Robust Pricing Research Articles

A robust optimization approach for an artillery fire-scheduling problem under uncertain threat

Abstract In this study, the determination of an artillery firing sequence is considered in order to minimize the enemy threat to friendly forces prior to the conclusion of the firing operation. A deterministic model and robust counterpart are developed to deal with the deterministic and uncertain enemy threat levels, respectively. In both cases, the optimal strategy is demonstrated to be firing based on the threat removal rate per unit time. Moreover, a two-phase approach is developed based on the concept of cardinality-constrained uncertainty. In this approach, the firing sequence is determined in the first phase and then, in the second phase, the robustness of the uncertain factors is adjusted by using the modified problem to evaluate the total threat exposure of friendly units to the enemy. A set of problems is generated and tested using the proposed model and strategy. The results of numerical experiments demonstrate that the proposed strategy outperforms conventional fire-scheduling approaches. Additionally, the price of robustness is considered by adjusting the value of uncertain factors.

The robust multiple-choice multidimensional knapsack problem

The multiple-choice multidimensional knapsack problem (MMKP) assumes n sets composed of mutually exclusive items. The goal is to select exactly one item per set, maximizing the overall utility, without violating a family of knapsack constraints. Motivated by recent applications of the MMKP to complex system reliability and quality of service management problems, we propose a robust version. More specifically, we relinquish the assumption that the problem parameters are deterministically known by limiting their values to a pre-specified uncertainty set. Depending on the structure of the variance−covariance matrix used to model the uncertainty, we identify four different cases, leading to robust formulations characterized by second order cone programs. We show how each of these programs is transformed into an equivalent linear program, implying that the use of a robust formulation for the MMKP comes with no extra computational complexity. Finally, using a novel matheuristic designed for the MMKP, we shed lights on the trade-off between the “price of robustness,” i.e., how much worse the objective function value of a robust solution is, compared with the deterministic one, and the “reliability,” i.e., the probability that a robust solution will lead to a feasible scenario for an arbitrary realization of the uncertain parameters.

Robust pricing\u2013hedging dualities in continuous time

We pursue a robust approach to pricing and hedging in mathematical finance. We consider a continuous-time setting in which some underlying assets and options, with continuous price paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland (Ann. Stat. 31:1413–1438, 2003), we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. superreplication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows us to quantify the impact of making assumptions or gaining information. We obtain a general pricing–hedging duality result: the infimum over superhedging prices of an exotic option with payoff G is equal to the supremum of expectations of G under calibrated martingale measures. Our results include in particular the martingale optimal transport duality of Dolinsky and Soner (Probab. Theory Relat. Fields 160:391–427, 2014) and extend it to multiple dimensions, multiple maturities and beliefs which are invariant under time-changes. In a general setting with arbitrary beliefs and for a uniformly continuous G, the asserted duality holds between limiting values of perturbed problems.

Robust Optimization for Hydroelectric System Operation Under Uncertainty

In this paper, we propose an optimal dispatch scheme for a cascade hydroelectric power system that maximizes the head levels of each dam, and minimizes the spillage effects taking into account uncertainty in the net load variations, i.e., the difference between the load and the renewable resources, and inflows to the cascade. By maximizing the head levels of each dam the volume of water stored, which is a metric of system resiliency, is maximized. In this regard, the operation of the cascade hydroelectric power system is robust to the variability and intermittency of renewable resources and increases system resilience to variations in climatic conditions. Thus, we demonstrate the benefits of coupling hydroelectric and photovoltaic resources. To this end, we introduce an approximate model for a cascade hydroelectric power system. We then develop correlated probabilistic forecasts for the uncertain output of renewable resources, e.g., solar generation, using historical data based on clustering and Markov chain techniques. We incorporate the generated forecast scenarios in the optimal dispatch of the cascade hydroelectric power system, and define a robust variant of the developed system. However, the robust variant is intractable due to the infinite number of constraints. Using tools from robust optimization, we reformulate the resulting problem in a tractable form that is amenable to existing numerical tools and show that the computed dispatch is immunized against uncertainty. The efficacy of the proposed approach is demonstrated by means of an actual case study involving the Seven Forks system located in Kenya, which consists of five cascaded hydroelectric power systems. With the case study, we demonstrate that the Seven Forks system may be coupled with solar generation since the “price of robustness” is small; thus, demonstrating the benefits of coupling hydroelectric systems with solar generation.

