Bayesian Nash equilibrium and variational inequalities

This paper provides a sufficient condition for the existence and uniqueness of a Bayesian Nash equilibrium by regarding it as a solution of a variational inequality. The payoff gradient of a game is defined as a vector whose component is a partial derivative of each player’s payoff function with respect to the player’s own action. If the Jacobian matrix of the payoff gradient is negative definite for each state, then a Bayesian Nash equilibrium is unique. This result unifies and generalizes the uniqueness of an equilibrium in a complete information game by Rosen (1965) and that in a team by Radner (1962). In a Bayesian game played on a network, the Jacobian matrix of the payoff gradient coincides with the weighted adjacency matrix of the underlying graph.     

Corporate investment,dividend decisions,differential taxation and the no-arbitrage condition

Masulis and Trueman (1988) investigated corporate investment and dividend decisions under differential personal taxation. They assumed investors in different tax brackets, a state-preference complete market (which includes pure securities for each state) with a ban on short-selling. They concluded that shareholders prefer non-zero dividend payment. In their model, the restrictions on short-sales were needed to bound tax arbitrage profits, among investors in different tax brackets, so that equilibrium could be reached. However, the joint assumptions of complete markets, and restrictions on short-selling, are inconsistent. By utilizing more recent results, from the tax arbitrage literature, we allow short-selling, and examine the role and implications of the no-arbitrage condition. We show that, with investors in different tax brackets, equilibrium is feasible. We conclude that a revised Masulis and Trueman type model does not explain a non-zero optimal dividend policy.     

FOREIGN EXCHANGE RATE EXPECTATIONS: SURVEY AND SYNTHESIS

Abstract.  This paper reviews the empirical literature on foreign exchange rate expectations. Prominent issues are the forward premium puzzle, expectations formation in financial markets, heterogeneity of expectations, market microstructure, time-varying risk premiums and forecast performance. Although much has been learned in each field, this survey highlights the areas of research in which our understanding of the mechanism of exchange rate expectations is still incomplete. Our survey suggests that both irrational expectations and time-varying risk premiums account for the forward discount anomaly, that long-term expectations reverse towards their long-run equilibrium values and that heterogeneous behaviour of market participants has the potential of explaining some of the empirical regularities in the international finance literature.     

Arbitrage and price revelation with asymmetric information and incomplete markets

This paper deals with the issue of arbitrage with differential information and incomplete financial markets, with a focus on information that no-arbitrage asset prices can reveal. Time and uncertainty are represented by two periods and a finite set S of states of nature, one of which will prevail at the second period. Agents may operate limited financial transfers across periods and states via finitely many nominal assets. Each agent i has a private information about which state will prevail at the second period; this information is represented by a subset S_i of S. Agents receive no wrong information in the sense that the “true state” belongs to the “pooled information” set ∩_iS_i, hence assumed to be non-empty.Our analysis is two-fold. We first extend the classical symmetric information analysis to the asymmetric setting, via a concept of no-arbitrage price. Second, we study how such no-arbitrage prices convey information to agents in a decentralized way. The main difference between the symmetric and the asymmetric settings stems from the fact that a classical no-arbitrage asset price (common to every agent) always exists in the first case, but no longer in the asymmetric one, thus allowing arbitrage opportunities. This is the main reason why agents may need to refine their information up to an information structure which precludes arbitrage.     

E-stability and stability of adaptive learning in models with private information

The paper demonstrates how the E-stability principle introduced by Evans and Honkapohja [2001. Learning and Expectations in Macroeconomics. Princeton University Press, Princeton, NJ] can be applied to models with heterogeneous and private information in order to assess the stability of rational expectations equilibria under learning. The paper extends already known stability results for the Grossman and Stiglitz [1980. On the impossibility of informationally efficient markets. American Economic Review 70, 393–408] model to a more general case with many differentially informed agents and to the case where information is endogenously acquired by optimizing agents. In both cases it turns out that the rational expectations equilibrium of the model is inherently E-stable and thus locally stable under recursive least squares learning.     

Recursive equilibrium in stochastic OLG economies: Incomplete markets

We prove generic existence of recursive equilibrium for overlapping generations economies with uncertainty and incomplete financial markets. Generic here means in a residual set of utilities and endowments. The result holds provided there is sufficient intragenerational household heterogeneity, and transition probabilities and the asset payoff matrix satisfy mild regularity conditions. The paper also provides a new methodological technique to establish comparative statics, or perturbation, properties in such environments.     

