In this paper we present some elements of calculus for economics: the chain rule and extended chain rule for calculation of derivatives of composite functions, and differentiation of functions defined implicitly.The emphasis, as always in this series, is in providing a pedagogical, intuitive presentation to these topics.
{"title":"Intuitive Mathematical Economics Series. Chain Rule and Derivatives of Functions Defined Implicitly","authors":"S. Pernice","doi":"10.2139/ssrn.3333441","DOIUrl":"https://doi.org/10.2139/ssrn.3333441","url":null,"abstract":"In this paper we present some elements of calculus for economics: the chain rule and extended chain rule for calculation of derivatives of composite functions, and differentiation of functions defined implicitly.The emphasis, as always in this series, is in providing a pedagogical, intuitive presentation to these topics.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130370809","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}
An alternative mathematical modeling approach is proposed for the Theory of Regulatory Compliance.
本文提出了一种可替代的数学建模方法来研究法规遵从性理论。
{"title":"Theory of Regulatory Compliance: Quadratic Regression","authors":"Richard Fiene","doi":"10.2139/ssrn.3306659","DOIUrl":"https://doi.org/10.2139/ssrn.3306659","url":null,"abstract":"An alternative mathematical modeling approach is proposed for the Theory of Regulatory Compliance.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130833045","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}
Babak Mahdavi-Damghani, Konul Mustafayeva, Cristin Buescu, S. Roberts
With the recent rise of Machine Learning (ML) as a candidate to partially replace classic Financial Mathematics (FM) methodologies, we investigate the performances of both in solving the problem of dynamic portfolio optimization in continuous-time, finite-horizon setting for a portfolio of two assets that are intertwined. In the Financial Mathematics approach we model the asset prices not via the common approaches used in pairs trading such as a high correlation or cointegration, but with the cointelation model in Mahdavi-Damghani (2013) that aims to reconcile both short-term risk and long-term equilibrium. We maximize the overall P&L with Financial Mathematics approach that dynamically switches between a mean-variance optimal strategy and a power utility maximizing strategy. We use a stochastic control formulation of the problem of power utility maximization and solve numerically the resulting HJB equation with the Deep Galerkin method introduced in Sirignano and Spiliopoulos (2018). We turn to Machine Learning for the same P&L maximization problem and use clustering analysis to devise bands, combined with in-band optimization. Although this approach is model agnostic, results obtained with data simulated from the same cointelation model gives a slight competitive advantage to the ML over the FM methodology1.
{"title":"Portfolio Optimization for Cointelated Pairs: SDEs vs Machine Learning","authors":"Babak Mahdavi-Damghani, Konul Mustafayeva, Cristin Buescu, S. Roberts","doi":"10.2139/ssrn.3474742","DOIUrl":"https://doi.org/10.2139/ssrn.3474742","url":null,"abstract":"With the recent rise of Machine Learning (ML) as a candidate to partially replace classic Financial Mathematics (FM) methodologies, we investigate the performances of both in solving the problem of dynamic portfolio optimization in continuous-time, finite-horizon setting for a portfolio of two assets that are intertwined. In the Financial Mathematics approach we model the asset prices not via the common approaches used in pairs trading such as a high correlation or cointegration, but with the cointelation model in Mahdavi-Damghani (2013) that aims to reconcile both short-term risk and long-term equilibrium. We maximize the overall P&L with Financial Mathematics approach that dynamically switches between a mean-variance optimal strategy and a power utility maximizing strategy. We use a stochastic control formulation of the problem of power utility maximization and solve numerically the resulting HJB equation with the Deep Galerkin method introduced in Sirignano and Spiliopoulos (2018). We turn to Machine Learning for the same P&L maximization problem and use clustering analysis to devise bands, combined with in-band optimization. Although this approach is model agnostic, results obtained with data simulated from the same cointelation model gives a slight competitive advantage to the ML over the FM methodology1.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130474698","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 modelling of energy systems often has to balance two aspects. High level of detail, e.g. technical constraints on the one hand and analysis of long-term system optimization on the other. When focusing on one of the two aspects, models can be solved in a reasonable time. In order to combine both aspects in one model we use a problem-specific iterative approach. A detailed system model is linked to iterative adjustments of investments. This is based on a subgradient method of optimization. The approach can be described as a detailed dispatch model with adjustments towards an investment model. The results show that the algorithm is quite efficient for a stylized model. For a larger model, performance is not yet sufficient for day-to-day practical use, but several elements for further improvement are identified.