The determinants of informal housing price in Beijing: Village power, informal institutions, and property security

Although the how's of the informal housing market in China has been discussed widely; the answers remain conflicted. Using the theory of new institutional economics, this paper aims to answer two questions: what informal institutions secure the de facto tenure and facilitate the market? and How do informal institutions affect the pricing mechanism? We propose a conceptual framework to investigate the links between informal institutions, de facto tenure, and the price of informal housing. It is argued that villages have the ability to seek political patronage and regulate the market transactions to secure the tenure of informal housing. This ability is authorized by several formalities: the dual land system and the regime of village autonomy. The growing powers of market agencies in local development help strengthen the village's ability. The variances in ability between villages lead to a heterogeneity of tenure security, thus creating price differentials in the market. A higher tenure security increases the housing price. In conclusion, the informal housing market in China is “regulated” by a “semi-formal” system in which both informality and formality work and intertwine. A robust pricing mechanism has been emerging within the context of transition.

Dynamic Incentive-Aware Learning: Robust Pricing in Contextual Auctions

Motivated by pricing in ad exchange markets, we consider the problem of robust learning of reserve prices against strategic buyers in repeated contextual second-price auctions. Buyers’ valuations for an item depend on the context that describes the item. However, the seller is not aware of the relationship between the context and buyers’ valuations, i.e., buyers’ preferences. The seller’s goal is to design a learning policy to set reserve prices via observing the past sales data, and her objective is to minimize her regret for revenue, where the regret is computed against a clairvoyant policy that knows buyers’ heterogeneous preferences. Given the seller’s goal, utility-maximizing buyers have the incentive to bid untruthfully in order to manipulate the seller’s learning policy. We propose learning policies that are robust to such strategic behavior. These policies use the outcomes of the auctions, rather than the submitted bids, to estimate the preferences while controlling the long-term effect of the outcome of each auction on the future reserve prices. When the market noise distribution is known to the seller, we propose a policy called Contextual Robust Pricing (CORP) that achieves a T-period regret of O(d log(Td)log(T)), where d is the dimension of the contextual information. When the market noise distribution is unknown to the seller, we propose two policies whose regrets are sublinear in T.

Uncertain production scheduling optimization in open-pit mines and its ellipsoidal robust counterpart

The block extraction sequence is among the most challenging and important issues that should be regarded through whole mining operations to reach the maximum amount of profit. Such open-pit production scheduling problems can be solved under either deterministic or non-deterministic states. This study aims to model the problem applying robust counterpart linear optimization that uses the ellipsoidal set based counterpart. Block economic value as the objective function coefficient, and block weights and operational capacity as the constraints coefficients, are considered as uncertainty sources. Besides the exact mathematical modelling, a genetic algorithm is applied to solve the nonlinear ellipsoidal counterpart. The model, which was solved in deterministic and uncertain conditions, terminated in results with some differences in scheduled plans. The price of robustness is obtained at different conservative levels.

Pricing a bivariate option with copulas

The diversity of exotic-option contracts available in the market has increased significantly in recent years, and has aroused the interest to develop alternative valuation methodological approaches. Among the most interesting innovations, copulas analysis represents a major contribution to improve valuation methodologies. This paper explores the pricing of a bivariate call option on the better-of-two-markets: Mexico's Stock Exchange index, and the Standard & Poor's 500. The approach consists of a GARCH process that combines with copulas analysis and the Black and Scholes classical European call option valuation model. Copulas from the elliptical and Archimedean families provide the dependence structure among the underlying assets, and the estimated prices prove significantly different from those obtained using a static dependence assumption. The study concludes that dynamic copulas produce more robust prices than static dependence models.

Pricing a bivariate option with copulas

The diversity of exotic-option contracts available in the market has increased significantly in recent years, and has aroused the interest to develop alternative valuation methodological approaches. Among the most interesting innovations, copulas analysis represents a major contribution to improve valuation methodologies. This paper explores the pricing of a bivariate call option on the better-of-two-markets: Mexico's Stock Exchange index, and the Standard & Poor's 500. The approach consists of a GARCH process that combines with copulas analysis and the Black and Scholes classical European call option valuation model. Copulas from the elliptical and Archimedean families provide the dependence structure among the underlying assets, and the estimated prices prove significantly different from those obtained using a static dependence assumption. The study concludes that dynamic copulas produce more robust prices than static dependence models.

