We study a large market model of dynamic matching with no monetary transfers and a continuum of agents. Time is discrete and horizon finite. Agents are in the market from the first date and, at each date, have to be assigned items (or bundles of items). When the social planner can only elicit ordinal preferences of agents over the sequences of items, we prove that, under a mild regularity assumption, incentive compatible and ordinally efficient allocation rules coincide with spot mechanisms. A spot mechanism specifies “virtual prices” for items at each date and, at the beginning of time, for each agent, randomly selects a budget of virtual money according to a (potentially non-uniform) distribution over [0,1]. Then, at each date, the agent is allocated the item of his choice among the affordable ones. Spot mechanisms impose a linear structure on prices and, perhaps surprisingly, our result shows that this linear structure is what is needed when one requires incentive compatibility and ordinal efficiency. When the social planner can elicit cardinal preferences, we prove that, under a similar regularity assumption, incentive compatible and Pareto efficient mechanisms coincide with a class of mechanisms we call Spot Menu of Random Budgets mechanisms. These mechanisms are similar to spot mechanisms except that, at the beginning of the time, each agent must pick a distribution in a menu. This distribution is used to initially draw the agent's budget of virtual money.
{"title":"Dynamic assignment without money: Optimality of spot mechanisms","authors":"Julien Combe, Vladyslav Nora, Olivier Tercieux","doi":"10.2139/ssrn.3894259","DOIUrl":"https://doi.org/10.2139/ssrn.3894259","url":null,"abstract":"We study a large market model of dynamic matching with no monetary transfers and a continuum of agents. Time is discrete and horizon finite. Agents are in the market from the first date and, at each date, have to be assigned items (or bundles of items). When the social planner can only elicit ordinal preferences of agents over the sequences of items, we prove that, under a mild regularity assumption, incentive compatible and ordinally efficient allocation rules coincide with spot mechanisms. A spot mechanism specifies “virtual prices” for items at each date and, at the beginning of time, for each agent, randomly selects a budget of virtual money according to a (potentially non-uniform) distribution over [0,1]. Then, at each date, the agent is allocated the item of his choice among the affordable ones. Spot mechanisms impose a linear structure on prices and, perhaps surprisingly, our result shows that this linear structure is what is needed when one requires incentive compatibility and ordinal efficiency. When the social planner can elicit cardinal preferences, we prove that, under a similar regularity assumption, incentive compatible and Pareto efficient mechanisms coincide with a class of mechanisms we call Spot Menu of Random Budgets mechanisms. These mechanisms are similar to spot mechanisms except that, at the beginning of the time, each agent must pick a distribution in a menu. This distribution is used to initially draw the agent's budget of virtual money.","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127358225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Optimal Design of Combined Contingent Claims: Theory and Applications. In “Combined Custom Hedging: Optimal Design, Noninsurable Exposure, and Operational Risk Management”, Paolo Guiotto a...
{"title":"Combined Custom Hedging: Optimal Design, Noninsurable Exposure, and Operational Risk Management","authors":"P. Guiotto, Andrea Roncoroni","doi":"10.2139/ssrn.3775182","DOIUrl":"https://doi.org/10.2139/ssrn.3775182","url":null,"abstract":"Abstract Optimal Design of Combined Contingent Claims: Theory and Applications. In “Combined Custom Hedging: Optimal Design, Noninsurable Exposure, and Operational Risk Management”, Paolo Guiotto a...","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"141 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122999937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The multinomial logit (MNL) model is a workhorse model for modeling customer demand in many fields including operations, econometrics and marketing. In this work, we present a fast algorithm for solving the likelihood maximization problem for the MNL model with product features. Our algorithm falls under the general framework of minorize-maximize (MM) procedures and we show that it results in an efficient iterative procedure with closed-form updates. We establish a necessary and sufficient condition under which the optimization problem has a unique and bounded solution and establish convergence of our proposed algorithm to the global optimal solution.
