Optimisers Everywhere - MATLAB & Simulink

Optimizers everywhere Optimization in Financial Applications with MATLABDr. Sarah DrewesMathWorks Consulting Services 2016 The MathWorks, Inc.1

Optimization2

Optimization in Financial Applications with MATLAB§Financial Optimization§Optimization Methods§Customized optimization models3

Financial Optimization4

Financial Applications and OptimizationAsset ManagementPortfolio OptimizationTradingMachine LearningRisk ManagementALMEconometricsPricing & ValuationRegressionMaximum Likelihood EstimationDistribution FittingCurve Fitting5

MATLAB – The Financial Development PlatformAccessResearch and QuantifyShareFilesData Analysis and VisualizationReportingDatabasesFinancial ModelingApplicationsDatafeedsApplication DevelopmentProductionFinancial DatafeedNeural Network el ComputingProduction ServerSpreadsheet Link EXEconometricsReport GeneratorTrading ToolboxMATLAB Distributed Computing Server6

Financial Optimization within MATLABFinancial–––Mean-Variance Portfolio OptimizationConditional Value-at-Risk Portfolio OptimizationMean-Absolute Deviation PortfoliosEconometrics–––Time Series Regression ModelsConditional Mean Variance ModelsMultivariate ModelsOptimizationStatistics–––Linear/ Nonlinear RegressionProbability disribution fittingMachine Learning, e.g., SVM, NN,.Neural Network–Nonlinear Regression, Convolutional NeuralNetworks7

Optimization Methods8

Optimization ProblemObjective FunctionTypically a linear or nonlinear functionmin f ( x)xDecision variables (can be discrete or integer)Subject to ConstraintsLinear constraints- inequalities- equalities- boundsAx bc( x) 0Aeq x beqcl x ueq( x) 0Nonlinear constraints- inequalities- equalities9

How to solve an optimization problem ?What do you know about your optimization problem ?accuracyruntimeinformationinformation10

Variables & 1520Integer 8Nonlinear10.60.80.60.40.40.20.20011

Objective 10.80.810.610.60.80.60.40.80.200f(x) ATxLinear0.40.20.200.60.40.40.20f(x) st-squares/curve fittingf(c) Σ [g(xk;c) – y k]212

Numerical optimizationWhenever possible, provide gradient/hessian information!f ʹ( x ) function [f,df] objective(x)f . % function valuedf . % gradient vectorf ( x Δx ) f ( x Δx )2Δxü Fewer function evaluationsü More accurate13

Derivative-Free Optimization f ?fminconRepeatedly sampleseveral pointsDirect SearchGenetic Algorithm14

Approaches in MATLAB§Local Optimization– Finds local minima/maxima- Uses supplied gradients or estimates them- Applicable for large scale problems withsmooth objective function– Faster/fewer function evaluations§Global Optimization– No gradient information required– Solve problems with non-smooth,discontinuous objective function15

02.560210.51.55001f(x) bjective function0Linear0.70.6f(x) 1001000510152080f(c) Σ [g(xk;c) – y x)fmincon-8212100-1-1-2-216

SolversVariablesConstraintsSolverObjective functionGlobal Optimization 010005101520Discrete17

Customized Optimization Models18

Supported Portfolio Optimization ModelsFinancial Toolbox§Mean-Variance Portfolio Optimization§Conditional Value-at-Risk Portfolio Optimization§Mean-Absolute Deviation Portfolio Optimization19

Customized Objective - Smart Beta20

Customized Portfolio Optimization Smart BetaSmart Beta Example – Risk Parity:Generate a portfolio where each asset’s marginal contribution to risk is equal§Marginal Contributions for N assets𝑀𝐶( 𝑥 𝑥 ( §Minimize distance between all contributions ),- 𝑥) 𝐶𝑜𝑣(𝑅 ( , 𝑅) )𝜎 𝑓 𝑥 𝑀𝐶( 𝑥 𝑀𝐶) 𝑥 ( ,- )*(21

Customized Portfolio Optimization Smart Beta§Create marginal risk contributions𝑀𝐶( 𝑥 𝑥 ( ),- 𝑥) 𝐶𝑜𝑣(𝑅 ( , 𝑅) )𝜎§Create a distance matrix§Minimize the total distancebetween all the contributions 𝑓 𝑥 𝑀𝐶( 𝑥 𝑀𝐶) 𝑥 ( ,- )*(22