{"title":"Optimal Capacity Adjustments in Electricity Market Models – An Iterative Approach Based on Operational Margins and the Relevant Supply Stack","authors":"B. Böcker, R. Leisen, C. Weber","doi":"10.2139/ssrn.3329411","DOIUrl":"https://doi.org/10.2139/ssrn.3329411","url":null,"abstract":"The modelling of energy systems often has to balance two aspects. High level of detail, e.g. technical constraints on the one hand and analysis of long-term system optimization on the other. When focusing on one of the two aspects, models can be solved in a reasonable time. In order to combine both aspects in one model we use a problem-specific iterative approach. A detailed system model is linked to iterative adjustments of investments. This is based on a subgradient method of optimization. The approach can be described as a detailed dispatch model with adjustments towards an investment model. The results show that the algorithm is quite efficient for a stylized model. For a larger model, performance is not yet sufficient for day-to-day practical use, but several elements for further improvement are identified.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123797643","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}
Javier Martínez-de-Albeniz, Carles Rafels, Neus Ybern
We show that the family of assignment matrices which give rise to the same nucleolus forms a compact join-semilattice with one maximal element. The above family is in general not a convex set, but path-connected.
{"title":"The Nucleolus of the Assignment Game: Structure of the Family","authors":"Javier Martínez-de-Albeniz, Carles Rafels, Neus Ybern","doi":"10.2139/ssrn.3320402","DOIUrl":"https://doi.org/10.2139/ssrn.3320402","url":null,"abstract":"We show that the family of assignment matrices which give rise to the same nucleolus forms a compact join-semilattice with one maximal element. The above family is in general not a convex set, but path-connected. <br>","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122754619","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 computational cost of estimating option valuation models is very high, due to model complexity and the abundance of available option data. We propose an approach that addresses these computational constraints by filtering the state variables using particle weights based on model-implied spot volatilities rather than model prices. We show that this approach is reliable. We illustrate our method by estimating the workhorse stochastic volatility and double-jump models using a big option data set. We obtain more precise estimates of variance risk premia and more plausible implied preference parameters, and we show that for these models moneyness and especially maturity restrictions may result in identification problems. The composition of the option sample affects parameter inference and the relative importance of options and returns in joint estimation.
{"title":"Estimation and Filtering With Big Option Data","authors":"Kris Jacobs, Yuguo Liu","doi":"10.2139/ssrn.3300564","DOIUrl":"https://doi.org/10.2139/ssrn.3300564","url":null,"abstract":"The computational cost of estimating option valuation models is very high, due to model complexity and the abundance of available option data. We propose an approach that addresses these computational constraints by filtering the state variables using particle weights based on model-implied spot volatilities rather than model prices. We show that this approach is reliable. We illustrate our method by estimating the workhorse stochastic volatility and double-jump models using a big option data set. We obtain more precise estimates of variance risk premia and more plausible implied preference parameters, and we show that for these models moneyness and especially maturity restrictions may result in identification problems. The composition of the option sample affects parameter inference and the relative importance of options and returns in joint estimation.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122610412","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 reinforcement learning (RL) in continuous time and study the problem of achieving the best trade-off between exploration of a black box environment and exploitation of current knowledge. We propose an entropy-regularized reward function involving the differential entropy of the distributions of actions, and motivate and devise an exploratory formulation for the feature dynamics that captures repetitive learning under exploration. The resulting optimization problem is a revitalization of the classical relaxed stochastic control. We carry out a complete analysis of the problem in the linear--quadratic (LQ) setting and deduce that the optimal feedback control distribution for balancing exploitation and exploration is Gaussian. This in turn interprets and justifies the widely adopted Gaussian exploration in RL, beyond its simplicity for sampling. Moreover, the exploitation and exploration are captured, respectively and mutual-exclusively, by the mean and variance of the Gaussian distribution. We also find that a more random environment contains more learning opportunities in the sense that less exploration is needed. We characterize the cost of exploration, which, for the LQ case, is shown to be proportional to the entropy regularization weight and inversely proportional to the discount rate. Finally, as the weight of exploration decays to zero, we prove the convergence of the solution of the entropy-regularized LQ problem to the one of the classical LQ problem.