{"title":"An MM Algorithm for Estimating the MNL Model with Product Features","authors":"Srikanth Jagabathula, Ashwin Venkataraman","doi":"10.2139/ssrn.3733971","DOIUrl":"https://doi.org/10.2139/ssrn.3733971","url":null,"abstract":"The multinomial logit (MNL) model is a workhorse model for modeling customer demand in many fields including operations, econometrics and marketing. In this work, we present a fast algorithm for solving the likelihood maximization problem for the MNL model with product features. Our algorithm falls under the general framework of minorize-maximize (MM) procedures and we show that it results in an efficient iterative procedure with closed-form updates. We establish a necessary and sufficient condition under which the optimization problem has a unique and bounded solution and establish convergence of our proposed algorithm to the global optimal solution.","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"392 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126744700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a service firm that caters to price and delay sensitive customers by offering a menu of service grades. Each service grade is associated with posted price and delay. Noting that an optimal menu size could be quite large when there are many classes, we study whether the firm can offer a simplified menu with a few service grades without significant revenue loss. There is a well developed stream of work that studies optimal menu design for price and delay sensitive customers. Most results show that the number of grades equal the number of classes. However, in practice, we observe a handful number of grades. This raises the question of how much is the optimality gap when firm employs a simple menu relative to the optimal one. Our analysis utilizes a large system approximations where we assume that the firm has ample capacity to serve the entire market. We set up an optimization model and make use of Taylor series and asymptotic arguments to obtain the solution. We show that, under a simplified menu, the firm could lose a significant fraction of its revenue in the worst case scenario. This happens when there is significant heterogeneity between the customer classes. In contrast, noting that customer heterogeneity may typically be less extreme, we show that the firm can in fact provide a simplified menu while providing a guarantee on worst case revenue that can be obtained as a fraction of the optimal. We characterize the worst case optimal menu and provide asymptotic bounds to the worst case revenue loss as the number of customer types grow without bound. Characterization of the firm's worst case revenue loss in terms of a measure of heterogeneity can be used to guide decision making when offering a simplified menu of service grades.
{"title":"Value of Simple Menus with Price and Delay Sensitive Customers","authors":"Abhishek Ghosh, Achal Bassamboo, R. Randhawa","doi":"10.2139/ssrn.3668071","DOIUrl":"https://doi.org/10.2139/ssrn.3668071","url":null,"abstract":"We consider a service firm that caters to price and delay sensitive customers by offering a menu of service grades. Each service grade is associated with posted price and delay. Noting that an optimal menu size could be quite large when there are many classes, we study whether the firm can offer a simplified menu with a few service grades without significant revenue loss. There is a well developed stream of work that studies optimal menu design for price and delay sensitive customers. Most results show that the number of grades equal the number of classes. However, in practice, we observe a handful number of grades. This raises the question of how much is the optimality gap when firm employs a simple menu relative to the optimal one. Our analysis utilizes a large system approximations where we assume that the firm has ample capacity to serve the entire market. We set up an optimization model and make use of Taylor series and asymptotic arguments to obtain the solution. We show that, under a simplified menu, the firm could lose a significant fraction of its revenue in the worst case scenario. This happens when there is significant heterogeneity between the customer classes. In contrast, noting that customer heterogeneity may typically be less extreme, we show that the firm can in fact provide a simplified menu while providing a guarantee on worst case revenue that can be obtained as a fraction of the optimal. We characterize the worst case optimal menu and provide asymptotic bounds to the worst case revenue loss as the number of customer types grow without bound. Characterization of the firm's worst case revenue loss in terms of a measure of heterogeneity can be used to guide decision making when offering a simplified menu of service grades.","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124229513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this research, we consider an online assortment optimization problem, where a decision-maker needs to sequentially offer assortments to users instantaneously upon their arrivals and users select products from offered assortments according to the contextual multinomial logit choice model. We propose a computationally efficient Lasso-RP-MNL algorithm for the online assortment optimization problem under the cardinality constraint in high-dimensional settings. The Lasso-RP-MNL algorithm combines the Lasso and random projection as dimension reduction techniques to alleviate the computational complexity and improve the learning and estimation accuracy under high-dimensional data with limited samples. For each arriving user, the Lasso-RP-MNL algorithm constructs an upper-confidence bound for each individual product's attraction parameter, based on which the optimistic assortment can be identified by solving a reformulated linear programming problem. We demonstrate that for the feature dimension $d$ and the sample size dimension $T$, the expected cumulative regret under the Lasso-RP-MNL algorithm is upper bounded by $tilde{mathcal{O}}(sqrt{T}log d)$ asymptotically, where $tilde{mathcal{O}}$ suppresses the logarithmic dependence on $T$. Furthermore, we show that even when available samples are extremely limited, the Lasso-RP-MNL algorithm continues to perform well with a regret upper bound of $tilde{mathcal{O}}( T^{frac{2}{3}}log d)$. Finally, through synthetic-data-based experiments and a high-dimensional XianYu assortment recommendation experiment, we show that the Lasso-RP-MNL algorithm is computationally efficient and outperforms other benchmarks in terms of the expected cumulative regret.
