We consider convex stochastic programs with an (approximate) initial probability distribution P having finite support supp P, i.e., finitely many scenarios. The behaviour of such stochastic programs is stable with respect to perturbations of P measured in terms of a Fortet-Mourier probability metric. The problem of optimal scenario reduction consists in determining a probability measure that is supported by a subset of supp P of prescribed cardinality and is closest to P in terms of such a probability metric. Two new versions of forward and backward type algorithms are presented for computing such optimally reduced probability measures approximately. Compared to earlier versions, the computational performance (accuracy, running time) of the new algorithms has been improved considerably. Numerical experience is reported for different instances of scenario trees with computable optimal lower bounds. The test examples also include a ternary scenario tree representing the weekly electrical load process in a power management model.

Scenario Reduction Algorithms in Stochastic Programming

Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics. Efficient algorithms are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty. For instance, it turns out that after 50% reduction of the scenario tree the optimal reduced tree still has about 90% relative accuracy.

/pdf/scenario-reduction-in-stochastic-programming-i0hwrxko63.pdf

Scenario Reduction in Stochastic Programming

Portfolio and risk management problems of power utilities may be modeled by multistage stochastic programs. These models use a set of scenarios and corresponding probabilities to model the multivariate random data process (electrical load, stream flows to hydro units, and fuel and electricity prices). For most practical problems the optimization problem that contains all possible scenarios is too large. Due to computational complexity and to time limitations this program is often approximated by a model involving a (much) smaller number of scenarios. The proposed reduction algorithms determine a subset of the initial scenario set and assign new probabilities to the preserved scenarios. The scenario tree construction algorithms successively reduce the number of nodes of a fan of individual scenarios by modifying the tree structure and by bundling similar scenarios. Numerical experience is reported for constructing scenario trees for the load and spot market prices entering a stochastic portfolio management model of a German utility.

/pdf/scenario-reduction-and-scenario-tree-construction-for-power-4c2omu498k.pdf

Scenario reduction and scenario tree construction for power management problems

https://www.cirrelt.ca/DocumentsTravail/CIRRELT-2008-36.pdf

The Design of Robust Value-Creating Supply Chain Networks: A Critical Review †

In models of decision making under uncertainty we often are faced with the problem of representing the uncertainties in a form suitable for quantitative models. If the uncertainties are expressed in terms of multivariate continuous distributions, or a discrete distribution with far too many outcomes, we normally face two possibilities: either creating a decision model with internal sampling, or trying to find a simple discrete approximation of the given distribution that serves as input to the model. This paper presents a method based on nonlinear programming that can be used to generate a limited number of discrete outcomes that satisfy specified statistical properties. Users are free to specify any statistical properties they find relevant, and the method can handle inconsistencies in the specifications. The basic idea is to minimize some measure of distance between the statistical properties of the generated outcomes and the specified properties. We illustrate the method by single- and multiple-period problems. The results are encouraging in that a limited number of generated outcomes indeed have statistical properties that are close to or equal to the specifications. We discuss how to verify that the relevant statistical properties are captured in these specifications, and argue that what are the relevant properties, will be problem dependent.

/pdf/generating-scenario-trees-for-multistage-decision-problems-3fcl9vpypi.pdf

Generating Scenario Trees for Multistage Decision Problems

A major issue in any application of multistage stochastic programming is the representation of the underlying random data process. We discuss the case when enough data paths can be generated according to an accepted parametric or nonparametric stochastic model. No assumptions on convexity with respect to the random parameters are required. We emphasize the notion of representative scenarios (or a representative scenario tree) relative to the problem being modeled.

