First Steps With The Black-Scholes Model - Magnimetrics

Black Scholes Formula · S0 is the stock price; · e is the exponential number; · q is the dividend yield percentage; · T is the term (one year will ... HomeFeaturesAboutTeamBlogGetinTouchSurveyOnlineCoursesFREEIntrotoGoogleDataStudioFREEIntroductiontoExcelBudgetModelinginExcelMenuHomeFeaturesAboutTeamBlogGetinTouchSurveyOnlineCoursesFREEIntrotoGoogleDataStudioFREEIntroductiontoExcelBudgetModelinginExcelBlack-ScholesModel:FirstStepsDobromirDikovJune12,2020Todaywetakealookatthemostpopularoptionspricingmodel.TheBlackScholesModel,alsoknownastheBlack-Scholes-Mertonmethod,isamathematicalmodelforpricingoptioncontracts.Itworksbyestimatingthevariationinfinancialinstruments.Thetechniquereliesontheassumptionthatpricesfollowalognormaldistribution.Basedonthis,itderivesthevalueofanoption.Itismoresuitableforpath-independentoptions,whichtheinvestorscannotexercisebeforetheirduedate.ThismakesitdifferentfromtheBinomialOptionPricingModel,whichisarisk-neutralmethodforvaluingpath-dependentoptions(e.g.,Americanoptions).AnalystsandinvestorsconsiderBlackScholestobeoneoftheessentialconceptsinmodernfinancialtheory.BlackScholesModelIntroducedin1973intheJournalofPoliticalEconomy,byFischerBlackandMyronScholes,andlaterbuiltuponbyRobertMerton,themodelwontheNobelPrizeineconomicsin1997.Themodelfollowsadifferentialequation,whichweusetosolveforoptions’prices.WeusethestandardvariationofthemodelforEuropeanstyleoptions.BlackScholesisnotsuitableforAmericanstyleoptions,asitdoesnotconsiderinvestorscanexercisetheseoptionsatanymomentbeforetheirexpirationdate.Themethodassumesthepriceoftheunderlyingstockfollowsalognormaldistribution,asitcannotbenegative.ThedistributionthemodelreliesonisbasedonBrownianmotion,atheoryforpredictingnaturallyoccurringrandomphenomena.Weusethemodeltocalculateanestimateforanoption’sprice,usingthefollowinginformation:Thecurrentprice;Thestrikeprice;Timetoexpiration;Expecteddividendyield;Expectedinterestrate;andExpectedvolatility.Initially,themodeldoesn’tconsidertheeffectofdividendpayouts.Thereforeit’sfrequentlyadaptedtodosobydeterminingtheex-dividenddatevalueoftheunderlyingasset.Thedividendyieldhastheeffectofreducingthecurrentstockpricebecausetheoptionholderforgoesthedividends.AssumptionsTheBlackScholesmodelonlyworksifwemakecertainassumptions.ThemodellooksatEuropeanstyleoptions,whichwecanonlyexecuteattheirexpirationdate;Nodividendpaymentsoccurduringtheoptionlifespan(thiswasanassumptionintheinitialmodel,whichanalystshavesinceeliminatedbyaddingthedividendyieldinthemodelformula);Themarketsarecompletelyefficient,meaningwecan’tpredicttheirmovements;Buyingandsellingoptionsinvolvenotransactioncosts;Boththevolatilityoftheunderlyingassetandtherisk-freerateremainconstantuntiltheexpirationdate,whichisnotcorrect,asinrealityvolatilityshiftswithsupplyanddemand;andThereturnsoftheunderlyingassetfollowanormaldistribution.BlackScholesFormulaThemodelformulaisquitecomplexmathematically,whichisnotaparticularissue,aswecanemployspecializedtoolstocalculateitforus.Butit’sstillessentialtounderstandhowitworks.Tocalculatethepriceofacalloption,undertheBlackScholesmodel,wecanusethefollowingequation:Where:S0isthestockprice;eistheexponentialnumber;qisthedividendyieldpercentage;Tistheterm(oneyearwillbeT=1,whilesixmonthswillbeT=0.5);N(d1)isthedeltaofthecalloption,meaningthechangeinthecallpriceovertheshiftinthestockprice;Kisthestrikeprice;ristherisk-freerate;andN(d2)istheprobabilitythatthefuturestockpricewillbehigherthanthestrikeprice,thelikelihoodwewillexercisetheoption.Onesideisthestockpricemultipliedbythecumulativestandardnormaldistributionofd1,andtheother–thestrikeprice,discountedtotimeTandthenmultipliedbythecumulativestandardnormaldistributionofd2.Wherewecancalculated1andd2withthefollowingformulae:Whereσisthevolatilityofthereturnsoftheunderlyingasset.Wecanapplythesamelogicforaputoption:ExampleBlackScholesCalculationTobetterillustratetheconceptbehindtheBlackScholesModel,wewilltakealookatthefollowingmodel.Westartwiththemodelinputs.Asdiscussedabove,weneedthefollowingsixvariablestocalculatethecallandputoptionvalues.Next,wewillcalculatetheoptions’parameters.Startingwithd1,weget0.39fromourmodelassumptions. Tocalculateit,wesimplytransfertheformulaford1toExcel. Followingthesameapproach,wecalculated2,N(d1)andN(d2).