Comonotonic Processes

We consider in this paper two Markovian processes X and Y, solutions of a stochastic differential equation with jumps, that are comonotonic, i.e., that are such that for all t, almost surely, X_{t} is greater in one state of the world than in another if and only if the same is true for Y_{t}. This notion of comonotonicity can be of great use for finance, insurance and actuarial issues. We show here that the assumption of comonotonicity imposes strong constraints on the coefficients of the diffusion part of X and Y.Comonotonicity, Comonotonic processes, Jump processes, Risk sharing schemes, Pareto optimal allocations

Consensus consumer and intertemporal asset pricing with heterogeneous beliefs

The aim of the paper is to analyze the impact of heterogeneous beliefs in an otherwise standard competitive complete market economy. The construction of a consensus probability belief, as well as a consensus consumer, are shown to be valid modulo an aggregation bias, which takes the form of a discount factor. In classical cases, the consensus probability belief is a risktolerance weighted average of the individual beliefs, and the discount factor is proportional to beliefs dispersion. This discount factor makes the heterogeneous beliefs setting fundamentally different from the homogeneous beliefs setting, and it is consistent with the interpretation ofbeliefs heterogeneity as a source of risk.We then use our construction to rewrite in a simple way the equilibrium characteristics (market price of risk, risk premium, risk-free rate) in a heterogeneous beliefs framework and to analyze the impact of beliefs heterogeneity. Finally, we show that it is possible to construct specific parametrizations of the heterogeneous beliefs model that lead to globally higher risk premia and lower risk-free rates.risk premium; beliefs heterogeneity; optimism; pessimism; consensus consumer

Unbiased Disagreement in financial markets, waves of pessimism and the risk return tradeoff

Can investors with irrational beliefs be neglected as long as they are rational on average ? Do their trades cancel out with no consequences on prices, as implicitly assumed by traditional models? We consider a model with irrational investors, who are rational on average. We obtain waves of pessimism and optimism that lead to countercyclical market prices of risk and procyclical risk-free rates. The variance of the state price density is greatly increased. The long run risk-return relation is mod- i…ed; in particular, the long run market price of risk might be higher than both the instantaneous and the rational ones.irrational investors, rational on average

Strategic Beliefs

We provide a discipline for beliefs formation through a model of subjective beliefs, in which agents hold incorrect but strategic beliefs. More precisely, we consider beliefs as a strategic variable that agents can manipulate to maximize their utility from trade. Our framework is therefore an imperfect competition framework, and the underlying concept is the concept of Nash equilibrium. We find that a strategic behavior leads to beliefs subjectivity and heterogeneity. Optimism (resp. overconfidence) as well as pessimism (resp. doubt) both emerge as optimal beliefs. Furthermore, we obtain a positive correlation between pessimism (resp. doubt) and risk-tolerance. The consensus belief is pessimistic and, as a consequence, the risk premium is higher than in a standard setting. Our model is embedded in a standard financial markets equilibrium problem and may be applied to several other situations in which agents have to choose the optimal exposure to a risk (choice of an optimal retention rate for an insurance company, choice of the optimal proportion of equity to retain for an entrepreneur and for a given project)Beliefs, Strategic, Pessimism, Consensus, Risk-premium, Heterogeneous, Doubt, Overconfidence

Arbitrage and state price deflators in a general intertemporal framework

In securities markets, the characterization of the absence of arbitrage by the existence of state price deflators is generally obtained through the use of the Kreps–Yan theorem.This paper deals with the validity of this theorem (see Kreps, D.M., 1981. Arbitrage and equilibrium in economies with infinitely many commodities. Journal of Mathematical Economics 8, 15–35; Yan, J.A., 1980. Caractérisation d'une classe d'ensembles convexes de L1 ou H1. Sém. de Probabilités XIV. Lecture Notes in Mathematics 784, 220–222) in a general framework. More precisely, we say that the Kreps–Yan theorem is valid for a locally convex topological space (X,?), endowed with an order structure, if for each closed convex cone C in X such that CX? and C?X+={0}, there exists a strictly positive continuous linear functional on X, whose restriction to C is non-positive.We first show that the Kreps–Yan theorem is not valid for spaces if fails to be sigma-finite.Then we prove that the Kreps–Yan theorem is valid for topological vector spaces in separating duality X,Y, provided Y satisfies both a “completeness condition” and a “Lindelöf-like condition”.We apply this result to the characterization of the no-arbitrage assumption in a general intertemporal framework.Arbitrage; State price deflators; Free lunch; Fundamental theorem of asset pricing; Investment opportunities

On Abel's Concept of Doubt and Pessimism

In this paper, we characterize subjective probability beliefs leading to a higher equilibrium market price of risk. We establish that Abel's result on the impact of doubt on the risk premium is not correct (see Abel, A., 2002. An exploration of the effects of pessimism and doubt on asset returns. Journal of Economic Dynamics and Control, 26, 1075-1092). We introduce, on the set of subjective probability beliefs, market price of risk dominance concepts and we relate them to well known dominance concepts used for comparative statics in portfolio choice analysis. In particular, the necessary first order conditions on subjective probability beliefs in order to increase the market price of risk for all nondecreasing utility functions appear as equivalent to the monotone likelihood ratio property.Pessimism, optimism, doubt, stochastic dominance, risk premium, market price of risk, riskiness, portfolio dominance, monotone likelihood ratio

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.Heterogeneous beliefs; Consensus belief; Aggregation of belief; Representative agents

On Abel's Concepts of Doubt and Pessimism.

In this paper, we characterize subjective probability beliefs leading to a higher equilibrium market price of risk. We establish that Abel's result on the impact of doubt on the risk premium is not correct in general; see Abel [2002. An exploration of the effects of pessimism and doubt on asset returns. Journal of Economic Dynamics and Control 26, 1075–1092]. We introduce, on the set of subjective probability beliefs, market-price-of-risk dominance concepts and we relate them to well-known dominance concepts used for comparative statics in portfolio choice analysis. In particular, the necessary first-order conditions on subjective probability beliefs in order to increase the market price of risk for all nondecreasing utility functions appear as equivalent to the monotone likelihood ratio property.Pessimism; Optimism; Doubt; Risk Premium; Stochastic Dominance; Market Price of Risk; Portfolio Dominance; Monotone Likelihood Ratio;