Customized Portfolio Optimization Smart BetaSmart Beta Example – Risk Parity: 𝑚𝑖𝑛 𝑓 𝑥 𝑀𝐶( 𝑥 𝑀𝐶) 𝑥 ;s.t.(,- )*(𝑥 e 1lb x ubà Nonlinear objective function with linear equality and bound constraints23

Customized Portfolio Optimization Smart Beta§Solve with fmincon from MATLAB Optimization Toolbox total distance betweenmarginal risk contributionsbudget constraintbounds24

Customized Portfolio Optimization Smart sTime25

Customized Constraints - Robust Constraints26

Customized Portfolio Optimization Robust ConstraintsMean-Variance Portfolio Optimizationmin 𝑥 𝐶 𝑥;𝑠. 𝑡. 𝑚 𝑥 𝑅𝑥 𝑒 1x 0Covariance matrixmean of asset returnsbudget constraintlower boundFinancial Toolbox27

Customized Portfolio Optimization Robust ConstraintsAdditional constraints supported by§§§§§§Financial ToolboxLinear inequality constraintsLinear equality constraintsGroup constraintsGroup ratio constraintsAverage turnover constraintsOne-way turnover constraints28

Customized Portfolio OptimizationRobust ConstraintsExtend standard model by individual constraints, e.g. robust constraints𝑃𝑟𝑜𝑏(𝜉 x 𝑅 ) prandom vector of returnsprobability levelreturn levelàDeterministic approximation (Chebychev’s inequality)𝑚 𝑥 T-UTmean vector of returns𝑥 𝐶𝑥 𝑅covariance of returnsP. Bonami, M.A. Lejeune, ‚An Exact Solution Approach for Portfolio Optimization Problems Under Stochastic and IntegerConstraints’, Operations Research 2009, Vol. 57,Issue 329

Customized Portfolio Optimization Robust ConstraintsMean-Variance Portfolio Optimization with robust constraintmin 𝑓 𝑥 𝑥 𝐶 𝑥;𝑠. 𝑡𝑐 𝑥 𝑅 𝑚 𝑥 𝑚 𝑥 𝑥 𝑒 1x 0𝑝T 𝑥 𝐶𝑥 𝑅-UT 𝑥 𝐶𝑥 01 𝑝budget constraintlower boundrobust constraintà Robust mean variance model is a nonlinear, convex NLP30

Customized Portfolio OptimizationRobust Constraints§Solve with fmincon from MATLAB Optimization Toolbox mean-variance riskproxybudget constraintrobust constraint31

Customized Portfolio OptimizationRobust ConstraintsProvide gradients𝑓 𝑥 𝑓 𝑥𝑐 𝑥 𝑐 𝑥32

Customized Portfolio OptimizationRobust ConstraintsProvide Hessian Z 𝑓 𝑥 Z 𝑐( 𝑥 [ 𝜆( Z 𝑐𝑒𝑞) 𝑥 [ 𝜆)33

Customized Portfolio Optimization Robust Constraintsruntimeaccuracyinformationinformation34

Customized Portfolio Optimization Robust ConstraintsWithout gradientsWith gradients35

Customized Portfolio Optimization Robust ConstraintsWithout gradientsWith gradients36

Customized Portfolio Optimization Robust ConstraintsR37

Customized Portfolio OptimizationR 38

MATLAB – The Financial Development PlatformAccessResearch and QuantifyShareFilesData Analysis and VisualizationReportingDatabasesFinancial ModelingApplicationsDatafeedsApplication DevelopmentProductionFinancial DatafeedNeural Network el ComputingProduction ServerSpreadsheet Link EXEconometricsReport GeneratorTrading ToolboxMATLAB Distributed Computing Server39

Customized Portfolio Optimization Deployment§Compile your MATLABoptimization model for yourdedicated platform§Make it available for yourenterprise environment40

Summary§Optimization for financial applications is built within MATLAB toolboxescovering many standard applications§A large variety of optimization algorithms available in MATLAB Optimization Toolbox and Global Optimization Toolbox §Customized optimization models made easy by quick modeling, advancedoptimization process diagnostics and rapid deployment.§Enhance optimization performance and accuracy by adding maximalinformation.41

Thank you !42

Pricing & Valuation. 6 MATLAB – The Financial Development Platform Financial Statistics Optimization Financial Instruments Econometrics MATLAB Parallel Computing MATLAB Distributed Computing Server Files Databases . MATLAB Compiler