Black-Scholes Option Valuation - 5-Minute Finance

Black-Scholes Formula: C0=S0N(d1)-Xe-rTN(d2) · C0 is the value of the call option at time 0. · S0: the value of the underlying stock at time 0. · N(): the ... menu HOME navigate_beforeBACK {{item.icon}} add_circle close arrow_forward search {{menu.icon}} {{menu.title}} playlist_add All{{menu.title}} {{item.icon}}{{item.title}} SearchResults {{item.icon}}{{item.title}} {{item.icon}}{{item.title}} {{item.icon}}{{item.title}} Noresultsfound. Clear email Contact close Contact Collaborate Courses Home Derivatives Current:BlackScholes SHARE:    OPENASSLIDESHOW BlackScholes     Black-ScholesOptionValuation TheBlack-ScholesWorld Ifweassumethatstockoptionsexistinaworldwhere… …themarketiscomplete,meaninggivenastockandabondwecanreplicatecallandputoptions, therisk-freeinterestrateandstockpricevolatilityarebothconstant, andstockpricesfollowalognormaldistribution… …thenwecanvalueEuropeanCallOptionsonNon-DividendPayingStocksusingtheBlack-ScholesFormulashownonthenextslide. Black-ScholesFormula:`C_0=S_0N(d_1)-Xe^{-rT}N(d_2)` where `d_1=\frac{ln(\frac{S_0}{X})+(r+\frac{\sigma^2}{2})T}{\sigma\sqrt(T)}` `d_2=d_1-\sigma\sqrt(T)` `C_0` isthevalueofthecalloptionattime0. `S_0`:thevalueoftheunderlyingstockattime0. `N()`:thecumulativestandardnormaldensityfunction(NORMSDIST()inExcel) `X`:theexerciseorstrikeprice. `r`:therisk-freeinterestrate(annualized). `T`:thetimeuntiloptionexpirationinyears. `\sigma`:theannualizedstandarddeviationsoflogreturns. `e`and`ln`aretheexponentialandnaturallogfunctionsrespectively(EXP()andLN()inExcel). WhereistheExpectedReturn? ThegroundbreakingfeatureoftheBlack-Scholesmodel,asopposedtoearlierattemptsatoptionvaluation,isthatitdoesnotrequirethestock’sexpectedreturnasaninput. Thisisimportant,becausewedon’tknowthestock’sexpectedreturn. Theideaisthatwedon’tneedtheexpectedreturnbecausewearegoingtohedgeoutriskfromthestock’sreturns(whatevertheyare). Youcanalsoviewtheexpectedreturnasbeingalreadyincludedinthestock’sprice,whichisdeterminedbythestock’slevelofriskandreturn. IntuitiveGraspoftheFormula Lookingattheoptionvalueformula:`C_0=S_0N(d_1)-Xe^{-rT}N(d_2)` Youcanlookatthe`N(d)`termsasthelikelihoodthattheoptionwillbeexercisedatexpirationthat(`S_T>X`). Ifboth`N(d)`termsarecloseto1,thentheoption`S_0-Xe^{-rT}`orthepresentstockpriceminusthepresentvalueofthestrikeprice.Thismakessensegiventhatatexpiration(if`S_0>X`)theoptionpays`S_T-X`. Iftheoptionhaslittlechanceofbeingexercised(thatis,both`N(d)`arenear0),thentheoptionwillhaveanear$0value. Forothervaluesinthe0to1range,thecallvaluecanbeviewedasthepayoff`S_0-Xe^{-rT}`weightedbytheprobabilitythatthecallisexercised. Black-ScholesApp ThefollowingappwillcalculatetheBlack-ScholesEuropeancalloptionpriceforasetofgiveninputs. Ifthestockpaysadividend,theninputthestock’sannualizedexpecteddividendyield. Thecalculatorwilladjustforthedividendbyloweringthestockpricebythepresentvalueoftheexpecteddividend.Inotherwordsthestockpriceusedintheformulawillbe:`S_0e^{-\deltaT}`where`\delta`istheexpectedannualizeddividendyield.Thisassumesdividendsarepaid continuously throughouttheyear.We’lldiscussdividendadjustmentslaterinthepresentation. Youcanusetheapptocheckyourowncalculations.Tohelp,youcanalsochoosetosee`d1`and`d2`toalsocheckthosevalues. HowDoWeCalculatetheInputParameters Thepresentstockpriceiseasilyobservable,andtheexercisepriceandtimetomaturityareaspectsoftheoptioncontract.Theparameterswhicharelesseasilyobservedare: Risk-freerate Dividendyield Volatility TheRiskFreeRate Theriskfreerateshouldbetheannualizedcontinuously-compoundedrateonadefaultfreesecurity withthesamematurity astheexpirationdataoftheoption. Forexample,iftheoptionexpiredin3months,youcanusethecontinuouslycompoundedannualratefora3-monthTreasuryBill. DividendsandStockandOptionPrices