Alan King
IBM Research, Math Sciences, Department Member
Many problem formulations in statistics and stochastic optimization generate estimates from data by selecting a LLbest" or L'optimal" point xV= xV (sl,..., s,), frequently by choosing x" to solve a generalized equation... more
Many problem formulations in statistics and stochastic optimization generate estimates from data by selecting a LLbest" or L'optimal" point xV= xV (sl,..., s,), frequently by choosing x" to solve a generalized equation in the form u (l. 1) Choose x E Rn such that 0 E f (x, s,)+ N (x), a= 1 where g: Rn x S--t Rn is a function that is continuous in the first argument and measurable in the second,{si) an iid sequence of random variables in a complete separable metric space S, and N: Rn 3 Rn a multifunction. In stochastic programming, for example, ...
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We explain our suggestions for standardizing input formats for computer codes which solve stochastic programs with recourse. The main reason to set some conventions is to allow programs implementing different methods of solution to be... more
We explain our suggestions for standardizing input formats for computer codes which solve stochastic programs with recourse. The main reason to set some conventions is to allow programs implementing different methods of solution to be used interchangeably. The general philosophy behind our design is a) to remain fairly faithful to the de facto standard for the statement of LP problems established by IBM for use with MPSX and subsequently adopted by the authors of MINOS, b) to provide sufficient flexibility so that a variety of problems may be expressed in the standard format, c) to allow problems originally formulated as deterministic LP to be converted to stochastic problems with a minimum of effort, d) to permit new options to be added as the need arises, and e) to provide some routines to facilitate the task of reading files specified in the standard format.
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Frequently it happens that the parameters needed to define a mathematical program are not precisely known, but can be estimated by sampling or derived from historical data. To be honest, the modeller should treat such parameters as... more
Frequently it happens that the parameters needed to define a mathematical program are not precisely known, but can be estimated by sampling or derived from historical data. To be honest, the modeller should treat such parameters as uncertain. This is simple to state but raises profound modelling, algorithmic and analytical issues.
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This chapter discusses the application of stochastic programming to the field of finance. The chief areas of application covered in this chapter are portfolio optimization and asset-liability management. Recent activity has explored the... more
This chapter discusses the application of stochastic programming to the field of finance. The chief areas of application covered in this chapter are portfolio optimization and asset-liability management. Recent activity has explored the relationship of stochastic programming models to options pricing and to portfolio selection using conditional Value at Risk criteria. We first develop single-period portfolio optimization from a stochastic programming perspective and link it to Markowitz's risk/reward framework and other ...
Abstract. This paper discusses the implementation in the Optimization Subroutine Library of dual decomposition, also known as Benders decomposition in mixed integer programming and the L-shaped method in stochastic programming. The... more
Abstract. This paper discusses the implementation in the Optimization Subroutine Library of dual decomposition, also known as Benders decomposition in mixed integer programming and the L-shaped method in stochastic programming. The formulation of optimality and feasibility cuts is explained and the correspondence to OSL variables is indicated.
Data conventions for the automatic input of multiperiod stochastic linear programs are described. The input format is based on the MPSX standard and is designed to promote the e cient conversion of originally deterministic problems by... more
Data conventions for the automatic input of multiperiod stochastic linear programs are described. The input format is based on the MPSX standard and is designed to promote the e cient conversion of originally deterministic problems by introducing stochastic variants in separate les. A exible\ header" syntax generates a useful variety of stochastic dependencies. An extension using the NETGEN format is proposed for stochastic network programs.
Research Interests:
Research Interests:
Abstract In this paper we identify important opportunities for parallelization in the least-squares Monte Carlo (LSM) algorithm, due to Longstaff and Schwartz, for the pricing of American options. The LSM method can be divided into three... more
Abstract In this paper we identify important opportunities for parallelization in the least-squares Monte Carlo (LSM) algorithm, due to Longstaff and Schwartz, for the pricing of American options. The LSM method can be divided into three phases: path-simulation, calibration and valuation. We describe how each of these phases can be parallelized, with more focus on the calibration phase, which is inherently more difficult to parallelize. We implemented these parallelization techniques on Blue Gene using the Quantlib open ...
