Convergence of the equilibrium prices in a family of financial models

In this paper, we consider a family of complete or incomplete Financial models such that the price processes of the Financial assets converge in distribution to those in a limit model. Different authors pointed out that we do not have necessarily convergence of the arbitrage pricing intervals in that context. We prove here that we have very good convergence properties for the equilibrium pricing interval as de_ned by Bizid, Jouini and Koehl (1998) in discrete time or Jouini and Napp (1999) in continuous time.equilibrium prices

Arbitrage and Control Problems in Finance. Presentation.

The theory of asset pricing takes its roots in the Arrow-Debreu model (see,for instance, Debreu 1959, Chap. 7), the Black and Scholes (1973) formula,and the Cox and Ross (1976) linear pricing model. This theory and its link to arbitrage has been formalized in a general framework by Harrison and Kreps (1979), Harrison and Pliska (1981, 1983), and Du¢e and Huang (1986). In these models, security markets are assumed to be frictionless: securities can be sold short in unlimited amounts, the borrowing and lending rates are equal, and there is no transaction cost. The main result is that the price process of traded securities is arbitrage free if and only if there exists some equivalent probability measure that transforms it into a martingale, when normalized by the numeraire. Contingent claims can then be priced by taking the expected value of their (normalized) payo§ with respect to any equivalent martingale measure. If this value is unique, the claim is said to be priced by arbitrage and it can be perfectly hedged (i.e. duplicated) by dynamic trading. When the markets are dynamically complete, there is only one such a and any contingent claim is priced by arbitrage. The of each state of the world for this probability measure can be interpreted as the state price of the economy (the prices of $1 tomorrow in that state of the world) as well as the marginal utilities (for consumption in that state of the world) of rational agents maximizing their expected utility.arbitrage, control problem

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 et al. (2002) 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 et al. (2002), 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

Efficient Trading Strategies

In this paper, we point out the role of anticomonotonicity in the characterization of efficient contingent claims, and in the measure of inefficiency size of financial strategies. Two random variables are said to be anticomonotonic if they move in opposite directions. We first provide necessary and sufficient conditions for a contingent claim to be efficient in markets, which might be with frictions in a quite general framework. We then compute a measure of inefficiency size for any contingent claim. We finally give several applications of these results, studying in particular the efficiency of superreplication strategies.anticomonotonicity, utility maximization, markets with frictions

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

Arbitrage with fixed costs and interest rate models

We study securities market models with fixed costs. We first characterize the absence of arbitrage opportunities and provide fair pricing rules. We then apply these results to extend some popular interest rate and option pricing models that present arbitrage opportunities in the absence of fixed costs. In particular, we prove that the quite striking result obtained by Dybvig, Ingersoll, and Ross (1996), which asserts that under the assumption of absence of arbitrage long zero-coupon rates can never fall, is no longer true in models with fixed costs, even arbitrarily small fixed costs. For instance, models in which the long-term rate follows a diffusion process are arbitrage-free in the presence of fixed costs (including arbitrarily small fixed costs). We also rationalize models with partially absorbing or reflecting barriers on the price processes. We propose a version of the Cox, Ingersoll, and Ross (1985) model which, consistent with Longstaff (1992), produces yield curves with realistic humps, but does not assume an absorbing barrier for the short-term rate. This is made possible by the presence of (even arbitrarily small) fixed costs.interest rates; pricing; fixed costs; arbitrage

A class of models satisfying a dynamical version of the CAPM

Under a comonotonicity assumption between aggregate dividends and the market portfolio, the CCAPM formula becomes more tractable and more easily testable. In this paper, we provide theoretical justifications for such an assumption.CAPM, CCAPM, market beta, equilibrium, financial markets