Core equivalence theorem: countably many types of agents and commodities in $\vec{L}^{1}(\mu)$

Abstract  We prove a core-Walras equivalence result for a finitely additive confederate economy with commodity space L¹(μ) and a measurable bounded map of extremely desirable commodities: when the map of extremely desirable commodities is simply bounded the properness of preferences is no longer equivalent to the existence of just one extremely desirable commodity as assumed in the countably additive model by Rustichini and Yannelis. Mathematics Subject Classification: 91B50, 91B54; 28B20, 46A22, 28A25 Journal of Economic Literature Classification: D51; C02     

<Emphasis Type="Italic">Notes and Comments:</Emphasis> The numeraire portfolio in financial markets modeled by a multi-dimensional jump diffusion process

In continuous time, we study a financial market which is free of arbitrage opportunities but incomplete under the physical probability measure P. Thus one has several choices of equivalent martingale measures. In the present paper, the (unique) martingale measure P ^* is studied which is defined by the concept of the numeraire portfolio. The choice of P ^* can be justified by a change of numeraire in place of a change of measure. Mathematics Subject Classification (2000): 90A09, 91B28, 91B62, 93E20, 62P05 Journal of Economic Literature Classification: G10, G12, G13     

Linear cumulative prospect theory with applications to portfolio selection and insurance demand

Abstract The present paper combines loss attitudes and linear utility by providing an axiomatic analysis of corresponding preferences in a cumulative prospect theory (CPT) framework. In a sense we derive a two-sided variant of Yaari’s dual theory, i.e., nonlinear probability weights in the presence of linear utility. The first important difference is that utility may have a kink at the status quo, which allows for the exhibition of loss aversion. Also, we may have different probability weighting functions for gains than for losses. We apply the model to both portfolio selection and insurance demand. Our results show that CPT with linear utility has more realistic implications than the dual theory since it implies only a weakened variant of plunging. Mathematics Subject Classification (2000): 91B08, 91B28, 91B30 Journal of Economic Literature Classification: D81, G11, G22     

Notes and Comments: Stochastic demand correspondences and their aggregation properties

Abstract Mathematics Subject Classification (2000): 91B06, 91B16 Journal of Economic Literature Classification: D11     

The origins of the mean-variance approach in finance: revisiting de Finetti 65 years later

Abstract In a recent critical review of de Finetti’s paper “Il problema dei pieni’’, the Nobel Prize winner Harry Markowitz recognized the primacy of de Finetti in applying the mean-variance approach to finance, but pointed out that de Finetti did not solve the problem for the general case of correlated risks. We argue in this paper that a more fair sentence would be: de Finetti did solve the general problem but under an implicit hypothesis of regularity which is not always satisfied. Moreover, a natural extension of de Finetti’s procedure to non-regular cases offers a general solution for the correlation case and shows that de Finetti anticipated a modern mathematical programming approach to mean-variance problems. Mathematics Subject Classification (2000): 91B30, 90C20 Journal of Economic Literature Classification: G11, C61, B23, D81, G22     

Homogeneous semi-Markov reliability models for credit risk management*

Abstract The credit risk problem is one of the most important issues of modern financial mathematics. Fundamentally it consists in computing the default probability of a company going into debt. The problem can be studied by means of Markov transition models. The generalization of the transition models by means of homogeneous semi-Markov models is presented in this paper. The idea is to consider the credit risk problem as a reliability problem. In a semi-Markov environment it is possible to consider transition probabilities that change as a function of waiting time inside a state. The paper also shows how to apply semi-Markov reliability models in a credit risk environment. In the last section an example of the model is provided. Mathematics Subject Classification (2000): 60K15, 60K20, 90B25, 91B28 Journal of Economic Literature Classification: G21, G33     

Notes and Comments: An approximation of caplet implied volatilities in Gaussian models

Abstract It is market practice to quote interest rate derivatives traded “over the counter” in terms of their implied volatility. For this reason, the term structure of at-the-money cap volatilities as well as the volatility surface of at-the-money swaptions are directly observed. This paper analyzes the case of caps. Any analysis of these markets would most likely report two main facts. The first is that the level of the volatility is inversely related to the level of the interest rates. The second is that the term structure is either a decreasing or a humped function of maturity. For a reference, see Rebonato (2003) and Brigo and Mercurio (2001). Rebonato (2003) suggests that the structure of implied volatility is humped in periods of normal market conditions and decreasing when markets are “excited”. Interpreting and explaining such phenomena is indeed an interesting and important issue. Mathematics Subject Classification (2000): 91B70 Journal of Economic Literature Classification: E43, C13     

