Introduction to the Black-Scholes formula (video) - Khan ...

In the BS option pricing formula why do we add sigma squared/2 to r for ... as an example) to the positive ... Ifyou'reseeingthismessage,itmeanswe'rehavingtroubleloadingexternalresourcesonourwebsite. Ifyou'rebehindawebfilter,pleasemakesurethatthedomains*.kastatic.organd*.kasandbox.orgareunblocked. CoursesSearchDonateLoginSignupSearchforcourses,skills,andvideosMaincontentEconomicsFinanceandcapitalmarketsOptions,swaps,futures,MBSs,CDOs,andotherderivativesBlack-ScholesformulaBlack-ScholesformulaIntroductiontotheBlack-ScholesformulaThisisthecurrentlyselecteditem.ImpliedvolatilityCurrenttime:0:00Totalduration:10:240energypointsEconomics·Financeandcapitalmarkets·Options,swaps,futures,MBSs,CDOs,andotherderivatives·Black-ScholesformulaIntroductiontotheBlack-Scholes formulaGoogleClassroomFacebookTwitterEmailBlack-ScholesformulaIntroductiontotheBlack-ScholesformulaThisisthecurrentlyselecteditem.ImpliedvolatilityVideotranscriptVoiceover:We'renowgonnatalkaboutprobablythemostfamous formulainalloffinance,andthat'stheBlack-ScholesFormula,sometimescalledthe Black-Scholes-MertonFormula,andit'snamedafterthesegentlemen.ThisrightoverhereisFischerBlack.ThisisMyronScholes.Theyreallylaidthe foundationforwhatledtotheBlack-ScholesModeland theBlack-ScholesFormulaandthat'swhyithastheirname.ThisisBobMerton,whoreally tookwhatBlack-Scholesdidandtookittoanotherleveltoreallygettoour moderninterpretationsoftheBlack-ScholesModel andtheBlack-ScholesFormula.Allthreeofthese gentlemenwouldhavewontheNobelPrizeinEconomics,exceptfortheunfortunatefactthatFischerBlackpassedaway beforetheawardwasgiven,butMyronScholesandBobMertondidgettheNobelPrizefortheirwork.Thereasonwhythisissuchabigdeal,whyitisNobelPrizeworthy,and,actually,there'smanyreasons.Icoulddoawhole seriesofvideosonthat,isthatpeoplehavebeen tradingstockoptions,orthey'vebeentradingoptions foravery,very,verylongtime.Theyhadbeentradingthem,theyhadbeenbuyingthem,theyhadbeensellingthem.Itwasamajorpartof financialmarketsalready,buttherewasnoreallygoodwayofputtingourmathematicalmindsaroundhowtovalueanoption.Peoplehadasenseofthe thingsthattheycaredabout,andIwouldassume especiallyoptionstradershadasenseofthethings thattheycaredaboutwhentheyweretradingoptions,butwereallydidn'thavean analyticalframeworkforit,andthat'swhatthe Black-ScholesFormulagaveus.Let'sjust,beforewediveinto thisseeminglyhairyformula,butthemorewetalkaboutit,hopefullyit'llstart toseemalotfriendlierthanitlooksrightnow.Let'sstarttogetanintuitionforthethingsthatwewouldcareaboutifwewerethinkingabout thepriceofastockoption.Youwouldcareaboutthestockprice.Youwouldcareabouttheexerciseprice.Youwouldespeciallycare abouthowmuchhigherorlowerthestockpriceisrelative totheexerciseprice.Youwouldcareaboutthe risk-freeinterestrate.Therisk-freeinterest ratekeepsshowingupwhenwethinkabouttakinga presentvalueofsomething,Ifwewanttodiscountthevalue ofsomethingbacktotoday.Youwould,ofcourse,think abouthowlongdoIhavetoactuallyexercisethisoption?Finally,thismightlooka littlebitbizarreatfirst,butwe'lltalkaboutitinasecond.Youwouldcareabouthow volatilethatstockis,andwemeasurevolatility asastandarddeviationoflogreturnsforthatsecurity.Thatseemsveryfancy,andwe'lltalkaboutthatin