List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

Lectures on Stochastic Programming: Modeling and Theory

Recent advances in the causal inference literature on mediation have extended traditional approaches to direct and indirect effects to settings that allow for interactions and non-linearities. In this paper, these approaches from causal inference are further extended to settings in which multiple mediators may be of interest. Two analytic approaches, one based on regression and one based on weighting are proposed to estimate the effect mediated through multiple mediators and the effects through other pathways. The approaches proposed here accommodate exposure-mediator interactions and, to a certain extent, mediator-mediator interactions as well. The methods handle binary or continuous mediators and binary, continuous or count outcomes. When the mediators affect one another, the strategy of trying to assess direct and indirect effects one mediator at a time will in general fail; the approach given in this paper can still be used. A characterization is moreover given as to when the sum of the mediated effects for multiple mediators considered separately will be equal to the mediated effect of all of the mediators considered jointly. The approach proposed in this paper is robust to unmeasured common causes of two or more mediators.

/pdf/mediation-analysis-with-multiple-mediators-2q2cgi37fe.pdf

Mediation Analysis with Multiple Mediators

Recent crises in the financial industry have shown weaknesses in the modeling of Risk-Weighted Assets (RWAs). Relatively minor model changes may lead to substantial changes in the RWA numbers. Similar problems are encountered in the Value-at-Risk (VaR)-aggregation of risks. In this article, we highlight some of the underlying issues, both methodologically, as well as through examples. In particular, we frame this discussion in the context of two recent regulatory documents we refer to as Basel 3.5.

https://sas.uwaterloo.ca/~wang/papers/2014Embrechts-Puccetti-Ruschendorf-Wang-Beleraj-Risks.pdf

An Academic Response to Basel 3.5

We introduce the concept of a Markov risk measure and we use it to formulate risk-averse control problems for two Markov decision models: a finite horizon model and a discounted infinite horizon model. For both models we derive risk-averse dynamic programming equations and a value iteration method. For the infinite horizon problem we develop a risk-averse policy iteration method and we prove its convergence. We also propose a version of the Newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence. Finally, we discuss relations to min---max Markov decision models.

/pdf/risk-averse-dynamic-programming-for-markov-decision-2n4jawrjd9.pdf

Risk-averse dynamic programming for Markov decision processes

We extend the definition of a convex risk measure to a conditional framework where additional information is available. We characterize these risk measures through the associated acceptance sets and prove a representation result in terms of conditional expectations. A suitable regularity property of conditional risk measures is defined and discussed. Finally, we introduce the concept of a dynamic convex risk measure as a family of successive conditional convex risk measures and characterize those satisfying some natural time consistency properties. As a reference example, illustrating all the proposed developments, we introduce a suitably defined dynamic version of the class of entropic risk measures.

/pdf/conditional-and-dynamic-convex-risk-measures-3ka6ggo893.pdf

Conditional and Dynamic Convex Risk Measures

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a ‘risk measure’ that summarizes the risk of the portfolio. We define the notion of ‘risk measurement procedure’, which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property.

http://www.cmap.polytechnique.fr/~rama/papers/robustrisk.pdf

Robustness and sensitivity analysis of risk measurement procedures

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a "risk measure" which summarizes the risk of the portfolio. We define the notion of "risk measurement procedure", which includes both of these steps, and study the robustness of risk measurement procedures and their sensitivity to a change in the data set. After introducing a rigorous definition of 'robustness' of a risk measurement procedure, we illustrate the presence of a conflict between subadditivity and robustness of risk measurement procedures. We propose a measure of sensitivity for risk measurement procedures and compute the sensitivity function of various examples of risk estimators used in financial risk management, showing that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate in particular that using historical Value at Risk leads to a more robust procedure for risk measurement than recently proposed alternatives like CVaR. We also propose other risk measurement procedures which possess the robustness property.

/pdf/robustness-and-sensitivity-analysis-of-risk-measurement-2rb6zjdef5.pdf

Robustness and Sensitivity Analysis of Risk Measurement Procedures

In this paper we propose a generalization of the concepts of convex and coherent risk measures to a multiperiod setting, in which payoffs are spread over different dates. To this end, a careful examination of the axiom of translation invariance and the related concept of capital requirement in the one-period model is performed. These two issues are then suitably extended to the multiperiod case, in a way that makes their operative financial meaning clear. A characterization in terms of expected values is derived for this class of risk measures and some examples are presented.

/pdf/risk-measures-and-capital-requirements-for-processes-2gsau5a5my.pdf

Risk measures and capital requirements for processes

We discuss liquidity risk from a pure risk - theoretical point of view in the axiomatic context of Coherent Measures of Risk. We propose a formalism for Liquidity Risk which is compatible with the axioms of coherency. We emphasize the difference between 'coherent risk measures',(CRM) defined on portfolio values as opposed to 'coherent portfolio risk measures' (CPRM) defined on the vector space of portfolios, and we observe that in presence of liquidity risk the value function on the space of portfolios is no more necessarily linear. We propose a new nonlinear 'Value' function which depends on a new notion of 'liquidity policy' based on a general description of the micro-structure of illiquid markets and the impact that this micro-structure has when marking to market a portfolio. We discuss the consequences of the introduction of the function in the coherency axioms and we study the properties induced on CPRMs. We show in particular that CPRMs are convex, finding a result that was proposed as a new axiom in the literature of so called 'convex measures of risk'. The framework we propose is not a model but rather a new formalism, in the sense that it is completely free from hypotheses on the dynamics of the market. We provide interpretation and characterization of the formalism as well as some stylized example.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1048322_code629430.pdf?abstractid=1048322&rulid=261261&mirid=3

Giacomo Scandolo

Papers

Conditional and Dynamic Convex Risk Measures

Robustness and sensitivity analysis of risk measurement procedures

Robustness and Sensitivity Analysis of Risk Measurement Procedures

Risk measures and capital requirements for processes

Liquidity Risk Theory and Coherent Measures of Risk