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Research Interests:
The aim of this article is to propose a general approach to link a stochastic programming enabler to a mathematical programming modeling language. Modelers often choose to formulate their problems in well-tested, general purpose modeling... more
The aim of this article is to propose a general approach to link a stochastic programming enabler to a mathematical programming modeling language. Modelers often choose to formulate their problems in well-tested, general purpose modeling languages such as GAMS and AMPL, but these modeling languages do not currently implement a natural syntax for stochastic programming. Specialized stochastic programming tools are available to efficiently generate and solve large-scale stochastic programs, but they lack many of the ...
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Research Interests:
Abstract Convex optimization provides a natural framework for pricing and hedging financial instruments in incomplete market models. Duality theory of convex optimization has been shown to yield elementary proofs of well-known... more
Abstract Convex optimization provides a natural framework for pricing and hedging financial instruments in incomplete market models. Duality theory of convex optimization has been shown to yield elementary proofs of well-known martingale-expressions for arbitrage-free prices of European contingent claims in incomplete markets described with finite probability spaces. This paper extends the analysis to American contingent claims. Hedging problems for an American contingent claim are first expressed as convex optimization problems, ...
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Man-made (or artificial) eutrophication has been considered as one of the most serious water quality problems of lakes during the last 20-plus years. Increasing discharges of domestic and industrial waste water and the intensive use of... more
Man-made (or artificial) eutrophication has been considered as one of the most serious water quality problems of lakes during the last 20-plus years. Increasing discharges of domestic and industrial waste water and the intensive use of crop fertilizers—all leading to growing nutrient loads of the recipients—can be mentioned among the major causes of this undesirable phenomenon. The typical symptoms of eutrophication are, among others, sudden algal blooms, water coloration, floating water plants and debris, excreation of toxic ...
We propose a new framework for analyzing pricing theory for incomplete markets and contingent claims, using conjugate duality and optimization theory. Various statements in the literature of the fundamental theorem of asset pricing give... more
We propose a new framework for analyzing pricing theory for incomplete markets and contingent claims, using conjugate duality and optimization theory. Various statements in the literature of the fundamental theorem of asset pricing give conditions under which an essentially arbitrage-free market is equivalent to the existence of an equivalent martingale measure, and a formula for the fair price of a contingent claim as an expectation with respect to such a measure. In the setting of incomplete markets, the fair price is not attainable as such a particular expectation, but rather as a supremum over an infinite set of equivalent martingale measures. Here, we consider the problem as a stochastic program and derive pricing results for quite general discrete time processes. It is shown that in its most general form, the martingale pricing measure is attainable if it is permitted to be finitely additive. This setup also gives rise to a natural way of analyzing models with risk preferences...
The objective function of a mathematical program is what an optimization procedure uses to select better solutions over poorer solutions. For example, if the objective is to maximize profit, then the procedure tries to move in the... more
The objective function of a mathematical program is what an optimization procedure uses to select better solutions over poorer solutions. For example, if the objective is to maximize profit, then the procedure tries to move in the direction of solutions that increase profit while still remaining feasible. But when the profit depends on a parameter that is uncertain (like prices tomorrow), then the notion of maximizing profit is no longer very simple.
As was illustrated in our News Mix example in Chap. 1, it is not straightforward to pass from a deterministic to a stochastic formulation. We need to rethink the whole model, very often by changing both variables and constraints. Although... more
As was illustrated in our News Mix example in Chap. 1, it is not straightforward to pass from a deterministic to a stochastic formulation. We need to rethink the whole model, very often by changing both variables and constraints. Although many reformulations may make sense mathematically, they may in fact be rather peculiar in terms of interpretations. The purpose of this section is to discuss some of these issues, partly in terms of examples. The goal is not to declare some formulations generally superior to others, but rather to help you think ...
