Stochastic optimization and worst-case analysis in monetary policy design

In this paper, we examine the cost of insurance against model uncertainty for the Euro area considering four alternative reference models, all of which are used for policy-analysis at the ECB.We find that maximal insurance across this model range in terms of aMinimax policy comes at moderate costs in terms of lower expected performance. We extract priors that would rationalize the Minimax policy from a Bayesian perspective. These priors indicate that full insurance is strongly oriented towards the model with highest baseline losses. Furthermore, this policy is not as tolerant towards small perturbations of policy parameters as the Bayesian policy rule. We propose to strike a compromise and use preferences for policy design that allow for intermediate degrees of ambiguity-aversion.These preferences allow the specification of priors but also give extra weight to the worst uncertain outcomes in a given context. JEL Klassifikation: E52, E58, E6

Portfolio Decisions with Higher Order Moments

In this paper, we address the global optimization of two interesting nonconvex problems in finance. We relax the normality assumption underlying the classical Markowitz mean-variance portfolio optimization model and consider the incorporation of skewness (third moment) and kurtosis (fourth moment). The investor seeks to maximize the expected return and the skewness of the portfolio and minimize its variance and kurtosis, subject to budget and no short selling constraints. In the first model, it is assumed that asset statistics are exact. The second model allows for uncertainty in asset statistics. We consider rival discrete estimates for the mean, variance, skewness and kurtosis of asset returns. A robust optimization framework is adopted to compute the best investment portfolio maximizing return, skewness and minimizing variance, kurtosis, in view of the worst-case asset statistics. In both models, the resulting optimization problems are nonconvex. We introduce a computational procedure for their global optimization.Mean-variance portfolio selection, Robust portfolio selection, Skewness, Kurtosis, Decomposition methods, Polynomial optimization problems

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 portfolio optimization model that provides additional strong performance guarantees for all possible realizations of the asset returns. This insurance is provided via optimally chosen derivatives on the assets in the portfolio. The resulting model constitutes a convex second- order cone program, which is amenable to efficient numerical solution. We evaluate the model using simulated and empirical backtests and conclude that it can out- perform standard robust portfolio optimization as well as classical mean-variance optimization.robust optimization, portfolio optimization, portfolio insurance, second order cone programming

Bounding Option Prices Using SDP With Change Of Numeraire

Recently, given the first few moments, tight upper and lower bounds of the no arbitrage prices can be obtained by solving semidefinite programming (SDP) or linear programming (LP) problems. In this paper, we compare SDP and LP formulations of the European-style options pricing problem and prefer SDP formulations due to the simplicity of moments constraints. We propose to employ the technique of change of numeraire when using SDP to bound the European type of options. In fact, this problem can then be cast as a truncated Hausdorff moment problem which has necessary and sufficient moment conditions expressed by positive semidefinite moment and localizing matrices. With four moments information we show stable numerical results for bounding European call options and exchange options. Moreover, A hedging strategy is also identified by the dual formulation.moments of measures, semidefinite programming, linear programming, options pricing, change of numeraire

Robust Resource Allocations in Temporal Networks

Temporal networks describe workflows of time-consuming tasks whose processing order is constrained by precedence relations. In many cases, the durations of the network tasks can be influenced by the assignment of resources. This leads to the problem of selecting an ‘optimal’ resource allocation, where optimality is measured by network characteristics such as the makespan (i.e., the time required to complete all tasks). In this paper, we study a robust resource allocation problem where the functional relationship between task durations and resource assignments is uncertain, and the goal is to minimise the worst-case makespan. We show that this problem is generically NP-hard. We then develop convergent bounds for the optimal objective value, as well as feasible allocations whose objective values are bracketed by these bounds. Numerical results provide empirical support for the proposed method.Robust Optimisation, Temporal Networks, Resource Allocation Problem

Mean Variance Optimization of Non-Linear Systems and Worst-case Analysis

In this paper, we consider expected value, variance and worst-case optimization of nonlinear models. We present algorithms for computing optimal expected values, and variance, based on iterative Taylor expansions. We establish convergence and consider the relative merits of policies beaded on expected value optimization and worst-case robustness. The latter is a minimax strategy and ensures optimal cover in view of the worst-case scenario(s) while the former is optimal expected performance in a stochastic setting. Both approaches are used with a macroeconomic policy model to illustrate relative performances, robustness and trade-offs between the strategies.

Robust Optimization of Currency Portfolios

We study a currency investment strategy, where we maximize the return on a portfolio of foreign currencies relative to any appreciation of the corresponding foreign exchange rates. Given the uncertainty in the estimation of the future currency values, we employ robust optimization techniques to maximize the return on the portfolio for the worst-case foreign exchange rate scenario. Currency portfolios differ from stock only portfolios in that a triangular relationship exists among foreign exchange rates to avoid arbitrage. Although the inclusion of such a constraint in the model would lead to a nonconvex problem, we show that by choosing appropriate uncertainty sets for the exchange and the cross exchange rates, we obtain a convex model that can be solved efficiently. Alongside robust optimization, an additional guarantee is explored by investing in currency options to cover the eventuality that foreign exchange rates materialize outside the specified uncertainty sets. We present numerical results that show the relationship between the size of the uncertainty sets and the distribution of the investment among currencies and options, and the overall performance of the model in a series of backtesting experiments.robust optimization, portfolio optimization, currency hedging, second-order cone programming

Worst-Case Value-at-Risk of Non-Linear 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 further 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 non-convex) quadratic functions of the returns of the derivative underliers. These models lead to new Worst-Case Polyhedral VaR (WCPVaR) and Worst-Case Quadratic VaR (WCQVaR) approximations, respectively. WCPVaR is a suitable VaR approximation for portfolios containing long positions in European options expiring at the end of the investment horizon, whereas WCQVaR is suitable for portfolios containing long and/or short positions in European and/or exotic options expiring beyond the investment horizon. We prove that WCPVaR and WCQVaR optimization can be formulated as tractable second-order cone and semidefinite programs, respectively, and reveal interesting connections to robust portfolio optimization. Numerical experiments demonstrate the benefits of incorporating non-linear relationships between the asset returns into a worst-case VaR model.Value-at-Risk, Derivatives, Robust Optimization, Second-Order Cone Programming, Semidefinite Programming

Mean variance optimization of non-linear systems and worst-case analysis

In this paper, we consider expected value, variance and worst-case optimization of nonlinear models. We present algorithms for computing optimal expected values, and variance, based on iterative Taylor expansions. We establish convergence and consider the relative merits of policies beaded on expected value optimization and worst-case robustness. The latter is a minimax strategy and ensures optimal cover in view of the worst-case scenario(s) while the former is optimal expected performance in a stochastic setting. Both approaches are used with a macroeconomic policy model to illustrate relative performances, robustness and trade-offs between the strategies. Klassifikation: C61, E4