Stochastic Programming Tools And ComponentsGautam Mitra

Outline

Scope and objectivesDecision making under uncertaintyStochastic processes and Scenario GenerationInformation systems architectureComputational systems architectureApplication to ALMConclusions

Scope and Objectives Descriptive Models as defined by a set of mathematical relations, which simply predicts how a physical, industrial or a social system may behave. Normative Models constitute the basis for (quantitative) decision making by a superhuman following an entirely rational that is, logically scrupulous set of arguments. Hence quantitative decision problems and idealised decision makers are postulated in order to define these models. Prescriptive Models involve systematic analysis of problems as carried out by normally intelligent persons who apply intuition and judgement. Two distinctive features of this approach are uncertainty analysis and preference (or value or utility) analysis. Decision Models are in some sense a derived category as they combine the concept underlying the normative models and prescriptive models.

Data ModelDecision Model Constrained optimisationDescriptive ModelSimulation and Evaluation

Ex ante decision Scope and Objectives Ex post evaluation(simulation)

Decision making under uncertaintyThere are three established approaches:Stochastic Programs : SP [Recourse Models, Chance constrained programming]Dynamic Programming: DPStochastic Control: SCN

After a broad overview we return to SP models for illustrations of our modelling concepts.

Stochastic Programming : SP

Given a general range of T stages, the stochastic data process is defined as: and the decision process is represented as . represents a sequence of random vectors and x represents a sequence of decisions where is of dimension Both TSP and MSP are defined on a probability space () and the data path corresponds with the -field which precedes the stage t of the data process. The observations (of the state of the world) and decisions are set out as a sequence TSP: Two stage stochastic programMSP: Multi stage stochastic program Decision making under uncertainty

Dynamic Programming: DP

The general set of notations adopted by us closely follows those set out by Dupacova and Sladky (2002) who investigated the connections between MSP and DP. Let denote a set of states and called State Space, let denote a set of permissible decisions called Decision Space, denote a set of parameter realisations such that . (see Bertsekas and Tsitsiklis (1996) and Powell and Carvalho (1998)).For every stage t = 1..T we define where We also note that

We assume that is given and is mapping from onto A sequence of decisions is called a policy thus optimal policy upto t is defined asOptimum_Policy (t)=[d(1),d(2),,d(t)]_optimum. Decision making under uncertainty

Stochastic Control: SCNDesign of feedback laws to modify behaviour of a dynamical system under uncertainty is an active and well-established area of research. While stochastic control methods were originally motivated by engineering applications, they have found applications in portfolio management and in supply chain planning. Portfolio optimisation under risk constraint involves the use of feedback, as the monies from the sale of units of one asset may be used to purchase units of another asset. For a family of continuous time stochastic processes indexed by a control parameter , stochastic control problem is defined by assigning a real-valued cost function

And minimising the expectation over admissible controls. For a broad class of cost functions and of stochastic differential systems, this problem reduces to solving a partial differential equation, called Hamilton-Jacobi-Bellman equation. Applications of stochastic control in finance are discussed in Korn (1997 ), Hansen and Sergent (2001) and in Fleming (1995). Similar feedback-based problems also arise in supply chain planning and control theory based methods may be used for systematic decision-making processes for the supply chain. Perea et al (2003, 2000) have proposed applying different control theoretic approaches to supply chain management. Decision making under uncertainty

The modelling paradigms revisitedStage 0 : Analyse historical data [ data model ]Stage 1 : Create data paths for random parameter values [ descriptive models ]Stage 2 : Make decision using SP or DP or SCN[ decision models ] {data processes and decision processes are inter-twined}Stage 3 : Test the decisions using simulationBack testing, stress testing (out of sample)[ descriptive models ] Decision making under uncertainty

Classes of SP Problems Decision making under uncertainty

SP Software Tools

SP Paradigm Decision making under uncertaintyStochastic programming seen as a combination of optimisation decision models and models of randomness