在本研究中，我们考虑了一个在线分类优化问题，其中决策者需要在用户到达时立即顺序地向用户提供分类，用户根据上下文多项逻辑选择模型从提供的分类中选择产品。针对高维环境下基数约束下的在线分类优化问题，提出了一种计算效率高的Lasso-RP-MNL算法。Lasso- rp - mnl算法结合Lasso和随机投影作为降维技术，在有限样本的高维数据下降低了计算复杂度，提高了学习和估计精度。对于每个到达的用户，Lasso-RP-MNL算法为每个单个产品的吸引力参数构建了一个上置信度界，在此基础上，可以通过求解一个重新表述的线性规划问题来识别乐观分类。我们证明了对于特征维$d$和样本量维$T$, Lasso-RP-MNL算法下的期望累积遗憾的上界渐近为$tilde{mathcal{O}}(sqrt{T}log d)$，其中$tilde{mathcal{O}}$抑制了对$T$的对数依赖。此外，我们表明，即使在可用样本非常有限的情况下，Lasso-RP-MNL算法仍然表现良好，遗憾上限为$tilde{mathcal{O}}( T^{frac{2}{3}}log d)$。最后，通过基于综合数据的实验和高维的XianYu分类推荐实验，我们证明了Lasso-RP-MNL算法的计算效率很高，并且在期望累积遗憾方面优于其他基准。
{"title":"Online Assortment Optimization with High-Dimensional Data","authors":"Xue Wang, Mike Mingcheng Wei, Tao Yao","doi":"10.2139/ssrn.3521843","DOIUrl":"https://doi.org/10.2139/ssrn.3521843","url":null,"abstract":"In this research, we consider an online assortment optimization problem, where a decision-maker needs to sequentially offer assortments to users instantaneously upon their arrivals and users select products from offered assortments according to the contextual multinomial logit choice model. We propose a computationally efficient Lasso-RP-MNL algorithm for the online assortment optimization problem under the cardinality constraint in high-dimensional settings. The Lasso-RP-MNL algorithm combines the Lasso and random projection as dimension reduction techniques to alleviate the computational complexity and improve the learning and estimation accuracy under high-dimensional data with limited samples. For each arriving user, the Lasso-RP-MNL algorithm constructs an upper-confidence bound for each individual product's attraction parameter, based on which the optimistic assortment can be identified by solving a reformulated linear programming problem. We demonstrate that for the feature dimension $d$ and the sample size dimension $T$, the expected cumulative regret under the Lasso-RP-MNL algorithm is upper bounded by $tilde{mathcal{O}}(sqrt{T}log d)$ asymptotically, where $tilde{mathcal{O}}$ suppresses the logarithmic dependence on $T$. Furthermore, we show that even when available samples are extremely limited, the Lasso-RP-MNL algorithm continues to perform well with a regret upper bound of $tilde{mathcal{O}}( T^{frac{2}{3}}log d)$. Finally, through synthetic-data-based experiments and a high-dimensional XianYu assortment recommendation experiment, we show that the Lasso-RP-MNL algorithm is computationally efficient and outperforms other benchmarks in terms of the expected cumulative regret. <br>","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125460997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a ride-hailing platform that provides free information to taxi drivers. Upon receiving a rider's request, the platform broadcasts the rider's origin and destination to idle drivers, who accept or ignore the request depending on the profitability considerations. We show that providing such information may reduce drivers' equilibrium profit. Hence information provision is a double-edged sword: the drivers may choose to take more profitable riders via "strategic idling." When multiple drivers compete for the same request, how the platform breaks the tie affects the incentives of the drivers. We propose a routing policy that can align the incentives and achieve the first-best outcome for large systems.
