Berç Rustem

Simulation and optimization approaches to scenario tree generation

In this paper, three approaches are presented for generating scenario trees for 3nancial portfolio problems. These are based on simulation, optimization and hybrid simulation/optimization. In the simulation approach, the price scenarios at each time period are generated as the centroids of random scenario simulations generated sequentially or in parallel. The optimization method generates a number of discrete outcomes which satisfy speci3ed statistical properties by solving either a sequence of non-linear optimization models (one at each node of the scenario tree) or onelargeoptimization proble m. In thehybrid approach, theoptimization proble m is re duce d in size by 3xing price variables to values obtained by simulation. These procedures are backtested using historical data and computational results are presented. ? 2003 Elsevier B.V. All rights reserved.

Worst-case robust decisions for multi-period mean-variance portfolio optimization

Abstract In this paper, we extend the multi-period mean–variance optimization framework to worst-case design with multiple rival return and risk scenarios. Our approach involves a min–max algorithm and a multi-period mean–variance optimization framework for the stochastic aspects of the scenario tree. Multi-period portfolio optimization entails the construction of a scenario tree representing a discretised estimate of uncertainties and associated probabilities in future stages. The expected value of the portfolio return is maximized simultaneously with the minimization of its variance. There are two sources of further uncertainty that might require a strengthening of the robustness of the decision. The first is that some rival uncertainty scenarios may be too critical to consider in terms of probabilities. The second is that the return variance estimate is usually inaccurate and there are different rival estimates, or scenarios. In either case, the best decision has the additional property that, in terms of risk and return, performance is guaranteed in view of all the rival scenarios. The ex-ante performance of min–max models is tested using historical data and backtesting results are presented.

Robust min}max portfolio strategies for rival forecast and risk scenarios

We consider an extension of the Markowitz mean}variance optimization framework to multiple return and risk scenarios. It is well known that asset return forecasts and risk estimates are inherently inaccurate. The method proposed provides a means for considering rival representations of the future. The optimal portfolio is computed, simultaneously with the worst case, to take account of all rival scenarios. This is a min-max strategy which is essentially equivalent to a robust pooling of the scenarios. Robustness is ensured by the noninferiority of min}max. For example, a basic worst-case optimal return is guaranteed in view of multiple return scenarios. If robustness happens to have too high a cost, guided by the min}max pooling, it is also possible to explore other pooling alternatives. A min}max algorithm is used to solve the problem and illustrate the robust character of min}max with return and risk scenarios. We study the properties of the min}max risk}return frontier and compare with the potentially suboptimal worst-case where the investment strategy and the worst case are computed separately. ( 2000 Elsevier Science B.V. All rights reserved. JEL classixcation: C44; C61; C63; G11

Brief An algorithm for constrained nonlinear optimization under uncertainty

This paper considers robust formulations for the constrained control of systems under uncertainty. The underlying model is nonlinear and stochastic. A mean-variance robustness framework is adopted. We consider formulations to ensure feasibility over the entire domain of the uncertain parameters. However, strict feasibility may not always be possible, and can also be very expensive. We consider two alternative approaches to address feasibility. Flexibility in the operational conditions is provided via a penalty framework. The robust strategies are tested on a dynamic optimization problem arising from a chemical engineering application.

Worst-Case Value at Risk of Nonlinear Portfolios

Portfolio optimization problems involving value at risk VaR are often computationally intractable and require complete information about the return distribution of the portfolio constituents, which is rarely available in practice. These difficulties are compounded when the portfolio contains derivatives. We develop two tractable conservative approximations for the VaR of a derivative portfolio by evaluating the worst-case VaR over all return distributions of the derivative underliers with given first-and second-order moments. The derivative returns are modelled as convex piecewise linear or---by using a delta--gamma approximation---as possibly nonconvex quadratic functions of the returns of the derivative underliers. These models lead to new worst-case polyhedral VaR WPVaR and worst-case quadratic VaR WQVaR approximations, respectively. WPVaR serves as a VaR approximation for portfolios containing long positions in European options expiring at the end of the investment horizon, whereas WQVaR is suitable for portfolios containing long and/or short positions in European and/or exotic options expiring beyond the investment horizon. We prove that---unlike VaR that may discourage diversification---WPVaR and WQVaR are in fact coherent risk measures. We also reveal connections to robust portfolio optimization. This paper was accepted by Dimitris Bertsimas, optimization.

Robust portfolio optimization with derivative insurance guarantees

Robust portfolio optimization aims to maximize the worst-case portfolio return given that the asset returns are allowed to vary within a prescribed uncertainty set. If the uncertainty set is not too large, the resulting portfolio performs well under normal market conditions. However, its performance may substantially degrade in the presence of market crashes, that is, if the asset returns materialize far outside of the uncertainty set. We propose a novel robust optimization model for designing portfolios that include European-style options. This model trades off weak and strong guarantees on the worst-case portfolio return. The weak guarantee applies as long as the asset returns are realized within the prescribed uncertainty set, while the strong guarantee applies for all possible asset returns. The resulting model constitutes a convex second-order cone program, which is amenable to efficient numerical solution procedures. We evaluate the model using simulated and empirical backtests and analyze the impact of the insurance guarantees on the portfolio performance.

An Algorithm for the Inequality-Constrained Discrete Min--Max Problem

In this paper we discuss an approach using an augmented Lagrangian formulation to directly solve the inequality constrained min--max problem. The algorithm involves a sequential quadratic programming subproblem, an adaptive penalty parameter selection rule, and a stepsize strategy, convergent to unit steps, that ensures progress toward optimality and feasibility of the inequality constraints. It is shown that the penalty parameter does not grow indefinitely. The convergence of the algorithm is established, and its numerical effectiveness is demonstrated with test examples.

Maximizing the net present value of a project under uncertainty

We address the maximization of a projects expected net present value when the activity durations and cash flows are described by a discrete set of alternative scenarios with associated occurrence probabilities. In this setting, the choice of scenario-independent activity start times frequently leads to infeasible schedules or severe losses in revenues. We suggest to determine an optimal target processing time policy for the project activities instead. Such a policy prescribes an activity to be started as early as possible in the realized scenario, but never before its (scenario-independent) target processing time. We formulate the resulting model as a global optimization problem and present a branch-and-bound algorithm for its solution. Extensive numerical results illustrate the suitability of the proposed policy class and the runtime behavior of the algorithm.

Semi-Infinite Programming and Applications to Minimax Problems

A minimisation problem with infinitely many constraints – semi-infinite programming problem (SIP) is considered. The problem is solved using a two stage procedure that searches for global maximum violation of the constraints. A version of the algorithm that searches for any violation of constraints is also considered, and the performance of the two algorithm is compared. An application to solving minimax problem (with and without coupled constraints) is given and a comparison with the algorithm for continuous minimax of Rustem and Howe (2001) is included. Finally, we consider an application to chemical engineering problems.

Pessimistic Bilevel Optimization

We study a variant of the pessimistic bilevel optimization problem, which comprises constraints that must be satisfied for any optimal solution of a subordinate (lower-level) optimization problem. We present conditions that guarantee the existence of optimal solutions in such a problem, and we characterize the computational complexity of various subclasses of the problem. We then focus on problem instances that may lack convexity, but that satisfy a certain independence property. We develop convergent approximations for these instances, and we derive an iterative solution scheme that is reminiscent of the discretization techniques used in semi-infinite programming. We also present a computational study that illustrates the numerical behavior of our algorithm on standard benchmark instances.