Alejandro Balbás

When can you immunize a bond portfolio

The object of this paper is to give conditions under which it is possible to immunize a bond portfolio. Maxmin strategies are also studied, as well as their relations with immunized ones. Some special shocks on the interest rate are analyzed, and general conditions about immunization are obtained. When immunization is not possible, capital losses are measured.

Extending pricing rules with general risk functions

The paper addresses pricing issues in imperfect and/or incomplete markets if the risk level of the hedging strategy is measured by a general risk function. Convex Optimization Theory is used in order to extend pricing rules for a wide family of risk functions, including Deviation Measures, Expectation Bounded Risk Measures and Coherent Measures of Risk. Necessary and sufficient optimality conditions are provided in a very general setting. For imperfect markets the extended pricing rules reduce the bid-ask spread. The findings are particularized so as to study with more detail some concrete examples, including the Conditional Value at Risk and some properties of the Standard Deviation. Applications dealing with the valuation of volatility linked derivatives are discussed.

Portfolio choice and optimal hedging with general risk functions: A simplex-like algorithm

The minimization of general risk functions is becoming more and more important in portfolio choice theory and optimal hedging. There are two major reasons. Firstly, heavy tails and the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses. This paper proposes a new method to simultaneously minimize several general risk or dispersion measures. The representation theorems of risk functions are applied to transform the general risk minimization problem in a minimax problem, and later in a linear programming problem between infinite-dimensional Banach spaces. Then, new necessary and sufficient optimality conditions are stated and a simplex-like algorithm is developed. The algorithm solves the dual problem and provides both optimal portfolios and their sensitivities. The approach is general enough and does not depend on any particular risk measure, but some of the most important cases are specially analyzed. A final real data numerical example illustrates the practical performance of the proposed methodology.

How Financial Theory Applies to Catastrophe-Linked Derivatives. An Empirical Test of Several Pricing Models

The paper focuses on the PCS Catastrophe Insurance Option Contracts and empirically tests the degree of agreement between their real quotes and the standard fmancial theory. The highest possible precision is incorporated since the real quotes are perfectly synchronized and the bid-ask spread is always considered. A static setting is assumed and the main topics of arbitrage, hedging and portfolio choice are involved in the analysis. Three significant conclusions are reached. First, the catastrophe derivatives may be very often priced by arbitrage methods, and the paper provides some examples of practical strategies that were available in the market. Second, hedging arguments also yield adequate criteria to price the derivatives and some real examples are provided as well. Third, in a variance aversion context many agents could be interested in selling derivatives to invest the money in stocks and bonds. These strategies show a suitable level in the variance for any desired expected return. Furthermore, the methodology here applied seems to be quite general and may be useful to price other derivative securities. Simple assumptions on the underlying asset behavior are the only required conditions.

Minimizing measures of risk by saddle point conditions

The minimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions. This new problem involves Banach spaces but a mean value theorem for risk measures is stated, and this simplifies the dual problem. Then, optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable. This characterization is significantly different if one compares it with previous literature. Furthermore, the saddle point condition very easily applies in practice. Four applications in finance and insurance are presented.

Dispersion measures as immunization risk measures

The quadratic and linear cash flow dispersion measures M 2 and ~ N are two immunization risk measures designed to build immunized bond portfolios. This paper generalizes these two measures by showing that any dispersion measure is an immunization risk measure and therefore, it sets up a tool to be used in empirical testing. Each new measure is derived from a different set of shocks (changes on the term structure of interest rates) and depends on the corresponding subset of worst shocks. Consequently, a criterion for choosing appropriate immunization risk measures is to take those developed from the most reasonable sets of shocks and the associated subset of worst shocks and then select those that work best empirically. Adopting this approach, this paper then explores both numerical examples and a short empirical study on the Spanish Bond Market in the mid-1990s to show that measures between linear and quadratic are the most appropriate, and amongst them, the linear measure has the best properties. This confirms previous studies on US and Canadian markets that maturityconstrained-duration-matched portfolios also have good empirical behavior. 2002 Elsevier Science B.V. All rights reserved.

Integration and arbitrage in the Spanish financial markets: An empirical approach*

Several authors have introduced different ways to measure the integration between fmancial markets. Most of them are derived from the basic assumptions to price assets, like the Law of One Price or the absence of arbitrage opportunities. Two perfectly integrated markets must give identical price to identical fmal payoffs, and a vector of positive discount factors, common to both markets, must exist. Therefore, if these properties do not hold, their degree of violation can be measured and considered as an integration measure. The present paper empirically test the integration measures in the Spanish fmancial markets. Hence, several interesting values are obtained, like for instance, the state prices or the risk-neutral probabilities. Furthermore, when the risk-neutral probabilities do not exist, explicit cross-market arbitrage portfolios are detected. The results of our test are surprising for several reasons. First of all, the arbitrage opportunities very often appear, and the bid-ask spread and the transaction costs are not able to avoid the arbitrage profits. Furthermore, the criticisms, which are usually argued when empirical papers show the existence of arbitrage opportunities, do not apply here, since we work with perfectly synchronized high frequency data. On the other hand, different integration measures show a similar evolution along the tested period, although these measures give different information about the markets efficiency and integration, and they do not have to be necessarily related.

Projective system approach to the martingale characterization of the absence of arbitrage

Abstract The equivalence between the absence of arbitrage and the existence of an equivalent martingale measure fails when an infinite number of trading dates is considered. By enlarging the set of states of nature and the probability measure through a projective system of perfect measure spaces, we characterize the absence of arbitrage when the time set is countable.

Density theorems for ideal points in vector optimization

Abstract Several results are established concerning the density of the set of ideal points in the set of minimal solutions of positive support functionals of sets in normed linear spaces. The above ideal points are defined and several characterizations and sufficient conditions for their existence are also stated.

Duality theory for infinite-dimensional multiobjective linear programming

Abstract The purpose of this paper is to study the relationship between an arbitrary-dimensional multiobjective linear program and its dual program. Dual solutions are obtained which, under certain conditions, measure the sensitivity of the proper primal solutions.