{"title":"Harnessing the Double-Edged Sword via Routing: Information Provision on Ride-Hailing Platforms","authors":"Leon Yang Chu, Zhixi Wan, Dongyuan Zhan","doi":"10.2139/ssrn.3266250","DOIUrl":"https://doi.org/10.2139/ssrn.3266250","url":null,"abstract":"We consider a ride-hailing platform that provides free information to taxi drivers. Upon receiving a rider's request, the platform broadcasts the rider's origin and destination to idle drivers, who accept or ignore the request depending on the profitability considerations. We show that providing such information may reduce drivers' equilibrium profit. Hence information provision is a double-edged sword: the drivers may choose to take more profitable riders via \"strategic idling.\" When multiple drivers compete for the same request, how the platform breaks the tie affects the incentives of the drivers. We propose a routing policy that can align the incentives and achieve the first-best outcome for large systems.","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134340856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a new approach that integrates empirical estimation and assortment optimization to achieve display personalization for e-commerce platforms. We propose a two-stage Multinomial Logit (MNL) based consider-then-choose model, which accurately captures the two stages of a consumer's decision-making process -- consideration set formation and purchase decision given a consideration set. To calibrate our model, we develop an empirical estimation method using views and sales data at the aggregate level. The accurate predictions of both view counts and sales numbers provide a solid basis for our assortment optimization. To maximize the expected revenue, we compute the optimal target assortment set based on each consumer’s taste. Then we adjust the display of items to induce this consumer to form her consideration set that coincides with the target assortment set. We formulate this consideration set induction process as a nonconvex optimization, for which we provide the sufficient and necessary condition for feasibility. This condition reveals that a consumer is willing to consider at most K(C) items given the viewing cost C incurred by considering and evaluating an item, which is intrinsic to consumers’ online shopping behavior. As such, we argue that the assortment capacity should not be imposed by the platform, but rather comes from the consumers due to limited time and cognitive capacity. We provide a simple closed-form relationship between the viewing cost and the number of items a consumer is willing to consider. To mitigate computational difficulties associated with nonconvexity, we develop an efficient heuristic to induce the optimal consideration set. We test the heuristic and show that it yields near-optimal solutions. Given accurate taste information, our approach can increase the revenue by up to 35%. Under noisy predictions of consumer taste, the revenue can still be increased by 1% to 2%. Our approach does not require a designated space within a webpage, and can be applied to virtually all webpages thereby generating site-wise revenue improvement.