• What will the weather be during the outdoor concert? • Will my technology capture the market, or will my competitor capture all the sales? • Will the government change the taxation rules? • Will the killer asteroid hit during lunch? •... more
• What will the weather be during the outdoor concert? • Will my technology capture the market, or will my competitor capture all the sales? • Will the government change the taxation rules? • Will the killer asteroid hit during lunch? • What will the weather be during the outdoor concert? • How cold will it be this winter (and hence what will be the energy demand)? • What will demand be for my product the next 10years (as I am making a major investment in production equipment)? • Will the Chinese currency become convertible? • What is the travel time to ...
While there are several texts on how to solve and analyze stochastic programs, this is the first text to address basic questions about how to model uncertainty, and how to reformulate a deterministic model so that it can be analyzed in a... more
While there are several texts on how to solve and analyze stochastic programs, this is the first text to address basic questions about how to model uncertainty, and how to reformulate a deterministic model so that it can be analyzed in a stochastic setting. This text would be suitable as a stand-alone or supplement for a second course in OR/MS or in optimization-oriented engineering disciplines where the instructor wants to explain where models come from and what the fundamental issues are.
Abstract The subprime financial crisis of 2008 exposed systemic weaknesses in the financial sector. Emergency liquidity on an enormous scale was required to prevent a collapse of the banking system. There is much discussion now concerning... more
Abstract The subprime financial crisis of 2008 exposed systemic weaknesses in the financial sector. Emergency liquidity on an enormous scale was required to prevent a collapse of the banking system. There is much discussion now concerning how to monitor and manage the risk of such systemic events.
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In this chapter, we discuss approaches to applying financial markets theory to address risk and uncertainty, such as uncertainty in pricing or production costs, in the formulation of supply chain models. Financial parameters such as... more
In this chapter, we discuss approaches to applying financial markets theory to address risk and uncertainty, such as uncertainty in pricing or production costs, in the formulation of supply chain models. Financial parameters such as revenues and costs are key factors that drive optimal decisions in production planning and supply chain problems. Yet typically, these factors are the most difficult to capture and quantify. The values of these parameters may depend on some future “state of the market,” such as exchange rates, interest rates, or consumer prices, or they ...
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We consider a problem faced by a supplier of custom products that have long production lead times. The problem is inherently multistage with a large number of stages. The majority of these products have no salvage value, so the supplier... more
We consider a problem faced by a supplier of custom products that have long production lead times. The problem is inherently multistage with a large number of stages. The majority of these products have no salvage value, so the supplier is exposed to significant risk of excess production. Moreover, customer forecasts will likely err on the upside because the option to purchase has value. This chapter describes a counterbalancing mechanism for the supplier to obtain some compensation for part of the inventory risk.
The importance of the risk-neutral measure as a pricing operator generates interest in the calibration problem: given observed market prices, retrieve a pricing measure compatible with them. It can be established through convex duality... more
The importance of the risk-neutral measure as a pricing operator generates interest in the calibration problem: given observed market prices, retrieve a pricing measure compatible with them. It can be established through convex duality arguments that the Arbitrage Pricing Theory calibration problem, ie, calibration to a set of benchmark securities, is equivalent to a certain portfolio optimization problem. We present an explicit duality argument to extend this result: every portfolio optimization problem in a market that allows liquid trading is ...
Abstract Stream processing systems are designed to support applications that use real time data. Examples of streaming applications include security agencies processing data from communications media, battlefield management systems for... more
Abstract Stream processing systems are designed to support applications that use real time data. Examples of streaming applications include security agencies processing data from communications media, battlefield management systems for military operations, consumer fraud detection based on online transactions, and automated trading based on financial market data. Many stream processing applications are faced with the challenge of increasingly large volumes of data and the requirement to deliver low-latency responses ...
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Research Interests:
Research Interests:
A computer program product stored on machine readable media including machine readable instructions for selecting a project portfolio from available projects, the instructions for implementing a method include inputting an investment cost... more
A computer program product stored on machine readable media including machine readable instructions for selecting a project portfolio from available projects, the instructions for implementing a method include inputting an investment cost and a return for each available project and user-specified constraints; creating a formulation, the formulation comprising the investment cost and the return for each available project, the constraints and variables, the variables comprising for each available project a threshold probability of ...
