Stochastic Programming Tools And Components

1. Stochastic Programming Tools And Components Gautam Mitra

2. Outline Scope and objectives Decision making under uncertainty Stochastic processes and Scenario Generation Information systems architecture Computational systems architecture Application to ALM Conclusions

3. 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.

4. Scope and Objectives Time 0 1 2 3 … T Ex post evaluation (simulation) Ex ante decision • Data Model • Decision Model • Constrained optimisation • Descriptive Model • Simulation and Evaluation

5. Decision making under uncertainty • There are three established approaches: • Stochastic Programs : SP [Recourse Models, Chance constrained programming] • Dynamic Programming: DP • Stochastic Control: SCN • After a broad overview we return to SP models for illustrations of our modelling concepts.

6. Decision making under uncertainty 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 ) and the data path Both TSP and MSP are defined on a probability space ( 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 program MSP: Multi stage stochastic program

7. 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 as Optimum_Policy (t)=[d(1),d(2),…,d(t)]_optimum.

8. Decision making under uncertainty Stochastic Control: SCN Design 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.

9. Decision making under uncertainty • The modelling paradigms revisited • Stage 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 simulation Back testing, stress testing (out of sample) [ descriptive models ]

10. Decision making under uncertainty Classes of SP Problems Stochastic Measures: EVPI and VSS

11. SP Software Tools

12. Decision making under uncertainty SP Paradigm Stochastic programming seen as a combination of optimisation decision models and models of randomness

13. Stochastic processes and SG • There 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, robust • SDEs, GBM • Forecasting: Parametric, nonparametric • Simulators: MCMC, HiddenMC, VECM, Bootstrap • Real applications use Domain specialist’s model/knowledge

14. Stochastic processes and SG

15. Stochastic processes and SG

16. Stochastic processes and SG

17. Stochastic processes and SG Mathematical structure • Stochastic processes generate data in form of fans/time series but SP needs trees

18. Stochastic processes and SG Mathematical structure • The research problems are investigated by: Georg Pflug, JitkaDupacova, Michael Dempster, Werner Romisch and others

19. Stochastic processes and SG Perspective on Scenario Generation Alternative Objective functions or constraint functions created using the scenario generator: • Consider events • Recall the Indicator Function • - stage SP: Find - measurable decisions • With • Recourse Objective • Constraint of the form

20. Stochastic processes and SG Remarks: • Objective • When is an indicator function this leads to probability objectives • Many distribution based relationships • Typically VaR and CVaR • Constraints • Chance Constraints: when constraints are required to hold with a prescribed probability over a certain interval we have chance constraints • Risk Constraints can be also be accommodated • VaR, CVaR, Expected Shortfall

21. Computational systems architecture • The overall system must be able to support data modelling • That is followed by the three stage process outlined earlier: • Scenario generation • Decision modelling • Simulation [back testing / stress testing]

22. Information systems architecture Multistage Tree ξ2 ξ3 ξ3 ξ3 ξT • Tree structure (number of stages, nodes, branches) defined as 

23. Information systems architecture SG Integration

24. Information systems architecture OLAP and Information Value Chain OLAP Viewer Transactional Data Analytic database Market Data Decision Models Information Data Mart Analysis Knowledge

25. Information systems architecture Generic System Overview

26. Information systems architecture A Modelling Analyst SG1 Parameters SG2 Parameters SG3 Parameters SG4 Parameters Interface B SG1 SG2 SG3 SG4 Scenario Set -> Ξ C Parameters of Objective Function + Resource Constraints (A(Ξ), b(Ξ), c(Ξ)) SP Recourse Model SP Solver Objective function, 1st stage decision data -> DECISION DATA ANALYTIC DATABASE

27. Information 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.

28. Information systems architecture SGi SP Decision Model Simulation (EV, 2S, MS) LP/IP/SP Solver Decision Scenario Reduction (a) Ex Ante Simulation (b) Ex Post Stochastic Measures StatisticalMeasures Risk Measures

29. Computing Systems: SPInE • AMLs allow to conveniently express MPs in a format both easy to understand and that can be processed by a solver • SPs have different requirements, both in language constructs and in coupling with data • The design of SAMPL extends an AML (in our case AMPL) to provide these additional constructs • SPInE is the framework that deals with the second requirement (interpreting SAMPL and coupling to SGs)

30. SPInE: Modelling Framework “Core Model” Random Parameters Dynamic Structure Model Instance Decision Model (SAMPL) Simulation Model Tree Generator Model Generator Scenario Generator INPUTS

31. SPInE SG Example

32. Example: SG selection

33. Example: SG specific settings

34. SPInE Simulation and testing

35. Simulation and testing • For the in-sample stability of a Scenario Generator, we repeatedly solve the model with different instances of the SG under test • Then we analyze the distribution of the objective values

36. Out of sample SG testing • “Real world” scenarios is a large scenario tree, which is assumed to be the best available approximation of the stochastic process • Alternative view in the next slide SG under test scenarios “Real World” scenarios Fix n-stages variables Solve sub-problems Results Analysis Solve HN Model Optimisation model Optimisation model

37. Out of sample SG testing (6) Collect and Analyze Results

38. Simulation and testing • The design of the framework to allow various kind of testing is still in progress • A worlflow design seems natural: • Few specialised modules • Organized in a user defined way to perform potentially complex tasks

39. Case Study ALM model Optimisation Stochastic Measures (EVPI, VSS) In-Sample Stability Risk Measures (VaR, CVaR)

40. Model Components

41. Model Components SETS set asset; scenariosetscenario; VARIA BLES random param prices{assets, ... }; paramprodreq{timep,scenario}; probability paramprob{scenario}; paramtsell; PARAMETERS varhold{assets, scenarios, time} >=0; varhold{assets, scenarios, time} >=0; varsell{assets, scenarios, time} >=0;

42. An ALM Stochastic Programming Model Surplus Wealth = assets – PV(liabilities) – PV(goals) t=1 t=2..T-1 t=T carry carry

43. An ALM Stochastic Programming Model Surplus Wealth = assets – PV(liabilities) – PV(goals) subjectto 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];

44. Optimisation and Stochastic Measures WS: 67189.4 HN: 61354.4       EV: 54846 EEV: 60567.2 EVPI = (WS-HN) = 5834.91774172 VSS = (HN-EEV) = 787.229642443

45. Stability Measures

46. Risk Measures

47. Conclusions • Emergence 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 Skills • Information Engineering Skills • Algorithm and Software Engineering Skills

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