{"title":"Integrating Empirical Estimation and Assortment Personalization for E-Commerce: A Consider-Then-Choose Model","authors":"M. Li, Xiang Liu, Y. Huang, Cong Shi","doi":"10.2139/ssrn.3247323","DOIUrl":"https://doi.org/10.2139/ssrn.3247323","url":null,"abstract":"We develop a new approach that integrates empirical estimation and assortment optimization to achieve display personalization for e-commerce platforms. We propose a two-stage Multinomial Logit (MNL) based consider-then-choose model, which accurately captures the two stages of a consumer's decision-making process -- consideration set formation and purchase decision given a consideration set. To calibrate our model, we develop an empirical estimation method using views and sales data at the aggregate level. The accurate predictions of both view counts and sales numbers provide a solid basis for our assortment optimization. To maximize the expected revenue, we compute the optimal target assortment set based on each consumer’s taste. Then we adjust the display of items to induce this consumer to form her consideration set that coincides with the target assortment set. We formulate this consideration set induction process as a nonconvex optimization, for which we provide the sufficient and necessary condition for feasibility. This condition reveals that a consumer is willing to consider at most K(C) items given the viewing cost C incurred by considering and evaluating an item, which is intrinsic to consumers’ online shopping behavior. As such, we argue that the assortment capacity should not be imposed by the platform, but rather comes from the consumers due to limited time and cognitive capacity. We provide a simple closed-form relationship between the viewing cost and the number of items a consumer is willing to consider. To mitigate computational difficulties associated with nonconvexity, we develop an efficient heuristic to induce the optimal consideration set. We test the heuristic and show that it yields near-optimal solutions. Given accurate taste information, our approach can increase the revenue by up to 35%. Under noisy predictions of consumer taste, the revenue can still be increased by 1% to 2%. Our approach does not require a designated space within a webpage, and can be applied to virtually all webpages thereby generating site-wise revenue improvement.","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131551304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The increased amount of electricity supply from intermittent renewable energy sources leads more and more to high price volatility in electricity spot markets. An increasing share of generation is less dispatchable than in the past, and therefore higher amounts of flexible demand, which can be adjusted towards supply, are required. Even residential consumers are potential market participants, if the smart equipment of buildings and the electricity grid are readily available. This paper investigates the possibility for heat-pump operators to participate in spot markets. Especially problems and possible benefits are investigated when uncertainties in ambient temperatures or prices are considered. Therefore an optimization model, including an air-to-water heat pump, a storage tank and the heated building is implemented in MATLAB. In order to investigate the heat-pumps operation according to optimized heat-supply schedules. Along different scenarios, an agent-based model is used. Namely operations with day-ahead and intraday market participation are investigated, using historical EPEX spot electricity prices for 2014. Results show that uncertainty is a critical issue when private consumers participate in electricity markets. Even with a certain amount of system flexibility, there are tight operational constraints for the heating device, which are hard to fulfill. Short-term decisions including responses to current information are required. The system behavior is acceptable with very shortterm decision making, namely a hourly reoptimization with intraday-market participation. Further on, benefits can be yielded, when a combination of procurement before (day-ahead) and adjustments in the very short term (intraday) are applied.
{"title":"Flexible Use of Residential Heat Pumps - Possibilities and Limits of Market Participation","authors":"Jessica Raasch","doi":"10.2139/ssrn.3136973","DOIUrl":"https://doi.org/10.2139/ssrn.3136973","url":null,"abstract":"The increased amount of electricity supply from intermittent renewable energy sources leads more and more to high price volatility in electricity spot markets. An increasing share of generation is less dispatchable than in the past, and therefore higher amounts of flexible demand, which can be adjusted towards supply, are required. Even residential consumers are potential market participants, if the smart equipment of buildings and the electricity grid are readily available. This paper investigates the possibility for heat-pump operators to participate in spot markets. Especially problems and possible benefits are investigated when uncertainties in ambient temperatures or prices are considered. Therefore an optimization model, including an air-to-water heat pump, a storage tank and the heated building is implemented in MATLAB. In order to investigate the heat-pumps operation according to optimized heat-supply schedules. Along different scenarios, an agent-based model is used. Namely operations with day-ahead and intraday market participation are investigated, using historical EPEX spot electricity prices for 2014. Results show that uncertainty is a critical issue when private consumers participate in electricity markets. Even with a certain amount of system flexibility, there are tight operational constraints for the heating device, which are hard to fulfill. Short-term decisions including responses to current information are required. The system behavior is acceptable with very shortterm decision making, namely a hourly reoptimization with intraday-market participation. Further on, benefits can be yielded, when a combination of procurement before (day-ahead) and adjustments in the very short term (intraday) are applied.","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122240736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stochastic control problems in finance having complex controls inevitably give rise to low order accuracy, usually at most second order. Fourier methods are efficient at advancing the solution between control monitoring dates, but are not monotone. This gives rise to possible violations of arbitrage inequalities. We devise a preprocessing step for Fourier methods which involves projecting the Green's function onto the set of linear basis functions. The resulting algorithm is guaranteed to be monotone (to within a tolerance), infinity norm stable and satisfies an epsilon-discrete comparison principle. The algorithm has the same complexity per step as a standard Fourier method and has second order accuracy for smooth problems.