Withthose,wecancalculatetheCallOptionPriceandthePutOptionPrice.WecanemploytheCall-PutParityruletocheckwhetherourformulasandcalculationsarecorrect.NowthatwehaveourmodeltocalculatetheCallandPutoptionsprices,wecangoastepfurtherandlookathowthedifferentinputparametersaffecttheresultsofourmodel.TheDataTablefunctionalityinExcelisagreattooltopreparesomesensitivityanalysistablesforourvariables.YoucanlearnhowtouseDataTablesinourSensitivityAnalysisarticle.Let’slookattheStockPriceifitchangesfrom€10to€100andhowitaffectstheCallandPutvalues.WenoticethattheCallOptionhasnovalueupuntilthestockpricehits€40.WeseethereverseforthePutOption,whereitsvaluehitszerooncethestockpricegoesabove€85.IfwepreparethesamesensitivitytablefortheStrikePrice,wecanobservethereverserelationship.AstheStrikePriceincreases,theCallValuegoesdown,andthePutValuegoesup.Let’salsotakealookattherisk-freerate.Asitincreases,thisalsoincreasestherequiredrateofreturns.Subsequently,thisdecreasesthevalueofstocks.Wenoticeadirectrelationship,astherisk-freerateincreases,theCallOptionvalueincreasesaswell,whilethePutOptionvaluedecreasesitsworth.Followingthesameapproach,youcangoaheadandlookattherestoftheparametersandhowtheyaffecttheOptionValues.YoucandownloadtheExcelmodelattheendofthearticleandgooverthesensitivityanalysisinthere.ConclusionItisimportanttorememberthattheBlackScholesMertonmodelisatheoreticalconcept.Weuseittotryandestimatethebehaviorofthemarket.However,whenweusetheresultsinourdecision-makingprocess,wehavetounderstandtheunderlyingassumptionsinthemodelandhowitdeviatesfromreality.Theformulahelpedoptionstradingtobecomemorepopular,asitmakesitlooklesslikegambling.Nowadays,variousmodificationsofBlackScholesarewidelypopularasfundamentalstrategiesforriskcontrol,associatedwithvolatility.Youcanshowyoursupportbysharingthearticlewithcolleaguesandfriends.Also,don’tforgettodownloadtheexampleExcelmodelbelow.BlackScholesModel_Magnimetrics.xlsxFREEDOWNLOADSubscribetoourNewsletterGetaFREEExcelBenchmarkAnalysisTemplateNameEmailOpt-inIagreetoreceiveMagnimatricsnewslettersandacceptthedataprivacystatement.Imayunsubscribeatanytimeusingthelinkinthenewsletter.SubscribeDobromirDikovFCCA,FMVAHi!Iamafinanceprofessionalwith10+yearsofexperienceinaudit,controlling,reporting,financialanalysisandmodeling.Iamexcitedtodelvedeepintospecificsofvariousindustries,whereIcanidentifythebestsolutionsforclientsIworkwith.Inmysparetime,Iamintoskiing,hikingandrunning.IamalsoactiveonInstagramandYouTube,whereItrydifferentwaystoexpressmycreativeside.TwitterInstagramYoutubeLinkedinTheinformationandviewssetoutinthispublicationarethoseoftheauthor(s)anddonotnecessarilyreflecttheofficialopinionofMagnimetrics.NeitherMagnimetricsnoranypersonactingontheirbehalfmaybeheldresponsiblefortheusewhichmaybemadeoftheinformationcontainedherein.Theinformationinthisarticleisforeducationalpurposesonlyandshouldnotbetreatedasprofessionaladvice.Magnimetricsandtheauthorofthispublicationacceptnoresponsibilityforanydamagesorlossessustainedintheresultofusingtheinformationpresentedinthepublication.AnyThoughts?CancelreplyYoumightalsolikeoneofthefollowingarticles:ExcelandVBAForecastswiththePolynomialRegressionModelinExcelRegressionanalysisaimstomodeltheexpectedvaluesforadependentvariable(y)basedonindependentvariables(x).ThepolynomialregressionisastatisticaltechniqueReadMore»DobromirDikov06/02/2021FinancialAnalysisUnderstandingtheCostofDebtRatioResearchintheSMEsectorshowsthataround40-50%ofcompaniesseekdebtfinancingatleastonceintheirlifecycle.It’sessentialtounderstandReadMore»DobromirDikov29/01/2021FinancialAnalysisUnderstandingtheGordonGrowthModelforStockValuationUnderstandingtheGordonGrowthModelforStockValuationTheGordonGrowthModel(GGM)isamethodforthevaluationofstocks.InvestorsuseittoReadMore»DobromirDikov11/12/2020HomeFeaturesAboutMagnimetricsMeettheTeamBlogGetinTouchSurveyExcelDownloadsPrivacyPolicyCookiePolicyTerms&conditionsMenuHomeFeaturesAboutMagnimetricsMeettheTeamBlogGetinTouchSurveyExcelDownloadsPrivacyPolicyCookiePolicyTerms&conditionsMagnimetricsismadewithinPlovdiv,Bulgaria.LinkedinFacebookTwitterYoutubeMediumWaitaminute!GetaFREEExcelBenchmarkAnalysisTemplatewhenyousubscribetoournewsletterNameEmailOpt-inIagreetoreceiveMagnimatricsnewslettersandacceptthedataprivacystatement.Imayunsubscribeatanytimeusingthelinkinthenewsletter.Iwantit