An overlapping generations model with non-ordered preferences and numeraire-incomplete markets*

Abstract This paper shows that, even with a life-cycle component, the standard model of competitive consumption and asset trading can be extended to encompass general preference relations, which do not necessarily hinge upon special assumptions such as time or state separability, or even completeness or transitivity. More precisely, this paper addresses the equilibrium existence for an overlapping generations pure-exchange economy with non-ordered preferences and incomplete financial markets of numeraire assets. Mathematics Subject Classification (2000): 91B50, 91B62 Journal of Economic Literature Classification: D52, D91     

<Emphasis Type="SmallCaps">Notes and Comments:</Emphasis> On the uniqueness of convex-ranged probabilities

Abstract In Marinacci (2000), the following theorem was proved. Theorem 1. (Marinacci (2000) Let P and Q be two finitely additive probabilities on a λ -system Σ . Suppose that P is convex-ranged and that Q is countably additive. If there exists an A ⁺ ∈ Σ with 0<P(A ⁺ )<1 such that whenever B∈ Σ , then P=Q. Mathematics Subject Classification (2000): 28A10, 91B06 Journal of Economic Literature Classification: C60, D81     

Bounds for the utility-indifference prices of non-traded assets in incomplete markets

Abstract We consider a special class of financial models with both traded and non-traded assets and show that the utility indifference (bid) price of a contingent claim on a non-traded asset is bounded above by the expectation under the minimal martingale measure. This bound also represents the marginal bid price for the claim. The key conclusion is that the bound and the marginal bid price are independent of both the utility function and initial wealth of the agent. Thus all utility maximising agents charge the same marginal price for the claim. This conclusion is in some sense the opposite conclusion to that of Hubalek and Schachermayer (2001), who show that any price is consistent with some equivalent martingale measure. Mathematics Subject Classification (2000): 91B28, 91B16, 60J70 Journal of Economic Literature Classification: G13     

Taxes and money in incomplete financial markets

Abstract We analyze a model with incomplete financial markets, where money is needed to pay taxes. Equilibria exist, are typically regular and not Pareto optimal. Moreover, generically, there exists a redistribution of money among households which leads to a Pareto superior equilibrium. The intervention occurs only in the first period and it does not require either closing markets or upper bounds on the number of households. Mathematics Subject Classification (2000): 91B50 Journal of Economic Literature Classification: D52, D60, E50, H20     

<Emphasis Type="Italic">Notes and Comments:</Emphasis> Profitability in a multiple strategy market

The link between martingales and arbitrage is well-known in financial theory: arbitrage is not available if and only if there exists an equivalent measure such that the discounted prices are martingales with respect to this measure (MME). As a consequence, under MME, any previsible (non-anticipative) strategy cannot have secure (without risk) profit. Moreover, a careful reading of a bootstrap proof of the first fundamental theorem of asset pricing (see Schachermeier (1992)) underlines the fact that, if there is no possibility of arbitrage during any unit interval, then no arbitrage is allowed with any finite temporal horizon strategy. Mathematics Subject Classification (2000): Primary: 60G48; Secondary: 60G40, 60G07 Journal of Economic Literature Classification: C50, C72, D84     

Shortfall risk minimisation versus symmetric (quadratic) hedging

Abstract We consider the seller of a contingent claim who wants to hedge against the corresponding risk by means of a self-financing strategy, endowing less initial capital than the one required for (super)hedging. Two classical criteria used in this situation are shortfall risk minimisation and symmetric hedging (a natural generalisation of the quadratic hedging problem). We show that these two problems are equivalent if the market is complete. The case when the market is incomplete is also discussed. Mathematics Subject Classification (2000): 91B28, 91B70, 93E20 Journal of Economic Literature Classification: D81, G11     

The Aubin private core of differential information economies

Abstract We establish characterizations of Radner equilibrium allocations via private core notions in the framework of differential information economies with a finite number of states of nature and a measure space of agents that may have atoms. The commodity space is an ordered separable Banach space whose positive cone admits an interior point. We introduce the notion of Aubin private core and prove that it provides a characterization of Radner equilibrium allocations. We show that the Aubin private core is equivalent to Edgeworth private equilibria and to privately non-dominated allocations of suitable associated economies. Mathematics Subject Classification (2000): 91B50, 91B44 Journal of Economic Literature Classification: D51, D82, D11     

