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1、Monte Carlo Simulations using MATLABVincent Leclercq, Application engineerEmail : vincent.leclercqmathworks.frAgendaPrinciples and uses cases for Monte Carlo methodsUsing MATLAB toolbox for Monte Carlo simulationsDevelop you own Monte Carlo engineA quick overview of Variance reduction technicsAn sim

2、ple exampleCompute the area of a lakeWe shot N cannon ballsn balls out of the “lake” N- n balls in the lakeOne getsTake outsResults depends on :random number generation (Mersenne Twister)Number of simulationsTypical uses cases of Monte carlo in financeDerivatives pricingRiskStructurerStochastic Asse

3、t Liability ManagementGeneral PrinciplesEstimation technique based on the simulation of a great number of random variablesLets consider This can be seen as the expectation , with U being a uniform random variable on (0,1), ie U(0,1)We can estimate tis expectation using an empiric mean from random dr

4、awsGeneral Principles, continuedWe need to generate a sequence Ui of random samples, which are independent (iid)Then we can compute the empiric mean:From the law of Great Numbers, one gets :Variance :Confidence interval : using a Gaussian approximation Good points and drawbacksGood pointsVarious app

5、lication areasFew hypothesisEasy to developDrawbacksDependency to random number generatorBig variability (accuracy)Computation timeWhy MATLAB ?Efficiency :1 000 000 paths in less than 1 s25 times fastest thanExcelState of the Art AlgorithmsMersenne TwisterLinear algebraLots of statistical distributi

6、ons supported (+ than 20)Easy deploymentAgendaPrinciples and uses cases for Monte Carlo methodsUsing MATLAB toolbox for Monte Carlo simulationsDevelop you own Monte Carlo engineA quick overview of Variance reduction technicsWhich tools for Monte Carlo simulations ?MATLAB : Core linear algebra engine

7、, matrix factorisation, Statistics toolbox : Random numbers, copulas, Financial toolbox :Portsim :” Monte Carlo simulation of correlated asset returns”GARCH ToolboxgarchsimFinancial toolbox : portsimOn a time interval, performances are driven by the following equation :Time basis must be consistent

8、for input parameters (drift and volatility)Annual time basis - dt in yearsDaily time basis- dt in daysDemo 1:Geometric brownian motion Lognormality of equity pricesHistorical data inputDrift and volatilityAnnually or Daily Simulate 10 000 paths on one year.Compare resultsDemo 2: Use the previous pat

9、hs to price aVanilla optionApply the option payoffVanilla - No path dependancyCompute the call price for different strikesCompute the confidence intervalsGARCH Toolbox : garchsimStochastic VolatilitySimulations of Auto Regressive models / GARCH“Perform Monte Carlo simulation of univariate returns, i

10、nnovations,and conditional volatilities”Fitting (Adjust the model, garchfit function) and SimulationSimulation , several possibilities:Use of historic data (bootstrapping)See Market Risk Using Bootstrapping and Filtered Historical SimulationUse of random variablesAgendaPrinciples and uses cases for

11、Monte Carlo methodsUsing MATLAB toolbox for Monte Carlo simulationsDevelop you own Monte Carlo engineA quick overview of Variance reduction technicsWhat do I need for Monte Carlo ?A good random number generatorRand, randn - several chocie possible for random number generationRandom (more than 20 dis

12、tributions), copularnd - Statistics toolboxLinear algebra functions: Cholesky factorizationcumsumProcessGenerate Random numbersDirectly from the statistical distributionThrough a uniform law- Allow the use of quasi random number generationApply the model (volatility, )Computation of the empiric mean

13、Confidence interval estimationDemoCorrelated Equities SimulationInput :Time : NDaysNumber of different paths : NSimulation Number of Assets : 2, NAssets with correlationWe know :VolatilityCorrelationsOutput :Matrice de NDays* NSimulation*NAssetsPreserved CorrelationsAgendaPrinciples and uses cases f

14、or Monte Carlo methodsUsing MATLAB toolbox for Monte Carlo simulationsDevelop you own Monte Carlo engineA quick overview of Variance reduction technicsVariance ReductionWhy ?Slow Convergence of Monte Carlo pricingNeed a great number of pathsSolution : Use if various variance reduction methodsSeveral

15、 possible methodsVariance Reduction : OverviewAntithetic VariablesEfficient, easy to implementEfficiency depends of the option (ex : Butterfly)Control VariablesUse of a variable correlated to the one we want to estimateEx : Vanilla option PricingWe canuse the close formula (Hulll) in order to comput

16、e the variance and the expected return of the underlying at maturityWe need to estimate the covariance between our control variable (the underlying) and the variable we want to estimate (option price)Variance Reduction Overview (2/3)Quasi Monte CarloUse of low discrepancy sequences“quasi random” seq

17、uencesHalton sequences, Sobol sequences, Better AccuracyVariance Reduction Overview (3/3)Variance reduction using conditionning Principle:Var(EX) Reduced VarianceOther techniques :Importance samplingStratified samplingDemonstrationVanilla option pricing using Variance ReductionSeveral methodology us

18、edAntithetic VariablesQuasi Monte Carlo (Halton / Sobol)Control VariableResults comparisonVariance Reduction, Key takeoutsEfficient, Generic methodConfidence intervalsVariance Reduction technics should be used wisely, depending on the product to priceExample : Antithetic for options Butterfly lead to an increase of the varianceLots of research papersGeneral ConclusionMATLAB allow users to quickly develop and test advanced Monte Carlo simulation Very generic solutionNew : a complete framework Monte Carlo