Optimal information exchange in contests

We study optimal exchange of private information in a two-player all-pay auction contest with independent private binary values. A benevolent information center who is informed about the players’ values facilitates the exchange of information by disclosing a signal publicly. The informativeness of the signal determines the monotonicity of the unique symmetric equilibrium and the players’ expected payoff. We characterize the upper bound of players’ expected payoff and the corresponding optimal signals utilizing such a relation between the informativeness and the payoff. When the players are ex ante sufficiently heterogeneous, the optimal signals work through an information-rent channel by inducing allocative efficient contests. When the players are ex ante sufficiently homogeneous, the optimal signals work through an unlevel-playing-field channel by inducing asymmetric contests. In order to guarantee efficient allocation, a regulator can punish any exchange of information when the players are sufficiently homogeneous and impose no restrictions when they are sufficiently heterogeneous.     

Equilibrium in incomplete markets: II : Generic existence in stochastic economies

We model multiperiod securities markets with differential information. A price system that admits no free lunches is related to martingales when agents have rational expectations. We introduce the concept of resolution time, and show that a better informed agent and a less informed agent must agree on the resolution times of commonly marketed events if they have rational expectations and if there are no free lunches. It then follows that if all elementary events are marketed for a less informed agent then any price system that admits no free lunches to a better informed agent must eliminate any private information asymmetry between the two. We provide an example of a dynamically fully revealing price system that is arbitrage free and yields elementarily complete markets.     

Rational expectations equilibrium and the strategic choice of costly information

This paper studies costly information acquisition in one-good production economies when agents acquire private information and prices transmit information. Before asset markets open, agents choose the quality of their private information. After this information stage, agents trade assets in sequentially complete markets taking into account their private information and the information revealed by equilibrium prices (rational expectations equilibrium, (Radner, R., 1979. Rational expectations equilibrium: generic existence and the information revealed by prices, Econometrica 47, 655–678.)). An overall equilibrium in asset and information market is defined as a Nash equilibrium of the information game in which agents’ actions are information choices and their utility payoffs are the ex-ante expected utilities of the corresponding rationale expectations equilibrium. This paper shows that for a generic set of economies parameterized by endowments and productivity shocks, an overall equilibrium in information and asset market (a Nash equilibrium of the induced information game) with costly information acquisition and fully-revealing prices exists. In other words, informational efficiency is in general consistent with costly information acquisition.     

Heterogeneity in Economics

This paper argues that heterogeneity of agents’ characteristics plays a fundamental role in the economy and should do so in economic models. Three aspects are considered. Firstly the notion that assuming heterogeneity was a solution to the problem posed by the results of Sonnenschein Mantel and Debreu is considered and it is suggested that the more pragmatic approach adopted by Hildenbrand is likely to be more productive. Next the role of adaptation and evolution which necessarily involve variety or heterogeneity is examined. It is suggested that heterogeneity will persist since agents will only slowly learn to adapt and that in the meantime the environment will change. Lastly the role of heterogeneity in financial markets is examined. It is suggested that heterogeneous and varying expectations may account for many of the stylised facts which do not seem to be consistent with the standard financial markets model.“Variety is the spice of life”     

Wavelet decomposition of heterogeneous investment horizon

Building on the assumptions that investors are heterogeneous and that not all of them are fully rational, the market for trading any financial instrument can be separated into several segments, each associated with a different investment horizon. Thus, the expected return on an asset for each horizon maintains a different functional relationship with an expected market return. In other words, the trading of an asset by investors with heterogeneous investment horizons results in the coexistence of multiple security market lines. This proposed theory, which offers an alternative interpretation of investment behavior from that of the capital asset pricing model (CAPM) and the efficient markets hypothesis (EMH), is verified by using the newly introduced amalgamated discrete wavelet transform.     

Aggregation of heterogeneous beliefs

This paper is a generalization of [Calvet, L., Grandmont, J.-M., Lemaire, I., 2002. Aggregation of heterogenous beliefs and asset pricing in complete financial markets. Working paper] to a dynamic setting. We propose a method to aggregate heterogeneous individual probability beliefs, in dynamic and complete asset markets, into a single consensus probability belief. This consensus probability belief, if commonly shared by all investors, generates the same equilibrium prices as well as the same individual marginal valuation as in the original heterogeneous probability beliefs setting. As in [Calvet, L., Grandmont, J.-M., Lemaire, I., 2002. Aggregation of heterogenous beliefs and asset pricing in complete financial markets. Working paper], the construction stands on a fictitious adjustment of the market portfolio. The adjustment process reflects the aggregation bias due to the diversity of beliefs. In this setting, the construction of a representative agent is shown to be also valid.     

The market for crash risk

This paper examines the equilibrium when stock market crashes can occur and investors have heterogeneous attitudes towards crash risk. The less crash averse insure the more crash averse through options markets that dynamically complete the economy. The resulting equilibrium is compared with various option pricing anomalies: the tendency of stock index options to overpredict volatility and jump risk, the Jackwerth [Recovering risk aversion from option prices and realized returns. Review of Financial Studies 13, 433–451] implicit pricing kernel puzzle, and the stochastic evolution of option prices. Crash aversion is compatible with some static option pricing puzzles, while heterogeneity partially explains dynamic puzzles. Heterogeneity also magnifies substantially the stock market impact of adverse news about fundamentals.     