In this chapter, we discuss ways to use information from financial markets to calibrate models for discounting future risks. This type of information is important for modeling the impact of future uncertainty on present decisions. In some... more
In this chapter, we discuss ways to use information from financial markets to calibrate models for discounting future risks. This type of information is important for modeling the impact of future uncertainty on present decisions. In some areas of activity, there exist well developed financial markets with hordes of traders using the tools and information available to them to decide the present value of future events. This chapter describes a methodology to use market information.
Why stochastic? Stochastic programming is the branch of mathematical programming that optimizes problems that have uncertainty in the data. Financial risk management, electric power generation planning, pollution remediation, structural... more
Why stochastic? Stochastic programming is the branch of mathematical programming that optimizes problems that have uncertainty in the data. Financial risk management, electric power generation planning, pollution remediation, structural engineering, industrial capacity planning, are examples of fields where stochastic programming models have been developed.
Abstract. This paper discusses the implementation in the Optimization Subroutine Library of dual decomposition, also known as Benders decomposition in mixed integer programming and the L-shaped method in stochastic programming. The... more
Abstract. This paper discusses the implementation in the Optimization Subroutine Library of dual decomposition, also known as Benders decomposition in mixed integer programming and the L-shaped method in stochastic programming. The formulation of optimality and feasibility cuts is explained and the correspondence to OSL variables is indicated.
Abstract We develop a Systems Dynamics model for capturing the key interactions involved in the evolution of the subprime mortgage crisis. In particular, we propose an aggregate modeling resolution that involves three main sub-systems,... more
Abstract We develop a Systems Dynamics model for capturing the key interactions involved in the evolution of the subprime mortgage crisis. In particular, we propose an aggregate modeling resolution that involves three main sub-systems, namely, an aggregate banking system, an aggregate housing market and an economic environment. The model exposes the physics of each individual system as well as influences and interactions among the three systems.
This paper proposes a model for a relatively simple Web hosting provider. The model assumes the existence of a load-dispatcher and a finite number of Web-servers. We quantify the quality of service towards the clients of this facility... more
This paper proposes a model for a relatively simple Web hosting provider. The model assumes the existence of a load-dispatcher and a finite number of Web-servers. We quantify the quality of service towards the clients of this facility based on a service level agreement between the two parts: the web hosting provider and the client. We assume that the client has the knowledge and resources to quantify its needs. Based on these quantifications, which in our model become parameters, the provider can establish a service offer.
ABSTRACT Data conventions for the automatic input of multiperiod stochastic linear programs are described. The input format is based on the MPSX standard and is designed to promote the e cient conversion of originally deterministic... more
ABSTRACT Data conventions for the automatic input of multiperiod stochastic linear programs are described. The input format is based on the MPSX standard and is designed to promote the e cient conversion of originally deterministic problems by introducing stochastic variants in separate les. A exible\ header" syntax generates a useful variety of stochastic dependencies. An extension using the NETGEN format is proposed for stochastic network programs.
In an earlier paper [I.] we gave conditions that described the asymptotic behaviour of selections from a sequence of random sets in a bite-dimensional Euclidean space X that were single-valued almost surely. The results of the present... more
In an earlier paper [I.] we gave conditions that described the asymptotic behaviour of selections from a sequence of random sets in a bite-dimensional Euclidean space X that were single-valued almost surely. The results of the present paper reveal that these conclusions may be derived from a much more general asymptotic result for truly multivalued mappings. The basic approach is the same, however: we consider the convergence in distribution of the sequence of" difference quotient.
Abstract This chapter discusses the application of stochastic programming to the field of finance. The chief areas of application covered in this chapter are portfolio optimization and asset-liability management. Recent activity has... more
Abstract This chapter discusses the application of stochastic programming to the field of finance. The chief areas of application covered in this chapter are portfolio optimization and asset-liability management. Recent activity has explored the relationship of stochastic programming models to options pricing and to portfolio selection using conditional Value at Risk criteria.