Stochastic processes and SGThere are many well established methods/ models for describing random parameters over a temporal setting.The models can be summarised as:AR, MA , ARMA, ARCH, GARCH, ...Regression: quantile, robustSDEs, GBMForecasting: Parametric, nonparametricSimulators: MCMC, HiddenMC, VECM, Bootstrap

Real applications use Domain specialists model/knowledge

Stochastic processes and SG

Stochastic processes and SG

Stochastic processes and SG

Stochastic processes and SG Stochastic processes generate data in form of fans/time series but SP needs trees

Stochastic processes and SG The research problems are investigated by: Georg Pflug, Jitka Dupacova, Michael Dempster, Werner Romisch and others

ARCH/GARCHROBUST REGRESSIONTIME SERIESBROWNIAN MOTIONSDEs

Consider events

Recall the Indicator Function

- stage SP: Find -measurable decisions

With Recourse Objective

Constraint of the form

Perspective on Scenario Generation

Alternative Objective functions or constraint functions created using the scenario generator:Stochastic processes and SG

Remarks:ObjectiveWhen is an indicator function this leads to probability objectives Many distribution based relationships Typically VaR and CVaR

ConstraintsChance Constraints: when constraints are required to hold with a prescribed probability over a certain interval we have chance constraintsRisk Constraints can be also be accommodated VaR, CVaR, Expected ShortfallStochastic processes and SG

Computational systems architectureThe overall system must be able to support data modellingThat is followed by the three stage process outlined earlier:Scenario generationDecision modellingSimulation [back testing / stress testing]

Multistage Tree333T2Information systems architectureTree structure (number of stages, nodes, branches) defined as

SG IntegrationInformation systems architecture

OLAP and Information Value ChainInformation systems architecture

Generic System OverviewInformation systems architecture

Modelling AnalystSG2ParametersSG3ParametersSG4ParametersSG1 ParametersInterfaceScenario Set -> Parameters of Objective Function + Resource ConstraintsObjective function, 1st stage decision data -> DECISION DATAANALYTIC DATABASEABC(A(), b(), c()) SP Recourse Model SP SolverSG2SG1SG3SG4Information systems architecture

Scenario generation (for the approximation of distributions and stochastic processes by discretisation)Scenario generation method evaluation Solution algorithm (Benders, Lagrange, also Stochastic Decomposition methods)Evaluation of solutions (robustness of a model or the robustness of a set of decisions)Analysis of the risk associated with a given set of decisions.Information systems architecture

Simulation (EV, 2S, MS)Stochastic MeasuresStatisticalMeasuresRiskMeasuresInformation systems architecture

Computing Systems: SPInEAMLs allow to conveniently express MPs in a format both easy to understand and that can be processed by a solverSPs have different requirements, both in language constructs and in coupling with dataThe design of SAMPL extends an AML (in our case AMPL) to provide these additional constructsSPInE is the framework that deals with the second requirement (interpreting SAMPL and coupling to SGs)

SPInE: Modelling FrameworkScenario GeneratorModel InstanceCore ModelINPUTS

SPInE SG Example

Example: SG selection

Example: SG specific settings

SPInE Simulation and testing

Simulation and testingFor the in-sample stability of a Scenario Generator, we repeatedly solve the model with different instances of the SG under testThen we analyze the distribution of the objective values

Out of sample SG testingSG under test scenariosSolve HN ModelReal WorldscenariosResults AnalysisOptimisation modelOptimisation modelReal world scenarios is a large scenario tree, which is assumed to be the best available approximation of the stochastic processAlternative view in the next slideSolve sub-problems

Out of sample SG testing

Simulation and testingThe design of the framework to allow various kind of testing is still in progressA worlflow design seems natural:Few specialised modules Organized in a user defined way to perform potentially complex tasks

Case StudyALM model Optimisation

Stochastic Measures (EVPI, VSS)

In-Sample Stability

Risk Measures (VaR, CVaR)

