Black-Scholes Model: Formula & Examples - Study.com

The Black-Scholes Formula · T = the time to maturity, which is how long until the option expires, in years · S = the current price, which is the ... forTeachers forSchools forWorkingScholars® forCollegeCredit Login SignUp Menu forWorkingScholars® forCollegeCredit Plans Plans Courses Courses FindCoursesbySubject Science Math Business Psychology History English SocialScience Humanities Spanish ProfessionalDevelopment ByEducationLevel College HighSchool MiddleSchool TestPrep PopularTests… PopularTests SAT ACT TOEFL GMAT GRE CLEP GED ASVAB RealEstate TeacherCertification… TeacherCertification AEPA CEOE CSET/CBEST FTCE GACE ILTS MTEL MTLE MTTC MoGEA/MEGA NES NYSTCE OAE Praxis RICA TExES VCLA WEST Nursing… NursingTests HESI TEAS NCLEX PAXRN CNE OtherTests… OtherTests AP ASCP ASWB Accuplacer CHSPE CTEL DSST Finance FSA HSPT ISEE MBLEx NCE PERT PHR SHSAT TASC TECEP UExcel Exploreover 4,600 videocourses BrowseAllCourses SAT ACT TOEFL GMAT GRE CLEP GED ASVAB RealEstate AEPA CEOE CSET/CBEST FTCE GACE ILTS MTEL MTLE MTTC MoGEA/MEGA NES NYSTCE OAE Praxis RICA TExES VCLA WEST HESI TEAS NCLEX PAXRN CNE AP ASCP ASWB Accuplacer CHSPE CTEL DSST Finance FSA HSPT ISEE MBLEx NCE PERT PHR SHSAT TASC TECEP UExcel Credit Credit CreditOptions CreditOptions OnlineCollegeCredit HighSchool&GED CertificatesofCompletion HowitWorks TransferCredit TransferCredit AllTransferCredit FindYourSchool CollegeCoursesforCredit TransferringCredit HowItWorks EarnTransferableCredit&GetyourDegreefast Learnmore ContactSupport Copyright LessonTranscript Instructor: DavidBartosiak Showbio DavedrawsoffhisyearsofexperienceasaFinancialAdvisorandAnalysttoteachothersallaboutfinanceandtheinvestingworld. TheBlack-Scholesmodelisamathematicalmodelforfinancialmarkets.Fromthislargermodel,theBlack-Scholesformulafortheoreticaloptionvalueisusedtodeterminewhatpriceputandcalloptionsshouldbetradingatbasedonassumptionsofvolatility Updated:02/15/2021 Createanaccount FinancialMarketMathematicalModelSincethebeginningofthestockmarket,investorshavebeentryingtogainanedge.Thenumber-crunchersouttherehavelongsoughtoutamathematicalmodel,whichcouldpredictpricemovement.Iftheycouldsomehowfigureoutthesecretformula,theycouldbecomerichbeyondtheirwildestdreams.Theteamofeconomists,FischerBlack,MyronScholes,andRobertCMerton,triedtodojustthat.Theycameupwithanover-archingmathematicalmodelforfinancialmarketsthatcontainderivativeinstruments.Lackingacreativename,thismodelbecameknownastheBlack-Scholes-Mertonmodel.Fromthislargermodel,smallermodelsandequationsweremadebasedonthesameassumptions.Afteryearsofdevelopingthemodel,RobertMertonisattributedwithfirstmentioningthe''Black-Scholesoptionspricingmodel''in1973.Thistheoreticalmodelcouldhelpoptionsmarket-makersproperlypriceoptionsonalltypesoffinancialinstruments.Theirworkwassoground-breakingthat24yearslaterin1997,RobertC.MertonandMyronScholeswontheNobelMemorialPrizeinEconomicStudiesfortheirwork.Understandingthevariousinputswhichgointothecalculationcanhelptradersunderstandwhetherthemarketisoverpricing,underpricing,orfairlypricinganoption.Herewe'lldiscussthevariousfactorsindetailandlearnhowtousethemodeltocalculateoptionprices. Anerroroccurredtryingtoloadthisvideo. Tryrefreshingthepage,orcontactcustomersupport. YoumustcCreateanaccounttocontinue watching Registertoviewthislesson Areyouastudentorateacher? Iamastudent Iamateacher CreateYourAccountToContinueWatching Asamember,you'llalsogetunlimitedaccesstoover84,000 lessonsinmath,English,science,history,andmore.Plus,getpracticetests,quizzes,andpersonalizedcoachingtohelpyou succeed. Getunlimitedaccesstoover84,000lessons. Tryitnow Itonlytakesafewminutestosetupandyoucancancelanytime. Alreadyregistered?Loginherefor access  Back WhatteachersaresayingaboutStudy.com