Distributionally Robust Pricing in Auctions

Suppose that in a second-price auction, a seller wishes to set an optimal reserve price, but the information about the distribution of bidders’ iid valuations is scarce: the seller knows only an upper bound for valuations, the distribution’s mean, and, possibly, variance. I find reserve prices optimal in the sense of worst-case expected revenue maximization, where worst case is with respect to the unknown distribution. The optimal reserve price may not be unique, but the set of optimal prices always includes zero in the case when only mean is known.

Distribution-Free Pricing

We study a monopolistic robust pricing problem in which the seller does not know the customers' valuation distribution for a product but knows its mean and variance. This minimal requirement for information means that the pricing managers only need to be able to answer two questions: How much will your targeted customers pay on average? And, to measure your confidence in the previous answer, what is the standard deviation of customer valuations. We first focus on the maximin profit criterion and derive distribution-free upper and lower bounds on the profit function. By maximizing the tight profit lower bound, we obtain the optimal robust price in closed form as well as its distribution-free, worst-case performance bound. We then extend the single-product result to study the robust pure bundle pricing problem where the seller only knows the mean and variance of each product, and provide easily verifiable, distribution-free, sufficient conditions that guarantee the pure bundle to be more robustly profitable than a la carte (i.e., separate) sales. We further derive a distribution-free, worst-case performance guarantee for a heuristic scheme in which customers choose between buying either a single product or a pure bundle. Moreover, we generalize separate sales and pure bundling to a scheme called clustered bundling that imposes a price for each mutually exclusive subset of products and allows customers to choose one or multiple subsets, and provide various algorithms to compute clustered bundling heuristics. In parallel, most of our results hold for the minimax relative regret criterion as well. The robust price for a single product is in closed form under the maximin profit or minimax relative regret criterion and hence is easily computable. Its interpretation can be easily explained to pricing managers. We also provide efficient algorithms to compute various mixed bundling heuristics for the multi-product problem.

Complexity of min–max–min robustness for combinatorial optimization under discrete uncertainty

We consider combinatorial optimization problems with uncertain linear objective functions. In the min–max–min robust optimization approach, a fixed number k of feasible solutions is computed such that the respective best of them is optimal in the worst case. The idea is to calculate the set of candidate solutions in a potentially expensive preprocessing phase and then select the best solution out of this set in real-time, once the actual scenario is known. In this paper, we investigate the complexity of the resulting min–max–min problem in the case of discrete uncertainty, as well as its connection to the classical min–max robust counterpart, for many classical combinatorial optimization problems. Essentially, it turns out that the min–max–min problem is not harder to solve than the min–max problem, while producing much better solutions in general for larger k. In particular, this approach may mitigate the so-called price of robustness, making it an attractive alternative to the classical robust optimization approach in many practical applications.

Robust Postdonation Blood Screening Under Prevalence Rate Uncertainty

Blood products are essential components of any healthcare system, and their safety, in terms of being free of transfusion-transmittable infections, is crucial. While the Food and Drug Administration (FDA) in the United States requires all blood donations to be tested for certain infection types, it does not dictate which particular tests should be used by blood centers. Multiple FDA-licensed blood screening tests are available for each infection type, and screening tests are imperfectly reliable and have different costs. In addition, infection prevalence rates within the donor population are uncertain for both emerging and established infection types. In this setting, the budget-constrained blood center’s objective is to devise a “robust” postdonation bloodscreening scheme that minimizes the risk of an infectious donation being released into the blood supply. Toward this goal, we study the minimization of the transfusion-transmittable infection risk considering regret- and expectation-based objectives, and we characterize structural properties of their optimal solutions. This allows us to gain insight, derive the price of robustness, and develop efficient algorithms. The proposed robust solution lowers the expected infection risk over various FDA-compliant testing schemes as well as the expectation-based scheme under forecast error. These findings have important public policy implications. The e-companion is available at https://doi.org/10.1287/opre.2017.1658 .

Robust Distributed Predictive Control of Waterborne AGVs-A Cooperative and Cost-Effective Approach.