Game contingent claims in complete and incomplete markets

A game contingent claim is a contract which enables both the buyer and the seller to terminate it before maturity. For complete markets Kifer [Finance and Stochastics 4 (2000) 443–463] shows a connection to a (zero-sum) Dynkin game whose value is the unique no-arbitrage price of the claim. But, for incomplete markets one needs a more general approach. We interpret the contract as a generalized non-zero-sum stopping game. For the complete case this leads to the same results as in Kifer [Finance and Stochastics 4 (2000) 443–463]. For the general case we show the existence of an equilibrium point under the condition that both the seller and the buyer have an exponential utility function. For other utility functions such a point need not exist in the context of incomplete markets.     

Nonlinear mean reversion in the term structure of interest rates

The expectations hypothesis implies that the yield curve provides information on the future change in the short-term interest rate. However, transaction costs exist in the financial market, which prevent investors from realizing the arbitrage opportunity, when the arbitrage does not fully cover the transaction costs. The purpose of this paper is to assess the effect of transaction costs on the predictability of the term structure by using the threshold vector error correction model, which allows for the nonlinear adjustment to the long-run equilibrium relationship. A significant amount of threshold effect is found, and the adjustment coefficients are regime-dependent. The empirical result supports the nonlinear mean reversion in the term structure of interest rates.     

The effects of dependent beliefs on endogenous leverage

The number of financial markets and the beliefs about the relation between markets can have large effects on the access to credit in a model with collateralized borrowing. In the model, investors have beliefs about the payout likelihoods for assets. I vary the degree of dependence between the likelihoods for the asset payouts and solve for the endogenous leverage ratios. When investors believe that the payouts of the assets are more dependent, the model predicts higher leverage ratios for all assets. When the number of financial markets available to investors increases, a condition in terms of the belief elasticity characterizes whether or not the leverage ratios increase.     

Expectations Formation and Risk in Three Financial Markets: Surveying What the Surveys Say

This paper attempts to provide a logical overview of the literature which exploits survey data to examine issues of expectations formation and risk aversion in financial markets. Our survey suggests that: short term expectations are excessively volatile and exhibit bandwagon effects, while longer term expectations appear to be regressive and therefore stabilising; in bond and foreign exchange markets the standard result of forward rate biasedness is due in part to time-varying premia; recent research using disaggregate foreign exchange survey data demonstrates the importance of heterogeneous expectations.     

Information asymmetry,sentiment interactions,and asset price

We develop an asset pricing model with sentiment interactions between institutional and individual investors under the condition of information asymmetry. Our model considers private information and investor sentiment, two imperfections in securities markets, and integrates them into a theoretical model to investigate the role of the interaction between information asymmetry and investor sentiment in asset pricing. We show that the joint effect of private information and investor sentiment deviate the price of risky assets and efficiently explains anomalies in the stock market. Investor sentiment changes the effect of information on the equilibrium price relative to a world where all investors are completely rational. Private information changes the effect of investor sentiment on the equilibrium price in comparison with a scenario with symmetric market information. In addition, the individual investors’ learning and the disclosure of information both allow private information to be better integrated into the price and simultaneously changes the effect of investor sentiment on the equilibrium price.     

Simple versus rich language in disclosure games

This paper studies strategic information disclosure when the sender may not observe the payoff-relevant state, and the receiver may interpret messages naively. We characterize equilibria as a function of the language available to the sender. The language is simple if an informed sender can either fully disclose the state or nothing. The language is rich if he can disclose any closed interval containing the true state. We show that an informed sender and a strategic receiver get a higher ex-ante equilibrium payoff when the language is rich. The reverse holds for a naive receiver and an uninformed sender. Overall, our work suggests that the design of language is key in situations where disclosure is voluntary.     

The generalized Stackelberg equilibrium of the all-pay auction with complete information

We characterize the equilibrium of the all-pay auction with general convex cost of effort and sequential effort choices. We consider a set of n players who are arbitrarily partitioned into a group of players who choose their efforts ‘early’ and a group of players who choose ‘late’. Only the player with the lowest cost of effort has a positive payoff in any equilibrium. This payoff depends on his own timing vis-a-vis the timing of others. We also show that the choice of timing can be endogenized, in which case the strongest player typically chooses ‘late’, whereas all other players are indifferent with respect to their choice of timing. In the most prominent equilibrium the player with the lowest cost of effort wins the auction at zero aggregate cost. We thank Dan Kovenock and Luis C. Corchón for discussion and helpful comments. The usual caveat applies. Wolfgang Leininger likes to express his gratitude to Wissenschaftszentrum Berlin (WZB) for its generous hospitality and financial support.