prices according to general pricing rules which are supposed to have bounded losses and may depend upon the actions of the other economic agents. The commodity space is L_∞(M,M,μ), the space of all μ-essentially bounded M-measurable functions on M.As for our existence result, we consider the framework of Bewley (1972). However, there are four major problems in using this technique. To overcome two of these difficulties, we impose strong lower hemi-continuity assumptions upon the economies. The remaining problems are removed when the finite economies are large enough.Our model encompasses previous works on the existence of general equilibria when there are externalities and non-convexities but the commodity space is finite dimensional and those on general equilibria in non-convex economies with infinitely many commodities when no external effect is taken into account.     

Existence of marginal pricing equilibria in economies with externalities and non-convexities

We consider a general equilibrium model with externalities and non-convexities in production. The consumption sets, the preferences of the consumers and the production possibilities are represented by set-valued mappings to take into account possibility of external effects. There is no convexity assumption on the correspondences of production. We propose a definition of the marginal pricing rule, which generalizes the one used in the model without externality and, which satisfies a continuity assumption with respect to the external effect.We prove the existence of general equilibria under assumptions which allow us to encompass together the works on economies with externalities and convex conditional production sets, and those on marginal pricing equilibria in economies without externalities. We provide examples to illustrate the definition of the marginal pricing rule and to show the difference with the standard case.     

An analytical solution for the robust investment-reinsurance strategy with general utilities

In financial markets, different investors have different attitudes or preferences on the investment policies and reinsurance problems. For investors with different investment utilities, how to provide an optimal investment strategy is not only a very hard problem, but also an urgent problem to be solved. In this paper, we derive an analytical solution for the optimal allocation problem of investment-reinsurance with general-form utility function. The general utility function allows for varying relative risk aversion coefficient, which is an important feature in finance theory. However, obtaining analytical solutions for general utility function has been difficult or impossible. The solution presented in this paper is constructed through the homotopy analysis method (HAM) and written in the form of a Taylor series expansion. The fully nonlinear Hamilton–Jacobi–Bellman (HJB) equation is decomposed into an infinite series of linear PDEs, which can be solved analytically. In the end, three examples are presented to illustrate the convergence and accuracy of the method, it also demonstrates that different risk reference investors have different investment-reinsurance strategies.