Tryitnow Alreadyregistered? Loginhereforaccess Comingupnext: OptimalCapitalStructure You'reonaroll.Keepupthegoodwork! TakeQuiz Watch NextLesson  Replay Justcheckingin.Areyoustillwatching? Yes!Keepplaying. Yournextlessonwillplayin 10seconds 0:04FinancialMarket… 1:28TheBlack-ScholesFormula 3:30Calculating… 5:58LessonSummary Save Save Save Wanttowatchthisagainlater? LoginorsignuptoaddthislessontoaCustomCourse. LoginorSignup Timeline Autoplay Autoplay Speed Speed 29Kviews Video Quiz Course 29K views TheBlack-ScholesFormulaTheBlack-Scholesformulaisamathematicalmodeltocalculatethepriceofputandcalloptions.Sinceputandcalloptionsaredistinctlydifferent,therearetwoformulas,whichaccountforeachoption.Calloptionsgivetheoptionholdertherighttobuytheunderlyingstockforanagreeduponpriceanytimebetweentodayandoptionexpiration.Tradersthatbelievetheunderlyingstockwillgoupovertimebuythesecalloptionsinthehopesofmakingmoney.Ontheflip-side,putoptionsgivetheoptionholdertherighttoselltheunderlyingstockforanagreeduponpriceanytimebetweentodayandoptionexpiration.Tradersthatthinkastockisgoingtogodowncanbuytheseputoptionsinthehopesofmakingmoneyifthestockgoesdoes.TheactualBlack-Sholesformulalookscomplicatedbutisactuallysimplewhenyoubreakitdowntothebasics.Themainfactorsintheequationare: T=thetimetomaturity,whichishowlonguntiltheoptionexpires,inyears S=thecurrentprice,whichisthepriceoftheunderlyingstock K=thestrikeprice,whichistheagreeduponpriceofoptionexecution r=therisk-freerate,whichistherateaninvestorcouldgetbytakingonnorisk;it'stypicallythe3-monthtreasurybillyield;andfinally σ=thepricevolatility,whichisthevolatilityofthereturnsoftheunderlyingstock Hereisthemathematicalnotationoftheformula: Now,there'salotthere,buttheimportantthingtofocusoninthisformulaistheinclusionofthefunctionN(x),whichisthecumulativestandardnormaldistributionfunction.Thecumulativestandardnormaldistributionfunctionisdefinedastheprobabilityofarandomvariablewithnormaldistribution,ameanof0,andvarianceof1/2fallingintherangeof{-x,x}.It'sacomplicatedcalculationdealingwiththeareaunderanormalizeddistributioncurve.CalculatingTheoreticalOptionsForthesakeofoptionscalculation,youdon'tneedtobecomeaPhDinmath.TherearehundredsoffreeonlinecalculatorsyoucanusetoplugineasilyaccessiblevaluestocalculationtheBlack-Scholesformula.Let'sdoasamplecalculationwithXYZCorpStockanddefinethesevalues:Timetomaturity(T)=1yearCurrentPrice(S)=$120StrikePrice(K)=$100Risk-FreeRate(r)=1%,andPriceVolatility(?)=50%Firstyouneedtocalculatevaluesford1andd2,soyoucanthrowthesevaluesintothecumulativestandardnormaldistributionfunction.UsingthesevaluesandacalculatorIfoundonline,Iwasabletocomeupwithatheoreticalvalueof$34.20forthecalloption.Thepriceoftheputoptionatthesamestrikeisderivedfromthecalculationofthecalloption. Usingthesameinputvaluesinthepreviousequationgivesusthetheoreticalpriceoftheputoptionatthesamestrike.Theonlinecalculatorweusedbeforegivesusavalueof$12.22fortheputoption.Whywouldanyonegothroughallthistroubletomakethesecalculations?Let'ssaySarahislookingtotradeinXYZCorpstockoptions,butshe'snotsurewhichoptionshewantstobuy.Eachoptionexpirationdatehasseveraloptionstochoosefrom.AssumingshethinksXYZCorpstockisgoingtogoupoverthenextyearandshewantstobuyacalloption,shehasmanydifferentstrikepricestolookat.Byknowingthetheoreticalvalueofseveraldifferentoptionsatvariousstrikes,shecancomparethesetothecurrenttradingpriceoftheoptionstofindopportunities.Lookingatthe$100strikeintheexamplewithatheoreticalvalueof$34.20,Sarahcancomparethecurrenttradingpriceoftheoption.Iftheoptionistradingbelow$34.20,thentheoptionmaybeundervaluedbythemarket.Byinvestinginunderpricedoptionslikethis,Sarahhasgivenherselfabitofanedge.Ifshecanuncoverhugedeviationsfromtheoreticalvalueintheoptionschain,shecangiveherselfanevengreateredgeinthemarket.Don'tgettooexcitedabouttryingtouncovergiantdeviationsfrommarketpricingandtheoreticaloptionspricing.Market-makersarewellawareoftheBlack-Scholesmethodandtendtopriceoptionsaccordingly.LessonSummaryAllright,let'stakeamomenttoreview.TheBlack-Scholesformulaisamathematicalmodeltocalculatethepriceofputandcalloptions.Wealsolearnedthatcalloptionsgivetheoptionholdertherighttobuytheunderlyingstockforanagreeduponpriceanytimebetweentodayandoptionexpiration,whileputoptionsgivetheoptionholdertherighttoselltheunderlyingstockforanagreeduponpriceanytimebetweentodayandoptionexpiration.Wealsolearnedthattheformulatakesintoaccountthefollowingthings: T,whichisthetimetomaturity,whichishowlonguntiltheoptionexpiration,inyears S,whichisthecurrentprice,whichisthepriceoftheunderlyingstock K,whichisthestrikeprice,whichistheagreeduponpriceofoptionexecution r,whichistherisk-freerate,whichistherateaninvestorcouldgetbytakingonnorisk;it'stypicallythe3-monthtreasurybillyield Andfinally,σ,whichisthepricevolatility,whichisthevolatilityofthereturnsoftheunderlyingstock,expressedasapercentage Wethenlearnedthattheformulausedisthecumulativestandardnormaldistributionfunction,whichwedefinedastheprobabilityofarandomvariablewithnormaldistribution,ameanof0andvarianceof1/2fallingtherangeof{-x,x}.Bycalculatingthesetheoreticalvalues,optionstradersmakeprobability-baseddecisionsonwhichoptionstobuyandsell.Thisiswhytheformulaissoimportantifyouwanttogetinvolvedintrades. TounlockthislessonyoumustbeaStudy.comMember. Createyouraccount Practice: Black-ScholesModel:Formula&ExamplesQuiz Instructions:Chooseananswerandclick'Next'.Youwillreceiveyourscoreandanswersattheend. 1/5completed WhichinputintheBlack-Scholesformulaisexpressedasapercentage? Timetomaturity StrikePrice CurrentPrice Pricevolatility Next CreateYourAccountToTakeThisQuiz Asamember,you'llalsogetunlimitedaccesstoover84,000lessonsinmath, English,science,history,andmore.Plus,getpracticetests,quizzes,andpersonalizedcoachingtohelpyousucceed. Tryitnow Itonlytakesafewminutestosetupandyoucancancelanytime. Alreadyregistered? Loginhereforaccess Registertoviewthislesson Areyouastudentorateacher? Iamastudent Iamateacher UnlockYourEducation Seeforyourselfwhy30millionpeopleuseStudy.com BecomeaStudy.commemberandstartlearningnow. BecomeaMember Alreadyamember?LogIn  Back WhatteachersaresayingaboutStudy.com Tryitnow Alreadyregistered? Loginhereforaccess Createanaccount Black-ScholesModel:Formula&Examples Starttoday.Tryitnow Business / FinancialAccounting:HelpandReview FinancialAccounting:HelpandReview 25chapters| 278lessons {{courseNav.course.topics.length}}chapters| {{courseNav.course.mDynamicIntFields.lessonCount}}lessons |{{course.flashcardSetCount}} flashcardset{{course.flashcardSetCoun>1?'s':''}} Ch1. 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WhatisaMedicalLien?-Definition&Requirements WhatisaSuspenseAccount?-Definition&Examples WhatisaPayableonDeathBankAccount? CapitalLeasevs.OperatingLeaseinAccounting Goto IntroductiontoAccounting:HelpandReview Ch2. FinancialStatementsinAccounting:... Goto FinancialStatementsinAccounting:HelpandReview Ch3. MechanicsoftheAccountingCycle:Help... Goto MechanicsoftheAccountingCycle:HelpandReview Ch4. ProjectEvaluation&Analysis Goto ProjectEvaluation&Analysis Ch5. PreparingFinancialStatements:Help... Goto PreparingFinancialStatements:HelpandReview Ch6. InternalControlsinAccounting:Help... Goto InternalControlsinAccounting:HelpandReview Ch7. MerchandisingOperationsandInventory... Goto MerchandisingOperationsandInventoryinAccounting:HelpandReview Ch8. CurrentandLong-TermLiabilitiesin... Goto CurrentandLong-TermLiabilitiesinAccounting:HelpandReview Ch9. AdjustingAccounts&Preparing... Goto AdjustingAccounts&PreparingFinancialStatements:HelpandReview Ch10. 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