moredepthinfuturevideos,butatjustanintuitivelevel,justthinkabout2stocks.Solet'ssaythatthisis stock1rightoverhere,anditjumpsaround,andI'llmakethemgoflat,justsowemakenojudgmentaboutwhetherit'sagoodinvestment.Youhaveonestockthatkindofdoesthat,andthenyouhaveanotherstock.Actually,I'lldrawthemonthesame,solet'ssaythatisstock1,andthenyouhavea stock2thatdoesthis,itjumpsaroundallovertheplace.Sothisgreenoneright overhereisstock2.Youcouldimaginestock2justinthewayweusetheword 'volatile'ismorevolatile.It'sawilderride.Also,ifyouwerelookingat howdispersedthereturnsareawayfromtheirmean,youseeithas,thereturnshavemoredispersion.It'llhaveahigherstandarddeviation.So,stock2willhaveahighervolatility,orahigherstandarddeviation oflogarithmicreturns,andinafuturevideo,we'lltalkaboutwhywecareaboutlogreturns,Stock1wouldhavealowervolatility,soyoucanimagine, optionsaremorevaluablewhenyou'redealingwith,orifyou'redealingwitha stockthathashighervolatility,thathashighersigmalikethis,thisfeelslikeitwoulddrive thevalueofanoptionup.Youwouldratherhaveanoptionwhenyouhavesomethinglikethis,because,look,ifyou'reowningthestock,man,youhavetogoafter, thisisawildride,butifyouhavetheoption, youcouldignorethewildness,andthenitmightactuallymake,andthenyoucouldexercisetheoptionifitseemsliketherighttimetodoit.Soitfeelslike,ifyou werejusttradingit,thatthemorevolatilesomethingis,themorevaluablean optionwouldbeonthat.Nowthatwe'vetalkedaboutthis,let'sactuallylookat theBlack-ScholesFormula.ThevarietythatIhaverightoverhere,thisisforaEuropeancalloption.Wecoulddosomethingvery similarforaEuropeanputoption,sothisisrightoverhere isaEuropeancalloption,andremember,Europeancalloption,it'smathematicallysimpler thananAmericancalloptioninthatthere'sonlyonetime atwhichyoucanexerciseitontheexercisedate.OnanAmericancalloption,youcanexerciseitananypoint.Withthatsaid,let'stryto atleastintuitivelydissecttheBlack-ScholesFormulaalittlebit.Sothefirstthingyouhavehere,youhavethistermthatinvolved thecurrentstockprice,andthenyou'remultiplying ittimesthisfunctionthat'stakingthisasaninput,andthisashowwedefinethatinput,andthenyouhaveminustheexercisepricediscountedback,thisdiscounts backtheexerciseprice,timesthatfunctionagain,andnowthatinputisslightlydifferentintothatfunction.Justsothatwehavea littlebitofbackgroundaboutwhatthisfunctionNis,Nisthecumulativedistributionfunctionforastandard,normaldistribution.Iknowthatseems,might seemalittlebitdaunting,butyoucanlookatthe statisticsplaylist,anditshouldn'tbethatbad.Thisisessentiallysayingfor astandard,normaldistribution,theprobabilitythatyour randomvariableislessthanorequaltox,andanotherwayofthinkingaboutthat,ifthatsoundsalittle,andit'sallexplainedin ourstatisticsplaylistifthatwasconfusing,butifyouwanttothinkabout italittlebitmathematically,youalsoknowthatthisisgoingtobe,it'saprobability.It'salwaysgoingtobegreaterthanzero,anditisgoingtobelessthanone.Withthatoutoftheway,let'sthinkaboutwhat thesepiecesaretellingus.This,rightoverhere,isdealingwith,it's thecurrentstockprice,andit'sbeingweightedby sometypeofaprobability,andsothisis,essentially, onewayofthinkingaboutit,inveryroughterms,isthis iswhatyou'regonnaget.You'regonnagetthestock,andit'skindofbeing weightedbytheprobabilitythatyou'reactually goingtodothisthing,andI'mspeakinginveryroughterms,andthenthistermright