Model Components

Model Componentsvar hold{assets, scenarios, time} >=0;var hold{assets, scenarios, time} >=0;var sell{assets, scenarios, time} >=0;random param prices{assets, ... };param prodreq{timep,scenario};probability param prob{scenario};param tsell;

NameiassetsscenarioPistPrices[assets, scenarios, timeperiod]LtsLiability[timeperiod,scenario]sprob[scenario]trSales transaction cost (%)HistAmounthold[assets, scenarios,timeperiod]BistAmountbuy[assets, scenarios,timeperiod]SistAmountsell[assets, scenarios,timeperiod]

An ALM Stochastic Programming ModelSurplus Wealth = assets PV(liabilities) PV(goals)carrycarryt=1t=2..T-1t=T

An ALM Stochastic Programming Modelsubject to fundbalance1{t in 2..T-1,s in scen}: sum{a in assets} amountbuy[t,a,s]*price[t,a,s]*tbuy-sum{a in assets} amountsell[t,a,s]*price[t,a,s]*tsell= income[t,s]-liabilities[t,s];

Optimisation and Stochastic MeasuresWS: 67189.4HN: 61354.4EV: 54846EEV: 60567.2

EVPI = (WS-HN) = 5834.91774172VSS = (HN-EEV) = 787.229642443

Stability Measures

Risk Measures

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Scenario Generation Stability

Input ParametersValueDescriptionSimulationsEVHN(MS)HN(TS)EVHN(MS)HN(TS)

Scenarios50 50stage1 stage2159571.6059219131916.769262136657.515934336639.742831635022.3600142451100

Stability Simulations:100n of sim258989.9447168232870.310848137135.584868337110.790341538127.950766394944

VaR beta0.95Percentile359501.3027679333028.724053737809.169421937784.022489441233.5415185351343434

VaR out of sample tree100 100out of sample sim433461.862968938092.935785738074.618881444339.13227068283126125

Histogram points20109959461.7236502634252.203932938407.67187638389.235735550550.31377497658430431

10060050.118546253655.904527115939801799

999593989.828202275901.918338875884.808805756761.49527926974970974

999694004.453146775928.56387475926.622952159867.086031405121312431246

Optimum HN (MS)59290.1754475999794022.0030876561.931243676559.97605862972.67678355139913421347

Optimum EV54846.1985686999894026.390563377441.993381977440.01837766078.267535695120117691773

Optimum HN (TS)61161.2918621999994028.58430577510.157164877511.618987769183.8582878495117051696

1000094033.33741279268.874508879266.858361772289.449039985542960953

Var scenarios1000075395.03979213301365370

78500.63054427520498

EVHN (MS)HN (TS)81606.2212964215900

VaR0.000.000.0084711.8120485659700

CVaR66330.4159156.0359144.3987817.4028007116300

Variance1012166067.24426140492.02426510209.5690922.99355285513400

Mean66330.4159156.0359144.3994028.58430510900

StabilityValue

Min58989.9447168

Max60050.1185462

Range1060.1738294

Mean59514.9391206

stdev376.9168026175

Relative Max Deviation1.78%

Relative Mean Deviation0.63%

MBD000000DC.unknown

MBD0000016C.unknown

MBD03DFE796.unknown

MBD01064F71.unknown

MBD00000128.unknown

MBD00000048.unknown

MBD00000090.unknown

ConclusionsEmergence of risk analysis has led to novel reuse of established modelling paradigms [data modelling, decison modelling, descriptive modelling]Ex-ante decisions coupled with ex-post evaluation (combined paradigm: optimisation and simulation) is a method of choice in many applications.The research challenge is to bring together :Quantitative Modelling and Financial engineering SkillsInformation Engineering SkillsAlgorithm and Software Engineering Skills

Thank Youhttp://carisma.brunel.ac.uk/http://www.optirisk-systems.com/

****************************I will present OUR SP modelling language and framework, that is constituted by the language SAMPL and the framework SPInE.For those of you that are not familiar with modelling languages, I will do a gentle introduction. ......But first, a bit of notation.

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