نقص درونی بازارهای مالی: نقش ابهام و ابهام گریزی

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.

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. Traditional finance theory assumes that agents are either expected (EU) or subjective expected (SEU) utility maximizers. That is, they choose among alternative investment opportunities by simply confronting the respective expected utility values, computed through a unique probability distribution, which might be objectively given (EU) or subjectively derived (SEU). Experimental works in finance and in decisions contradict both EU and SEU predictions. In particular, one of the most popular evidence of people’s systematic violation of (subjective) expected utility is described by Ellsberg’s (1961) paradox, that provides a comparison of different attitudes of the same agent when facing alternative sources of uncertainty. Broadly speaking, Ellsberg’s experiment shows that people do not generally like situations in which they are not able to derive a unique probability distribution over the reference state space. These situations have became known as situations of ambiguity, and the general “dislike” for them as ambiguity aversion. This attitude cannot be reflected by SEU or EU models, since they do not allow agents to express their own degree of confidence about a probability distribution. In fact, under ambiguity, not only is the payoff deriving from the choice of an act uncertain, but also its expected value, since it can be evaluated using different probability distributions that are all plausible. Ambiguity aversion induces agents’ prudent behavior which is reflected in the functional that represents each agent’s preferences. A wide body of literature in decisions is dedicated to multiple priors approaches towards ambiguity, the most popular example of these being probably the multiple priors preferences (MEU) axiomatized by Gilboa and Schmeidler (1989). Gilboa and Schmeidler extend standard expected utility by representing preferences through a utility index and a set of additive probabilities, instead of a unique one, on the state space. Agents with MEU preferences rank payoffs according to the criterion: View the MathML sourcef≽g   ⇔   min⁡p∈C∫u(f)dp≥min⁡p∈C∫u(g)dp Turn MathJax on where u(⋅)u(⋅) is a standard utility index, and C is a convex subset of the standard simplex over the state space View the MathML sourceΩ. C is interpreted as the set of effective priors considered by the agents, and ambiguity is reflected in its multivalued nature. Decision makers express ambiguity aversion by assigning higher probabilities to unfavorable states, as reflected in the min operator. Recent developments in the multiple prior approach involve variations of the Gilboa and Schmeidler’s functional, including models that combine pessimism and optimism (Ghirardato et al., 2004), representations that account for a concern for robustness against model misspecification (Hansen and Sargent, 2001), and preferences that generalize MEU by weakening the set of underlying axioms (Maccheroni et al., 2006). The emergence of decision theoretic models that are less narrow than (subjective) expected utility has induced the growth of new fields of research in which these models are applied in standard macroeconomic and finance contests, with the aim of achieving a better representation of reality. In particular, behavioral finance tries to explain some financial phenomena that contradict standard theory by considering agents whose choices are normatively questionable, in the sense that they are incompatible with SEU (and obviously with EU as well). One of these particular phenomena is considered here. More specifically, we generalize Mukerji and Tallon’s analysis on endogenous incompleteness of financial markets with CEU maximizing agents (Mukerji and Tallon, 1999 and Mukerji and Tallon, 2001), to the case of variational preferences (Maccheroni et al., 2006). As it is well known, in finance theory it is common to distinguish between the risk of price change due to the unique circumstances of a specific security (idiosyncratic risk), and those correlated to the overall market (systematic risk). Consequently, in the analysis that follows, we assume that assets’ payoffs have an idiosyncratic component, meaning that the payoff of each risky asset is affected by the same shocks that hit the other assets and the endowment processes, and also by other factors that are specific to that particular asset. The idea behind this assumption is that, in real economies, firms’ profits are typically affected, not only by aggregate (or at least sectorial) shocks, but also by other circumstances that are more peculiar to each individual firm. The idiosyncratic risk is firm-specific, and should be at least reducible to arbitrarily low levels through diversification. In fact, standard diversification arguments (see for example Chamberlein, 1983, Chamberlein et al., 1983 and Reisman, 1988) show that, in a (incomplete) typical bond-equity finance economy, the equilibrium allocation would approximate a complete market allocation. The possibility of hedging financial risk should induce agents with random income streams to trade, in order to reduce their exposure to the economic risk, and to obtain a smoother consumption profile across time and contingencies. Nevertheless, empirical evidence contradicts this