{"title":"Exploration versus Exploitation in Reinforcement Learning: A Stochastic Control Approach","authors":"Haoran Wang, T. Zariphopoulou, X. Zhou","doi":"10.2139/ssrn.3316387","DOIUrl":"https://doi.org/10.2139/ssrn.3316387","url":null,"abstract":"We consider reinforcement learning (RL) in continuous time and study the problem of achieving the best trade-off between exploration of a black box environment and exploitation of current knowledge. We propose an entropy-regularized reward function involving the differential entropy of the distributions of actions, and motivate and devise an exploratory formulation for the feature dynamics that captures repetitive learning under exploration. The resulting optimization problem is a revitalization of the classical relaxed stochastic control. We carry out a complete analysis of the problem in the linear--quadratic (LQ) setting and deduce that the optimal feedback control distribution for balancing exploitation and exploration is Gaussian. This in turn interprets and justifies the widely adopted Gaussian exploration in RL, beyond its simplicity for sampling. Moreover, the exploitation and exploration are captured, respectively and mutual-exclusively, by the mean and variance of the Gaussian distribution. We also find that a more random environment contains more learning opportunities in the sense that less exploration is needed. We characterize the cost of exploration, which, for the LQ case, is shown to be proportional to the entropy regularization weight and inversely proportional to the discount rate. Finally, as the weight of exploration decays to zero, we prove the convergence of the solution of the entropy-regularized LQ problem to the one of the classical LQ problem.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126226567","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}
This paper is part I of a two-part paper. It proposes a two-stage game to analyze imperfect competition of producers in zonal power markets with a day-ahead and a real-time market. We consider strategic producers in both markets. They need to take both markets into account when deciding what to bid in each market. The demand shocks between these markets are modeled by several scenarios. The two-stage game is formulated as a Twostage Stochastic Equilibrium Problem with Equilibrium Constraints (TS-EPEC). Then it is further reformulated as a two-stage stochastic Mixed-Integer Linear Program (MILP). The solution of this MILP gives the Subgame Perfect Nash Equilibrium (SPNE). To tackle multiple SPNE, we design a procedure which finds all SPNE with different total dispatch costs. The proposed MILP model is solved using Benders decomposition embedded in the CPLEX solver. The proposed MILP model is demonstrated on the 6-node and the IEEE 30-node example systems.
{"title":"Increase-Decrease Game Under Imperfect Competition in Two-stage Zonal Power Markets – Part I: Concept Analysis","authors":"M. Sarfati, M. Hesamzadeh, P. Holmberg","doi":"10.17863/CAM.33975","DOIUrl":"https://doi.org/10.17863/CAM.33975","url":null,"abstract":"This paper is part I of a two-part paper. It proposes a two-stage game to analyze imperfect competition of producers in zonal power markets with a day-ahead and a real-time market. We consider strategic producers in both markets. They need to take both markets into account when deciding what to bid in each market. The demand shocks between these markets are modeled by several scenarios. The two-stage game is formulated as a Twostage Stochastic Equilibrium Problem with Equilibrium Constraints (TS-EPEC). Then it is further reformulated as a two-stage stochastic Mixed-Integer Linear Program (MILP). The solution of this MILP gives the Subgame Perfect Nash Equilibrium (SPNE). To tackle multiple SPNE, we design a procedure which finds all SPNE with different total dispatch costs. The proposed MILP model is solved using Benders decomposition embedded in the CPLEX solver. The proposed MILP model is demonstrated on the 6-node and the IEEE 30-node example systems.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115583598","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 differentiate even and odd numbers into various groups and subgroups. We provide the properties of the forms of numbers which fall into each groups and subgroups. We expound on the relationship of a special group of even numbers and the collatz conjecture, we also derive an accurate formula to calculate the steps involved when an even number of the group is the initial value of the collatz operation. For each group and subgroup of odd and even numbers, we discuss the observed pattern of their sequences and also derive accurate formulas for each sequence. Throughout, b, d, k, N, n, x, m, and z all denote positive integers, with d, and N denoting odd numbers, x and z denoting even numbers, and b denoting special even-even numbers The order of priority of the properties of each group is key in the differentiation of the numbers into their various groups and subgroups.
