Elyès Jouini

Law Invariant Risk Measures Have the Fatou Property

S. Kusuoka [K 01, Theorem 4] gave an interesting dual characterizationof law invariant coherent risk measures, satisfying the Fatou property.The latter property was introduced by F. Delbaen [D 02]. In thepresent note we extend Kusuokas characterization in two directions, thefirst one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M. Fritelli and E. Rossaza Gianin [FG05] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. F¨ollmer and A. Schied [FS 04]. Secondly — and more importantly — we show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance.We also introduce the notion of the Lebesgue property of a convex risk measure, where the inequality in the definition of the Fatou property is replaced by an equality, and give some dual characterizations of this property.

Viability and equilibrium in securities markets with frictions

In this paper we study some foundational issues in the theory of asset pricing with market frictions. We model market frictions by letting the set of marketed contingent claims (the opportunity set) be a convex set, and the pricing rule at which these claims are available be convex. This is the reduced form of multiperiod securities price models incorporating a large class of market frictions. It is said to be viable as a model of economic equilibrium if there exist price-taking maximizing agents who are happy with their initial endowment, given the opportunity set, and hence for whom supply equals demand. This is equivalent to the existence of a positive linear pricing rule on the entire space of contingent claims - an underlying frictionless linear pricing rule - that lies below the convex pricing rule on the set of marketed claims. This is also equivalent to the absence of asymptotic free lunches - a generalization of opportunities of arbitrage. When a market for a non marketed contingent claim opens, a bid-ask price pair for this claim is said to be consistent if it is a bid-ask price pair in at least a viable economy with this extended opportunity set. If the set of marketed contingent claims is a convex cone and the pricing rule is convex and sublinear, we show that the set of consistent prices of a claim is a closed interval and is equal (up to its boundary) to the set of its prices for all the underlying frictionless pricing rules. We also show that there exists a unique extended consistent sublinear pricing rule - the supremum of the underlying frictionless linear pricing rules - for which the original equilibrium does not collapse, when a new market opens, regardless of preferences and endowments. If the opportunity set is the reduced form of a multiperiod securities market model, we study the closedness of the interval of prices of a contingent claim for the underlying frictionless pricing rules.

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 modified; in particular, the long run market price of risk might be higher than both the instantaneous and the rational ones.

Arbitrage and Viability in Securities Markets with Fixed Trading Costs

This paper studies foundational issues in securities markets models with fixed costs of trading, i.e. transactions costs that are bounded regardless of the transaction size, such as fixed brokerage fees, investment taxes, operational, and processing costs or opportunity costs. We show that the absence of free lunches in such models is equivalent to the existence of a family of absolutely continuous probability measures for which the normalized securities price processes are martingales. This is a weaker condition than the absence of free lunch in frictionless models, which is equivalent to the existence of an equivalent martingale measure. We also show that the only arbitrage-free pricing rules on the set of attainable contingent claims are those that are equal to the sum of an expected value with respect to any absolutely continuous martingale measure and of a bounded fixed cost functional. Moreover, these pricing rules are the only ones to be viable as models of economic equilibrium.

Optimal Investment with Taxes: An Existence Result

We study the deterministic control problem of maximizing utility from consumption of an agent who seeks to optimally allocate his wealth between consumption and investment in a financial asset subject to taxes on benefits with first-in–first-out priority rule on sales. Short sales are prohibited and consumption is restricted to be non-negative. Such a problem has been introduced in a previous paper by the same authors where the first-order conditions have been derived. In this paper, we establish an existence result for this non-classical optimal control problem.

Is There a Pessimistic Bias in Individual Beliefs? Evidence from a Simple Survey

Cf. papier Is There a Pessimistic Bias in Individual Beliefs? Evidence from a Simple Survey

Discounting and Divergence of Opinion

The objective of this paper is to adopt a general equilibrium model and determine the socially efficient discount factor, risk free rate and discount rate when there are heterogeneous anticipations about the growth of the economy as well as heterogeneous time preference rates. Among others we tackle the following questions. Is the socially efficient discount factor an arithmetic average of the individual subjectively anticipated discount factors‘ More generally, can the Arrow-Debreu prices, the risk free rates, the subjectively expected socially efficient discount factors and discount rates be obtained as an average of the individual subjectively anticipated ones‘ Can beliefs dispersion be analyzed as a sort of additional risk or uncertainty leading to possibly lower discount rates‘ Is it socially efficient, when diversity of opinion is taken into account, to reduce the discount rate per year for more distant horizons‘ If so, what is the trajectory of the decline‘

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 Abels 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.

Are risk-averse agents more optimistic? A Bayesian estimation approach

Our aim is to analyze the link between optimism and risk aversion in a subjective expected utility setting and to estimate the average level of optimism when weighted by risk tolerance. This quantity is of particular importance since it characterizes the consensus belief in risk-taking situations with heterogeneous beliefs. Its estimation leads to a nontrivial statistical problem. We start from a large lottery survey (1,536 individuals). We assume that individuals have true unobservable characteristics and that their answers in the survey are noisy realizations of these characteristics. We adopt a Bayesian approach for the statistical analysis of this problem and use an hybrid MCMC approximation method to numerically estimate the distributions of the unobservable characteristics. We obtain that individuals are on average pessimistic and that pessimism and risk tolerance are positively correlated. As a consequence, we conclude that the consensus belief is biased towards pessimism.

Investment and Arbitrage Opportunities with Short Sales Constraints

In this paper we consider a family of investment projects defined by their deterministic cash flows. We assume stationarity—that is, projects available today are the same as those available in the past. In this framework, we prove that the absence of arbitrage opportunities is equivalent to the existence of a discount rate such that the net present value of all projects is nonpositive if the projects cannot be sold short and is equal to zero otherwise. Our result allows for an infinite number of projects and for continuous as well as discrete cash flows, generalizing similar results established by Cantor and Lippman (1983, 1995) and Adler and Gale (1997) in a discrete time framework and for a finite number of projects.