Arbitrage and completeness in financial markets with given <Emphasis Type="Italic">N</Emphasis>-dimensional distributions

Abstract In this paper, we focus on the following problem: given a financial market, modelled by a process , and a family of probability measures on , with N a positive integer and the time space, we search for financially meaningful conditions which are equivalent to the existence and uniqueness of an equivalent (local) martingale measure (EMM) Q such that the price process S has under Q the pre-specified finite-dimensional distributions of order N (N-dds) . We call these two equivalent properties, respectively, N -mixed no free lunch and market N -completeness. They are based on a classification of contingent claims with respect to their path-dependence on S and on the related notion of N-mixed strategy. Finally, we apply this approach to the Black-Scholes model with jumps, by showing a uniqueness result for its equivalent martingale measures set. Mathematics Subject Classification (2000): 60G48, 91B28 Journal of Economic Literature Classification: G12, D52     

The completion of security markets

Abstract In this article we study the completion by options of a two-period security market in which the space of marketed securities is a subspace X of . Although there are important results about the completion (by options) Z of X, the problem of the determination of Z in its general form is still open. In this paper we solve this problem by determining a positive basis of Z. This method of positive bases simplifies the theory of security markets and also answers other open problems of this theory. In the classical papers of this subject, call and put options are taken with respect to the riskless bond 1 of . In this article we generalize this theory by taking call and put options with respect to different risky vectors u from a fixed vector subspace U of . This generalization was inspired by certain types of exotic option in finance. Mathematics Subject Classification (2000): 46B40, 46A35, 91B28, 91B30 Journal of Economic Literature Classification: G190, D520     

Conditional comonotonicity

Abstract In this paper we propose a generalization of the notion of comonotonicity by introducing and exploring the concept of conditional comonotonicity. We characterize this notion and we show by examples that conditional comonotonicity is the natural extension of the concept of comonotonicity to dynamic settings. Mathematics Subject Classification (2000): 90A05, 60Gxx Journal of Economic Literature Classification: D910, C610, G120     

Weak convergence of tree methods,to price options on defaultable assets

Abstract We discuss a practical method to price and hedge European contingent claims on assets with price processes which follow a jump-diffusion. The method consists of a sequence of trinomial models for the asset price and option price processes which are shown to converge weakly to the corresponding continuous time jump-diffusion processes. The main difference with many existing methods is that our approach ensures that the intermediate discrete time approximations generate models which are themselves complete, just as in the Black-Scholes binomial approximations. This is only possible by dropping the assumption that the approximations of increments of the Wiener and Poisson processes on our trinomial tree are independent, but we show that the dependence between these processes disappears in the weak limit. The approximations thus define an easy and flexible method for pricing and hedging in jump-diffusion models using explicit trees for hedging and pricing. Mathematics Subject Classification (2000): 60B10, 60H35 Journal of Economic Literature Classification: G13     

On the smoothness of optimal paths

Abstract The aim of this paper is to study the differentiability property of optimal paths in dynamic economic models. We address this problem from the point of view of the differential calculus in sequence spaces which are infinite-dimensional Banach spaces. We assume that the return or utility function is concave, and that optimal paths are interior and bounded. We study the C ^r differentiability of optimal paths vis-à-vis different parameters. These parameters are: the initial vector of capital stock, the discount rate and a parameter which lies in a Banach space (which could be the utility function itself). The method consists of applying an implicit function theorem on the Euler–Lagrange equation. In order to do this, we make use of classical conditions (i.e., the dominant diagonal block assumption) and we provide new ones. Mathematics Subject Classification (2000): 90A16, 49K40, 93C55 Journal of Economic Literature Classification: C161, D99, O41     

Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff

Abstract We analyze the Galerkin infinite element method for pricing European barrier options and, more generally, options with discontinuous payoff. The infinite element method is a simple and efficient modification of the more common finite element method. It keeps the best features of finite elements, i.e., bandedness, ease of programming, accuracy. Three main aspects are considered: (i) the degeneracy of the pricing PDE models at hand; (ii) the presence of discontinuities at the barriers or in the payoff clause and their effects on the numerical approximation process; (iii) the need for resorting to suitable numerical methods for unbounded domains when appropriate asymptotic conditions are not specified. The numerical stability and convergence of the proposed method are proved. Mathematics Subject Classification (2000): 65N30, 65J10 Journal of Economic Literature Classification: G13, C63