蒙特卡羅在金融中模擬方法實(shí)現(xiàn)_english

1、Typical uses cases of Monte carlo in financeAgendaPrinciples and uses cases for Monte Carlo methodsUsing MATLAB toolbox for Monte Carlo simulationsDevelop you own Monte Carlo engineA quick overview of Variance reduction technicsAn simple exampleCompute the area of a lakeWe shot N cannon ballsn balls

2、 out of the “l(fā)ake” N- n balls in the lakeOne gets01020304050607080901000102030405060708090100SquareLakeLakeSquareAreaNnNAreanNNAreaArea*01020304050607080901000102030405060708090100Take outsResults depends on :random number generation (Mersenne Twister)Number of simulations010203040506070809010001020

3、3040506070809010001020304050607080901000102030405060708090100Typical uses cases of Monte carlo in financeDerivatives pricingRiskStructurerStochastic Asset Liability ManagementGeneral PrinciplesEstimation technique based on the simulation of a great number of random variablesLets consider This can be

4、 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 draws10)(dxxgI)(UgGeneral Principles, continuedWe need to generate a sequence Ui of random samples, which are independent (iid)Then we can compute th

5、e empiric mean:From the law of Great Numbers, one gets :Variance :Confidence interval : using a Gaussian approximation ImmlimmiimUgm1)(1mUgVarVarim) )()(Good points and drawbacksGood pointsVarious application areasFew hypothesisEasy to developDrawbacksDependency to random number generatorBig variabi

6、lity (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 distributions supported (+ than 20)Easy deploymentAgendaPrinciples and uses cases for Monte Carlo methodsUsing

7、 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, matrix factorisation, Statistics toolbox : Random numbers, copulas, Financial toolbox :Portsim :”

8、Monte Carlo simulation of correlated asset returns”GARCH ToolboxgarchsimFinancial toolboxOn a time interval, performances are driven by the following equation :Time basis must be consistent for input parameters (drift and volatility)Annual time basis - dt in yearsDaily time basis- dt in daysdtdtSdsD

9、emo 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 paths to price aVanilla optionApply the option payoffVanilla - No path dependancyCompute the call price fo

10、r different strikesCompute the confidence intervalsAgendaPrinciples and uses cases for Monte Carlo methodsUsing MATLAB toolbox for Monte Carlo simulationsDevelop you own Monte Carlo engineA quick overview of Variance reduction technicsProcessGenerate Random numbersDirectly from the statistical distributionThrough a uniform law- Allow the use of quasi random number generationApply the model (volatility, )Computation of the empiric meanConfidence interval estimationDemoCorrelated Equities SimulationInput :Time : NDaysNumber of diff