prediction, showing that nonparticipation in financial markets is extremely relevant (see for example Campbell, 2006). In this paper, we provide an explanation for under-diversification. More specifically, we show that, if agents exhibit variational preferences, ambiguity and ambiguity aversion may lead to a collapse in the trade of financial assets whose payoff is greatly affected by idiosyncratic risk. In particular, it is ambiguity about the idiosyncratic component of the risky payoffs which is responsible for the possible break down. It is worth noting that it is not ambiguity per se that generates no-trade, but the fact that agents evaluate the acts of selling and buying according to different probabilities. In fact, if agents are identical (same utility function and same attitude towards ambiguity), and, furthermore, if they consider the same prior, regardless the position held in the asset, trade will in principle occur, as in the SEU case. Finally, by appropriately restricting the class of preferences under consideration, we show that, even when the number of available assets becomes arbitrarily large, agents with these particular variational preferences cannot benefit from diversification, and the market breaks down. We emphasize that what is crucial in determining endogenous incompleteness is the fact that the set of effective priors is multivalued. More precisely, in a simplified two agents economy that we are going to consider, the sufficient condition for endogenous incompleteness is characterized through non-differentiability of the functional representation for variational preferences. Our result strongly relies on non-differentiability also in the more general economy. In particular, we show that for a specific class of differentiable variational preferences, namely the multiplier preferences introduced by Hansen and Sargent (2001), the sufficient condition for the absence of trade cannot hold. In the simpler setting, when the analysis is restricted to the subclass of variational preferences that are compatible with the sufficient condition for trade-breakdown, two opposite tendencies can be identified. From one side, risk aversion and the great variation of endowment across states tend to generate trade for insurance purposes. On the other side, the great variation of the idiosyncratic component lets the agents feel less confident about their probability assignment, and, therefore, it prevents trade. At empirical level, many studies have shown the existence of general uncertainty about dividend processes over the last century, and, therefore, the ambiguity assumption seems to be realistic. In general, the introduction of ambiguity and ambiguity aversion in financial models is highly justified. History of financial markets is in fact full of episodes of increase in uncertainty that have led to liquidity problems, especially in emerging markets. Further, pricing models’ uncertainty, resulting from the poor quality of information on which agents base the choice of their model, is acknowledged by most operators as a cause of serious mispricing errors, especially for derivative instruments. In this paper, we introduce ambiguity and ambiguity aversion through variational preferences (VP), which are a class of ambiguity averse preferences characterized by the functional representation: View the MathML sourceV(f)=min⁡p∈Δ(Ω)[Ep[u(f)]+c(p)] Turn MathJax on where c(⋅)c(⋅) is an ambiguity index. Different specifications of the function c lead to different preferences. Intuitively, each p in View the MathML sourceΔ is a possible probabilistic model, and the agents consider all possible models in View the MathML sourceΔ, giving weight c(p)c(p) to each of them. Therefore, the minimization over View the MathML sourcep∈Δ is aimed at seeking robustness against the possibility of a mistake in the choice of the probabilistic model. The main reason why we focus on variational preferences is that they comprehend as subcases other widely used and well-known classes of ambiguity averse preferences (including CEU preferences). Further, they are tractable and a very straightforward criterion for ambiguity comparisons can be easily implemented. Since the class of variational preferences is extremely wide, it is clear that such a generalization comes at some cost: in particular, we are not able to derive the analogues to all the results provided in Mukerji and Tallon (2001). For example, Mukerji and Tallon explicitly determine the minimal level of ambiguity required for trade to collapse, while, in our main theorem, we are only able to show that it is possible to characterize agents’ ambiguity attitude in such a way that ambiguity has the effect of preventing trade in the risky assets. Further, our findings concerning the failure of the usual diversification arguments are based on a strong assumption on the function c, 2 and, at least to us, it seems not possible to obtain these results in the general case. Nevertheless, we believe that variational preferences may be more suitable than CEU preferences in reflecting investors’ behavior under uncertainty. Indeed, the high degree of generality provided in the VP-framework is achieved through a relaxation of the underlying axioms, allowing for a more realistic representation of individuals’ choices. More specifically, in the contest of an investment decision problem, the comonotonic