overhereiswhatyoupay.Thisiswhatyoupay.Thisisyourexercise pricediscountedback,somewhatbeingweighted,andI'mspeaking,onceagain,I'mhand-weavingalotofthemathematics,bylikeareweactually goingtodothisthing?Areweactuallygoing toexerciseouroption?Thatmakessenserightoverthere,anditmakessenseifthe stockpriceisworthalotmorethantheexerciseprice,andifwe'redefinitelygoingtodothis,let'ssaythatD1andD2are very,verylargenumbers,we'redefinitelygoingtodo thisatsomepointintime,thatitmakessensethat thevalueofthecalloptionwouldbethevalueofthe stockminustheexercisepricediscountedbacktotoday.Thisrightoverhere, thisisthediscounting,kindofgivingusthepresent valueoftheexerciseprice.Wehavevideosondiscounting andpresentvalue,ifyoufindthatalittlebitdaunting.Italsomakessensethatthemore,thehigherthestockpriceis,soweseethatrightoverhere,relativetotheexerciseprice,themorethattheoptionwouldbeworth,italsomakessensethat thehigherthestockpricerelativetotheexerciseprice,themorelikelythatwewill actuallyexercisetheoption.Youseethatinbothof thesetermsrightoverhere.Youhavetheratioofthestock pricetotheexerciseprice.Aratioofthestockprice totheexerciseprice.We'retakinganaturallogofit,butthehigherthisratio is,thelargerD1orD2is,sothatmeansthelargertheinputintothecumulative distributionfunctionis,whichmeansthehigher probabilitieswe'regonnaget,andsoit'sahigherchance we'regonnaexercisethisprice,anditmakessensethatthenthisisactuallygoingtohavesomevalue.Sothatmakessense,therelationshipbetweenthestockpriceandtheexerciseprice.TheotherthingIwillfocuson,becausethistendstobeadeepfocusofpeoplewhooperatewithoptions,isthevolatility.Wealreadyhadanintuition,thatthehigherthevolatility,thehighertheoptionprice,solet'sseewherethisfactors intothisequation,here.Wedon'tseeitatthisfirstlevel,butitdefinitelyfactorsintoD1andD2.InD1,thehigheryourstandard deviationofyourlogreturns,sothehighersigma,wehaveasigmainthe numeratorandthedenominator,butinthenumerator,we'resquaringit.SoahighersigmawillmakeD1goup,sosigmagoesup,D1willgoup.Let'sthinkaboutwhat'shappeninghere.Well,herewehaveasigma.It'sstillsquared.It'sinthenumerator,butwe'resubtractingit.Thisisgoingtogrowfasterthanthis,butwe'resubtractingitnow,soforD2,ahighersigma isgoingtomakeD2godownbecausewearesubtractingit.Thiswillactuallymake,canweactuallysaythisisgoingtomake,ahighersigma'sgoingtomakethevalueofourcalloptionhigher.Well,let'slookatit.Ifthevalueofoursigmagoesup,thenD1willgoup,thenthisinput,thisinputgoesup.Ifthatinputgoesup,ourcumulativedistribution functionofthatinputisgoingtogoup,andsothisterm,thiswholetermisgonna drivethiswholetermup.Now,what'sgoingtohappenhere.Well,ifD2goesdown,thenourcumulativedistribution functionevaluatedthereisgoingtogodown,andsothiswholething isgoingtobelowerandsowe'regoingtohavetopayless.Ifwegetmoreandpayless,andI'mspeakinginveryhand-wavyterms,butthisisjusttounderstandthatthisisasintuitively dauntingasyoumightthink,butitlooksdefinitively,thatifthestandarddeviation,ifthestandarddeviation ofourlogreturnsorifourvolatilitygoesup,thevalueofourcalloption,thevalueofourEuropean calloptiongoesup.Likewise,usingthesamelogic,ifourvolatilityweretobelower,thenthevalueofour calloptionwouldgodown.I'llleaveyouthere.Infuturevideos,we'llthinkaboutthisinalittlebitmoredepth.ImpliedvolatilityUpNextImpliedvolatility