{"title":"On the Fundamentals of Collatz Conjecture","authors":"Joseph Olloh","doi":"10.2139/ssrn.3302210","DOIUrl":"https://doi.org/10.2139/ssrn.3302210","url":null,"abstract":"We differentiate even and odd numbers into various groups and subgroups. We provide the properties of the forms of numbers which fall into each groups and subgroups. We expound on the relationship of a special group of even numbers and the collatz conjecture, we also derive an accurate formula to calculate the steps involved when an even number of the group is the initial value of the collatz operation. For each group and subgroup of odd and even numbers, we discuss the observed pattern of their sequences and also derive accurate formulas for each sequence. Throughout, b, d, k, N, n, x, m, and z all denote positive integers, with d, and N denoting odd numbers, x and z denoting even numbers, and b denoting special even-even numbers The order of priority of the properties of each group is key in the differentiation of the numbers into their various groups and subgroups.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130191279","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 bilateral gamma model for returns is naturally derived from the lognormal model. Maximizing entropy in a random time change delivers the symmetric variance gamma model. The asymmetric variance gamma follows on incorporating skewness. Differential speeds for the upward and downward motions lead to the bilateral gamma. A further generalizations results in the bilateral double gamma model when the speed parameter of the bilateral gamma model is itself taken to be gamma distributed on entropy maximization. A rich five to seven parameter specification of preferences renders possible the extraction of physical densities from option prices. The quality of such extraction is measured by examining the uniformity of the estimated distribution functions evaluated at realized forward returns. The economic value of risky returns is seen to embed three/five risk premia for the bilateral gamma/bilateral double gamma. For the bilateral gamma they are up and down side volatilities compensated in up and down side drifts, and the down side drift compensated in the up side drift. For the bilateral double gamma one adds in addition compensations for skewness. Results reveal a drop in the down side risk premium since the crisis with an increase in the recent period.
{"title":"Bilateral Multiple Gamma Returns: Their Risks and Rewards","authors":"D. Madan, W. Schoutens, King Wang","doi":"10.2139/ssrn.3230196","DOIUrl":"https://doi.org/10.2139/ssrn.3230196","url":null,"abstract":"The bilateral gamma model for returns is naturally derived from the lognormal model. Maximizing entropy in a random time change delivers the symmetric variance gamma model. The asymmetric variance gamma follows on incorporating skewness. Differential speeds for the upward and downward motions lead to the bilateral gamma. A further generalizations results in the bilateral double gamma model when the speed parameter of the bilateral gamma model is itself taken to be gamma distributed on entropy maximization. A rich five to seven parameter specification of preferences renders possible the extraction of physical densities from option prices. The quality of such extraction is measured by examining the uniformity of the estimated distribution functions evaluated at realized forward returns. The economic value of risky returns is seen to embed three/five risk premia for the bilateral gamma/bilateral double gamma. For the bilateral gamma they are up and down side volatilities compensated in up and down side drifts, and the down side drift compensated in the up side drift. For the bilateral double gamma one adds in addition compensations for skewness. Results reveal a drop in the down side risk premium since the crisis with an increase in the recent period.","PeriodicalId":299310,"journal":{"name":"Econometrics: Mathematical Methods & Programming eJournal","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125249878","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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