Scenarios for Multistage Stochastic Programs

Multistage stochastic programming (in contrast to stochastic control) has found wide application in the formulation and solution of financial problems characterized by a large number of state variables and a generally low number of possible decision stages. The literature on the use of multistage recourse modelling to formalize complex portfolio optimization problems dates back to the early seventies, when the technique was first adopted to solve a fixed interest security portfolio problem. We present here the CALM model which has been designed to deal with uncertainty affecting both assets (in either the portfolio or the market) and liabilities (in the form of scenario dependent payments or borrowing costs). We consider as an instance a pension fund problem in which portfolio rebalancing is allowed over a long-term horizon at discrete time points and where liabilities refer to five different classes of pension contracts. The portfolio manager, given an initial wealth, seeks the maximization of terminal wealth at the horizon, with investment returns modelled as discrete state random vectors. Decision vectors represent possible investments in the market and holding or selling assets in the portfolio, as well as borrowing decisions from a credit line or deposits with a bank. Computational results are presented for a set of 10-stage portfolio problems using different solution methods and libraries (OSL,CPLEX,OB1). The portfolio problem with an underlying vector data process which allows up to 2688 realizations at the 10 year horizon is solved on an IBM RS6000/590 for a set of twenty four large scale test problems using the simplex and barrier methods provided by CPLEX (the latter for either linear or quadratic objective), the predictor/corrector interior point method provided in OB1, the simplex method of OSL, the MSLiP-OSL code instantiating nested Benders decomposition with subproblem solution using OSL simplex and the current version of MSLiP.

https://papers.ssrn.com/sol3/Delivery.cfm/9707271.pdf?abstractid=34780&rulid=145901&mirid=2

Dynamic Stochastic Programming For Asset-liability Management

Multistage stochastic programming - in contrast to stochastic control - has found wideapplication in the formulation and solution of financial problems characterized by a largenumber of state variables and a generally low number of possible decision stages. Theliterature on the use of multistage recourse modelling to formalize complex portfolio optimizationproblems dates back to the early seventies, when the technique was first adopted tosolve a fixed income security portfolio problem. We present here the CALM model, whichhas been designed to deal with uncertainty affecting both assets (in either the portfolio orthe market) and liabilities (in the form of scenario dependent payments or borrowing costs).We consider as an instance a pension fund problem in which portfolio rebalancing is allowedover a long-term horizon at discrete time points and where liabilities refer to five differentclasses of pension contracts. The portfolio manager, given an initial wealth, seeks the maximizationof terminal wealth at the horizon, with investment returns modelled as discretestate random vectors. Decision vectors represent possible investments in the market andholding or selling assets in the portfolio, as well as borrowing decisions from a credit lineor deposits with a bank. Computational results are presented for a set of 10-stage portfolioproblems using different solution methods and libraries (OSL, CPLEX, OB1). The portfolioproblem, with an underlying vector data process which allows up to 2688 realizations at the10-year horizon, is solved on an IBM RS6000y590 for a set of twenty-four large-scale testproblems using the simplex and barrier methods provided by CPLEX (the latter for eitherlinear or quadratic objective), the predictorycorrector interior point method provided in OB1,the simplex method of OSL, the MSLiP-OSL code instantiating nested Benders decompositionwith subproblem solution using OSL simplex, and the current version of MSLiP.

Dynamic stochastic programmingfor asset-liability management

Risk managers are increasingly required by international Regulatory Institutions to adopt accurate techniques for the measurement and control of portfolios financial risks. The task requires first the identification of the different risk sources affecting the portfolio and the measurement of their impact, then after: the adoption of appropriate portfolio strategies aimed at neutralising these risks. The comprehensive concept of Value-at-Risk (VaR) as a maximum tolerable loss, with a given confidence interval, has become in this regard the industry standard in risk management. In the paper we focus on the implications of different risk measurement techniques and portfolio optimisation strategies in presence of markets subject to periods of severe instability, resulting in significant deviations of financial returns from the Normality assumption typically adopted in mainstream finance. Comparative results on 1 day-VaR(99%) estimation are presented over a range of bond and equity markets with different risk profiles. The reference period of our analysis includes several market shocks and in particular the Argentinean Eurobond crisis of July 2001. The solution of an optimal portfolio problem over the crisis period is discussed within a [mean, variance, VaR99%] portfolio space, emphasising the difficulty of the portfolio's relative return maximisation problem faced by fund managers.

Tail estimation and mean–VaR portfolio selection in markets subject to financial instability

The second edition of Optimization Methods in Finance comes 11 years after the successful first edition with a relevant extension to include material on mean-variance portfolio selection and multi-...

Giorgio Consigli

Papers

Scenarios for Multistage Stochastic Programs

Dynamic Stochastic Programming For Asset-liability Management

Dynamic stochastic programmingfor asset-liability management

Tail estimation and mean–VaR portfolio selection in markets subject to financial instability

Optimization Methods in Finance