{"title":"ε-Monotone Fourier Methods for Optimal Stochastic Control in Finance","authors":"P. Forsyth, G. Labahn","doi":"10.21314/JCF.2018.361","DOIUrl":"https://doi.org/10.21314/JCF.2018.361","url":null,"abstract":"Stochastic control problems in finance having complex controls inevitably give rise to low order accuracy, usually at most second order. Fourier methods are efficient at advancing the solution between control monitoring dates, but are not monotone. This gives rise to possible violations of arbitrage inequalities. We devise a preprocessing step for Fourier methods which involves projecting the Green's function onto the set of linear basis functions. The resulting algorithm is guaranteed to be monotone (to within a tolerance), infinity norm stable and satisfies an epsilon-discrete comparison principle. The algorithm has the same complexity per step as a standard Fourier method and has second order accuracy for smooth problems.","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133116602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The theory of convex risk functions has now been well established as the basis for identifying the families of risk functions that should be used in risk-averse optimization problems. Despite its theoretical appeal, the implementation of a convex risk function remains difficult, because there is little guidance regarding how a convex risk function should be chosen so that it also well represents a decision maker’s subjective risk preference. In this paper, we address this issue through the lens of inverse optimization. Specifically, given solution data from some (forward) risk-averse optimization problem (i.e., a risk minimization problem with known constraints), we develop an inverse optimization framework that generates a risk function that renders the solutions optimal for the forward problem. The framework incorporates the well-known properties of convex risk functions—namely, monotonicity, convexity, translation invariance, and law invariance—as the general information about candidate risk functions, as well as feedback from individuals—which include an initial estimate of the risk function and pairwise comparisons among random losses—as the more specific information. Our framework is particularly novel in that unlike classical inverse optimization, it does not require making any parametric assumption about the risk function (i.e., it is nonparametric). We show how the resulting inverse optimization problems can be reformulated as convex programs and are polynomially solvable if the corresponding forward problems are polynomially solvable. We illustrate the imputed risk functions in a portfolio selection problem and demonstrate their practical value using real-life data. This paper was accepted by Yinyu Ye, optimization.
{"title":"Inverse Optimization of Convex Risk Functions","authors":"Jonathan Yu-Meng Li","doi":"10.2139/ssrn.3697392","DOIUrl":"https://doi.org/10.2139/ssrn.3697392","url":null,"abstract":"The theory of convex risk functions has now been well established as the basis for identifying the families of risk functions that should be used in risk-averse optimization problems. Despite its theoretical appeal, the implementation of a convex risk function remains difficult, because there is little guidance regarding how a convex risk function should be chosen so that it also well represents a decision maker’s subjective risk preference. In this paper, we address this issue through the lens of inverse optimization. Specifically, given solution data from some (forward) risk-averse optimization problem (i.e., a risk minimization problem with known constraints), we develop an inverse optimization framework that generates a risk function that renders the solutions optimal for the forward problem. The framework incorporates the well-known properties of convex risk functions—namely, monotonicity, convexity, translation invariance, and law invariance—as the general information about candidate risk functions, as well as feedback from individuals—which include an initial estimate of the risk function and pairwise comparisons among random losses—as the more specific information. Our framework is particularly novel in that unlike classical inverse optimization, it does not require making any parametric assumption about the risk function (i.e., it is nonparametric). We show how the resulting inverse optimization problems can be reformulated as convex programs and are polynomially solvable if the corresponding forward problems are polynomially solvable. We illustrate the imputed risk functions in a portfolio selection problem and demonstrate their practical value using real-life data. This paper was accepted by Yinyu Ye, optimization.","PeriodicalId":236552,"journal":{"name":"DecisionSciRN: Other Decision-Making in Operations Research (Topic)","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123841697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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