Raquel Balbás

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.

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.

Good deals in markets with friction

This paper studies a portfolio choice problem such that the pricing rule may incorporate transaction costs and the risk measure is coherent and expectation bounded. We will prove the necessity of dealing with pricing rules such that there are essentially bounded stochastic discount factors, which must be also bounded from below by a strictly positive value. Otherwise good deals will be available to traders, i.e., depending on the selected risk measure, investors can build portfolios whose (risk, return) will be as close as desired to (- infinite, + infinite) or (0, infinite). This pathologic property still holds for vector risk measures (i.e., if we minimize a vector valued function whose components are risk measures). It is worthwhile to point out that essentially bounded stochastic discount factors are not usual in financial literature. In particular, the most famous frictionless, complete and arbitrage free pricing models imply the existence of good deals for every coherent and expectation bounded measure of risk, and the incorporation of transaction costs will no guarantee the solution of this caveat

Good deals and benchmarks in robust portfolio selection

This paper deals with portfolio selection problems under risk and ambiguity. The investor may be ambiguous with respect to the set of states of nature and their probabilities. Both static and discrete or continuous time dynamic pricing models are included in the analysis. Risk and ambiguity are measured in general settings. The considered risk measures contain, as particular cases, the usual deviations and the coherent and expectation bounded measures of risk.

Compatibility between pricing rules and risk measures: the CCVaR

This paper has considered a risk measure ρ and a (maybe incomplete and/or imperfect) arbitrage-free market with pricing rule Π. They are said to be compatible if there are no reachable strategies y such that Π(y) is bounded and ρ(y) is close to −∞. We show that the lack of compatibility leads to meaningless situations in financial or actuarial applications.The presence of compatibility is characterized by properties connecting the Stochastic Discount Factor of Π and the sub-gradient of ρ. Consequently, several examples pointing out that the lack of compatibility may occur in very important pricing models are yielded. For instance the CVaR is not compatible with the Black and Scholes model or the CAPM.We prove that for a given incompatible couple (Π, ρ) we can construct a minimal risk measureM(Π,ρ) compatible with ρ and such that ρ≤M(Π,ρ). This result is particularized for the CVaR and the CAPM and the Black and Scholes model. Therefore we construct the Compatible Conditional Value at Risk (CCVaR). It seems that the CCVaR preserves the good properties of the CVaR and overcomes its shortcomings.ResumenConsideraremos una medida de riesgo ρ y un mercado libre de arbitraje (puede ser que incompleto o imperfecto) con regla de valoración Π. Éstos serán compatibles si no hay estrategias alcanzablesy tales que Π(y) permanece acotado y ρ(y) se acerca a −∞. Veremos que la falta de compatibilidad conduce a situaciones sin sentido económico en las aplicaciones actuariales o financieras.La compatibilidad será caracterizada mediante propiedades que ligan al Factor de Descuento Estocástico de Π y al sub-gradiente de ρ. Consecuentemente, se podrán dar importantes ejemplos en los que hay falta de compatibilidad. Por ejemplo, el CVaR no es compatible con el modelo de Black-Scholes o el CAPM.Probaremos que para cualquier par incompatible (Π, ρ) se puede construir una medida de riesgo minimalM(Π,ρ) compatible con ρ, y tal que ρ≤M(Π,ρ). Este resultado se particularizará para el CVaR y el CAPM y el modelo de Black-Scholes. Por tanto, construiremos el CVaR Compatible (CCVaR). El CCVaR parece preservar las buenas propiedades del CVaR y superar sus deficiencias.

Outperforming Benchmarks with Their Derivatives: Theory and Empirical Evidence

Recent literature has demonstrated the existence of an unbounded risk premium if one combines the most important models for pricing and hedging derivatives with coherent risk measures. There may exist combinations of derivatives (good deals) whose pair (return risk) converges to the pair (+∞, −∞). This paper goes beyond existence properties and looks for optimal explicit constructions and empirical tests. It will be shown that the optimal good deal above may be a simple portfolio of options. This theoretical finding will enable us to implement empirical experiments involving three international stock index futures (Standard & Poors 500, Eurostoxx 50 and DAX 30) and three commodity futures (gold, Brent and the Dow Jones-UBS Commodity Index). According to the empirical results, the good deal always outperforms the underlying index/commodity. The good deal is built in full compliance with the standard derivative pricing theory. Properties of classical pricing models totally inspire the good deal construction. This is a very interesting difference in our paper with respect to previous literature attempting to outperform a benchmark.

VaR as the CVaR sensitivity

VaR minimization is a complex problem playing a critical role in many actuarial and financial applications of mathematical programming. The usual methods of convex programming do not apply due to the lack of sub-additivity. The usual methods of differentiable programming do not apply either, due to the lack of continuity. Taking into account that the CVaR may be given as an integral of VaR, one has that VaR becomes a first order mathematical derivative of CVaR. This property will enable us to give accurate approximations in VaR optimization, since the optimization VaR and CVaR will become quite closely related topics. Applications in both finance and insurance will be given.

Differential equations connecting VaR and CVaR

The Value at Risk (VaR) is a very important risk measure for practitioners, supervisors and researchers. Many practitioners draw on VaR as a critical instrument in Risk Management and other Actuarial/Financial problems, while super- visors and regulators must deal with VaR due to the Basel Accords and Solvency II, among other reasons. From a theoretical point of view VaR presents some drawbacks overcome by other risk measures such as the Conditional Value at Risk (CVaR). VaR is neither differentiable nor sub-additive because it is neither continuous nor convex. On the contrary, CVaR satis es all of these properties, and this simpli es many ana- lytical studies if VaR is replaced by CVaR. In this paper several differential equations connecting both VaR and CVaR will be presented. They will allow us to address several important issues involving VaR with the help of the CVaR properties. This new methodology seems to be very efficient. In particular, a new VaR Representation Theorem may be found, and optimization problems involving VaR or probabilistic constraints always have an equivalent differentiable optimization problem. Applications in VaR, marginal VaR, CVaR and marginal CVaR estimates will be addressed as well. An illustrative actuarial numerical example will be given.

Minimizing Vector Risk Measures

The minimization of risk functions is becoming very important due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Vector optimization problems involving many types of risk functions are studied. The “balance space approach” of multiobjective optimization and a general representation theorem of risk functions is used in order to transform the initial minimization problem in an equivalent one that is convex and usually linear. This new problem permits us to characterize optimality by saddle point properties that easily apply in practice. Applications in finance and insurance are presented.