Distributionally robust optimization with applications to risk management

Many decision problems can be formulated as mathematical optimization models. While deterministic optimization problems include only known parameters, real-life decision problems almost invariably involve parameters that are subject to uncertainty. Failure to take this uncertainty under consideration may yield decisions which can lead to unexpected or even catastrophic results if certain scenarios are realized. While stochastic programming is a sound approach to decision making under uncertainty, it assumes that the decision maker has complete knowledge about the probability distribution that governs the uncertain parameters. This assumption is usually unjustified as, for most realistic problems, the probability distribution must be estimated from historical data and is therefore itself uncertain. Failure to take this distributional modeling risk into account can result in unduly optimistic risk assessment and suboptimal decisions. Furthermore, for most distributions, stochastic programs involving chance constraints cannot be solved using polynomial-time algorithms. In contrast to stochastic programming, distributionally robust optimization explicitly accounts for distributional uncertainty. In this framework, it is assumed that the decision maker has access to only partial distributional information, such as the first- and second-order moments as well as the support. Subsequently, the problem is solved under the worst-case distribution that complies with this partial information. This worst-case approach effectively immunizes the problem against distributional modeling risk. The objective of this thesis is to investigate how robust optimization techniques can be used for quantitative risk management. In particular, we study how the risk of large-scale derivative portfolios can be computed as well as minimized, while making minimal assumptions about the probability distribution of the underlying asset returns. Our interest in derivative portfolios stems from the fact that careless investment in derivatives can yield large losses or even bankruptcy. We show that by employing robust optimization techniques we are able to capture the substantial risks involved in derivative investments. Furthermore, we investigate how distributionally robust chance constrained programs can be reformulated or approximated as tractable optimization problems. Throughout the thesis, we aim to derive tractable models that are scalable to industrial-size 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

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

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

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

Distributionally Robust Joint Chance Constraints with Second-Order Moment Information

We develop tractable semidefinite programming based approximations for distributionally robust individual and joint chance constraints, assuming that only the first- and second-order moments as well as the support of the uncertain parameters are given. It is known that robust chance constraints can be conservatively approximated by Worst-Case Conditional Value-at-Risk (CVaR) constraints. We first prove that this approximation is exact for robust individual chance constraints with concave or (not necessarily concave) quadratic constraint functions, and we demonstrate that the Worst-Case CVaR can be computed efficiently for these classes of constraint functions. Next, we study the Worst-Case CVaR approximation for joint chance constraints. This approximation affords intuitive dual interpretations and is provably tighter than two popular benchmark approximations. The tightness depends on a set of scaling parameters, which can be tuned via a sequential convex optimization algorithm. We show that the approximation becomes essentially exact when the scaling parameters are chosen optimally and that the Worst-Case CVaR can be evaluated efficiently if the scaling parameters are kept constant. We evaluate our joint chance constraint approximation in the context of a dynamic water reservoir control problem and numerically demonstrate its superiority over the two benchmark approximations

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

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.Robust optimization Portfolio optimization Portfolio insurance Second-order cone programming