Rememberthatastockpriceisadjusteddownwardbythedividendamountwhenthedividendispaid.Forexample,saybeforea$1dividendispaidthestockis$50.Immediatelyafterthedividendispaidthestock’spricewillbe$49(otherwisetherewouldbeanarbitrage). However,stockoptioncontractterms(suchasthestrikeprice)arenotadjustedforcashdividends,orstockdividendsunder10%. Sopayingdividendreducesthevalueofacalloptionandincreasesthevalueofaput. Note,becausefirmsoftenincreaseordecreasetheirdividendpayments,wecanonlyestimatean expected dividendyieldorpayment. DividendAdjustments IntheBlack-Scholesworld(wheretheoptionisEuropean)wecanreducethestockpricebythepresentvalueofallthedividends duringthelifeoftheoption. Thediscountingisdonefromtheex-dividenddatetothepresent. Wecanusetherisk-freerate,thoughthisassumeswearecertainabouttheamountofthedividendpayment. Note,weonlyincludethedividendtobepaidduringthelifeoftheoption.Soiftheoptionexpiresinamonthandthenextdividendpaidbythestockisintwomonths,wedo not includeadividendadjustment. Volatility Volatility(thestandarddeviationoflog-returns)isnotdirectlyobservable,anditisthetoughestinputtodetermine.Twocommonwaystoestimatevolatility: Usehistoricaldata Extractingvolatilityfromotheroptions ImportantNote: VolatilityisassumedtobeconstantintheBlack-Scholesmodel.Thisiswhyyoucanestimatevolatilityoverahistoricalperiodandusethatvolatilityoveralaterperiod.Butthisassumptionwasmadeformathematicalease,anditisnotrealistic. Sointhemodel’sworld,usinghistoricalvolatilityisfine,eventhoughintherealworlditisapoorapproach. HistoricalData Wecanusehistoricalstockpricedatatocalculatecontinuouslycompoundedreturns(log-returns).Wecanthentakethestandarddeviationofthesereturns(using STDEV() inExcel,forexample)andannualizethestandarddeviation,affordinganestimateofannualvolatility. Todoso,wemustchooseoursamplingfrequency(daily,weekly,ormonthlyprices)andtheamountofhistorytouse. Dailypricesoverthelast100daysarecommonlyused. AnnualizingtheStandardDeviation Oncewehavethestandarddeviationoflogreturns,wemustannualizeit.Todosoweusetheequationbelow,where`\sigma_a`and`sigma_p`aretheannualandsampleperiodstandarddeviations: `\sigma_a=\sigma_p\sqrt{\text{#periodsinayear}}` Forexample,ifweareusing100daysofdailypricedata,andthestandarddeviationoverthosedaysis0.05%,then:`\sigma_a=0.05%(252)=12.6%` Aboveweassume252tradingdaysinayear. InteractiveApp Thefollowingappwillcalculateannualizedhistoricalvolatilityforanystockandchoiceofsamplingfrequencyandlengthofhistory. Changethedaterangeandseeifthehistoricalvolatilitychanges–rememberBlack-Scholesassumesconstantvolatility. ExtractingVolatilityfromOtherOptions Wecanalsoextractvolatilityfromsimilaroptionsandusethatnumberasthevolatilityforouroptions. Thisisoftenreferredtoas‘calibratingtothemarket’. Ithasthebenefitofbeingaforward-lookingmeasure,whichwilltakeintoaccountmarketexpectationsofthevolatilityfromfutureevents.Historicalvolatility,ontheotherhand,onlylooksbackwards. ImpliedvsHistoricalVolatility Thisdistinctionisparticularlyimportantifthereisaneventwhichwilltakeplaceduringthelifeoftheoption,whichhasn’thappenedhistorically. Consider,forexample,whattheconsequencesmightbeifyouownoptionsexpiringin3monthsonChevron(CVX)andinonemonthCongresswillvoteonlegislationtoallowunfetteredexportsofcrudeoilfromtheU.S.(exportsarenowsubstantiallylimited). ImpliedVolatility WecanextractavolatilityestimatefromtradedoptionsbypluggingtheoptionpriceintotheBlack-Scholesformulaandsolvingforvolatility.Thisvolatilityestimateiscalledtheoption’s‘impliedvolatility’. ButWhatAboutPutOptions? Sofarwehaveonlylookedatcalloptions,butcanBlack-Scholesalsovalueputoptions? Yes,wecanwriteanexplicitformulafortheBlack-ScholesvalueofaEuropeanputoption. Howeveritismuchmoreconvenienttosimplyuse put-callparity. 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