independence axiom underlying the CEU model appears to be too demanding for representing investors’ actual behavior, in comparison with the weak certainty independence axiom that characterizes variational preferences. To discuss this, we report Example 2 in Maccheroni et al. (2006), adapted to a financial problem (we refer to Maccheroni et al., 2006 for a detailed discussion of the example).3 Example An agent is considering the possibility of investing a fixed amount ww in a risky stock whose gross real return can be either R or r, with 0<r<1<R0<r<1<R, according to some unknown probability distribution. Alternatively, he can deposit the amount into a bank account with zero real return. The investor displays constant relative risk aversion, and he is ambiguity averse. As discussed in Maccheroni et al. (2006), a CEU-agent would prefer one of the two alternatives, independently from the amount to be invested ww. Vice versa, for an agent with variational preferences, there might be a threshold level View the MathML sourcew¯, such that he prefers the risky investment if View the MathML sourcew≤w¯, and the bank account otherwise. A further motivation supports the introduction of variational preferences in our model. Since some of the results concerning the behavior of financial markets under ambiguity are closely related with the differential properties of the functional representation (including Mukerji and Tallon’s no-trade result), it seems to us that the role played by non-differentiability should be deeply investigated. Variational preferences include subclasses of differentiable (e.g. multiplier) and non-differentiable (e.g. MEU, CEU) preferences, hence they seem to be extremely suitable for the achievement of this goal. The paper proceeds as follows. Section 2 presents some relevant results of the existing literature concerning the absence of trade under ambiguity. Section 3 introduces the economic setup. Section 4 roughly describes the decision theoretic setting. Section 5 defines the concept of equilibrium for the economy. Section 6 examines the ambiguity and ambiguity aversion effects on the general economy. Section 7 specializes to a simpler economy in order to provide additional insights to the analysis. Given the relevance of non-differentiability in determining the no-trade result of Section 7, Section 8 goes back to the general model introduced in Section 6 to discuss this issue. Section 9 concludes. All proofs are collected in Appendix A.

In this paper, we have generalized Mukerji and Tallon’s analysis of endogenous incompleteness of financial markets under ambiguity, by considering agents with variational preferences. Our findings only partially confirm Mukerji and Tallon’s results, since the sufficient condition for the absence of trade provided in our main theorem is crucially related with non-differentiability of the functional representation of preferences. More specifically, we show that such a condition cannot be satisfied when agents display multiplier preferences. Hence, the no-trade result can be extended only to a specific subclass of variational preferences, namely the class characterized by non-essentially strictly convex ambiguity index. For this specific subclass, some interesting results can be derived. In particular, trade of financial assets whose payoff is greatly affected by idiosyncratic risk may break down, so that agents are suboptimally left exposed to risk. Further, even when the number of available assets becomes arbitrarily large, agents cannot benefit from diversification, so that trade collapses in contradiction to what is prescribed by subjective expected utility theory. Finally, when the analysis is restricted to an extremely simple set up, two opposite tendencies are in act. From one side, risk aversion and the great variation of endowment across states tend to generate trade for insurance purposes. On the other side, the great variation of the idiosyncratic component lets the agents feel less confident about their probability assignment, and, therefore, it prevents trade. Since ambiguity and ambiguity aversion have been shown to be empirically relevant in financial markets, it is of crucial importance to distinguish between market anomalies that can be fully explained through kinks in otherwise standard preferences, and anomalies that instead are a more general consequence of ambiguity. Indeed, as discussed in Maccheroni et al. (2006), any variational preference can be arbitrarily well approximated by an everywhere differentiable variational preference, so that results that rely on non-differentiability only are in general not robust. In particular, for what concerns under-participation, such a distinction would have deep implications on financial regulation and assets design. In fact, if the reduction of the effects of asymmetric information and transaction costs is not by itself sufficient to guarantee completeness of risk sharing under ambiguity, then the introduction of new derivative securities will be necessary to offer alternative hedging possibilities. The clarification of the roles played by ambiguity and non-differentiability in determining some puzzling financial phenomena is also the aim of our future research, and we believe that the generality and the properties of variational preferences render them extremely suitable for the achievement of this result.