Waterborne autonomous guided vessels (waterborne AGVs) moving over open waters experience environmental uncertainties. This paper proposes a novel cost-effective robust distributed control approach for waterborne AGVs. The overall system is uncertain and has independent subsystem dynamics but coupling objectives and state constraints. Waterborne AGVs determine their actions in a parallel way, while still minimizing an overall cost function and respecting coupling constraints robustly by communicating within a neighborhood. Our first contribution is the proposal of the system robustness level for the cost-effective robust distributed model predictive control (RDMPC) for waterborne AGVs. Cost-effective RDMPC models the price of robustness by explicitly considering uncertainty and system characteristics in a tube-based robust control framework. The second contribution is an efficient integrated branch & bound (B&B) and the alternating direction method of multipliers (ADMMs) algorithm for solving the cost-effective RDMPC problem. The algorithm exploits special ordered variable sets and combining branching criteria with intermediate ADMM results conducting smart search in B&B. Simulation results demonstrate the effectiveness of the proposed approach for cooperative distributed waterborne AGVs with cost-effective robustness.

Semi-static completeness and robust pricing by informed investors

We consider a continuous-time financial market that consists of securities available for dynamic trading, and securities only available for static trading. We work in a robust framework where a set of nondominated models is given. The concept of semi-static completeness is introduced: it corresponds to having exact replication by means of semi-static strategies. We show that semi-static completeness is equivalent to an extremality property, and give a characterization of the induced filtration structure. Furthermore, we consider investors with additional information and, for specific types of extra information, we characterize the models that are semi-statically complete for the informed investors. Finally, we provide some examples where robust pricing for informed and uninformed agents can be done over semi-statically complete models.

Robust pricing and hedging around the globe

We consider the martingale optimal transport duality for cadlag processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semistatic superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part.

Robust Pricing and Hedging around the Globe

We consider the martingale optimal transport duality for cadlag processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semi-static superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan, and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part. In the case of finitely supported marginal laws, solving for the static part reduces to a semi-infinite linear program.

Robust barrier option pricing by frame projection under exponential Lévy dynamics

ABSTRACTWe present an efficient method for robustly pricing discretely monitored barrier and occupation time derivatives under exponential Lévy models. This includes ordinary barrier options, as well as (resetting) Parisian options, delayed barrier options (also known as cumulative Parisian or Parasian options), fader options and step options (soft-barriers), all with single and double barriers, which have yet to be priced with more general Lévy processes, including KoBoL (CGMY), Merton’s jump diffusion and NIG. The method’s efficiency is derived in part from the use of frame-projected transition densities, which transform the problem into the Fourier domain and accelerate the convergence of intermediate expectations. Moreover, these expectations are approximated by Toeplitz matrix-vector multiplications, resulting in a fast implementation. We devise an augmentation approach that contributes to the method’s robustness, adding protection against mis-specifying a proper truncation support of the transition density. Theoretical convergence is verified by a series of numerical experiments which demonstrate the method’s efficiency and accuracy.

An iterated Azéma–Yor type embedding for finitely many marginals

We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $\mu_{1},\ldots,\mu_{n}$ which are in convex order and satisfy an additional technical assumption. Our construction is explicit and is a multiple marginal generalization of the Azema and Yor [In Seminaire de Probabilites, XIII (Univ. Strasbourg, Strasbourg, 1977/78) (1979) 90–115 Springer] solution. In particular, we recover the stopping boundaries obtained by Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558–578] and Madan and Yor [Bernoulli 8 (2002) 509–536]. Our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when $n=3$. In our analysis we compute the law of the maximum at each of the $n$ stopping times. This is used in Henry-Labordere et al. [Ann. Appl. Probab. 26 (2016) 1–44] to show that the construction maximizes the distribution of the maximum among all solutions to the $n$-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.

Robust production scheduling in open-pit mining under uncertainty: a box counterpart approach

Open-Pit Production Scheduling (OPPS) problem focuses on determining a block sequencing and scheduling to maximize Net Present Value (NPV) of the venture under constraints. The scheduling model is critically sensitive to the economic value volatility of block, block weight, and operational capacity. In order to deal with the OPPS uncertainties, various approaches can be recommended. Robust optimization is one of the most applicable methods in this area used in this study. Robust optimization based on the box counterpart formulation is applied to deal with the OPPS problem. To have a comparison between the solutions of the box counterpart optimization model and the deterministic model, a Two-Dimensional (2D) numerical study of a hypothetical open-pit mine is conducted followed by additional computations on the actual large-scale instances (Marvin orebody). This investigation shows that the different features of the robust planning under uncertainty can be scheduled. Also the price of robustness is obtained in different levels of conservatism.