Binomial Trees | AnalystPrep - FRM Part 1 Study Notes and ...

Binomial models with one or two steps are unrealistically simple. Assuming only one or two steps would yield a very rough approximation of the option price. In ... LimitedTimeOffer:Save10%onall2021and2022PremiumStudyPackageswithpromocode:BLOG10  SelectyourPremiumPackage » part-1valuation-and-risk-management BinomialTrees 20Sep2019 Aftercompletingthisreadingyoushouldbeableto: CalculatethevalueofanAmericanandaEuropeancallorputoptionusingaone-stepandtwo-stepbinomialmodel. Describehowvolatilityiscapturedinthebinomialmodel. Describehowthevaluecalculatedusingabinomialmodelconvergesastimeperiodsareadded. Defineandcalculatethedeltaofastockoption. Explainhowthebinomialmodelcanbealteredtopriceoptionsonstockswithdividends,stockindices,currencies,andfutures. PricingOptionsUsingtheBinomialModel Thebinomialoptionpricingmodelisasimpleapproximationofreturnswhich,uponrefining,convergestotheanalyticpricingformulaforvanillaoptions.ThemodelisalsousefulforvaluingAmericanoptionsthatcanbeexercisedbeforeexpiry. Themodelcanberepresentedas: $$ \begin{array} \hline {}&{\smallP}&{S}_{0}u\\ {S}_{0}&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\ {}&{\small1-P}&{S}_{0}d\\ \end{array}$$ Thenotationusedisasfollows: \({S}_{0}\)=stockpricetoday \(P\)=probabilityofapricerise \(u\)=Thefactorbywhichthepricerises \(d\)=Thefactorbywhichthepricefalls Overasmalltimeinterval\(\Deltat\),thepricetodayrisesorfallstooneofonlytwopotentialfuturevalues:\({S}_{0}u\),and\({S}_{0}d\). Theunderlyingpriceisassumedtofollowarandomwalk. Risk-neutralValuation Thefollowingformulaareusedtopriceoptionsinthebinomialmodel: \(u\)=sizeoftheupmovefactor=\({e}^{\sigma\sqrt{t}}\),and \(d\)=sizeofthedownmovefactor=\({e}^{-\sigma\sqrt{t}}=\frac{1}{{e}^{\sigma\sqrt{t}}}=\frac{1}{u}\) \(\sigma\)istheannualvolatilityoftheunderlyingasset’sreturnsand\(t\)isthelengthofthestepinthebinomialmodel. \({\pi}_{u}=\)probabilityofanupmove=\(\frac{{e}^{rt}-d}{u-d}\) \({\pi}_{d}\)=probabilityofadownmove=\(1-{\pi}_{u}\) Let\(f_u\)bethevalueofanoptionwhenthepricegoesupand\(f_d\)thevaluewhenthepricegoesdown.  Thevalue,\(f\)oftheoption,foronestep-binomialisgivenby: $$f=e^{-rt}\left(\pif_u+(1-\pi)f_d\right)$$ Where, $${\pi}=\frac{{e}^{rt}-d}{u-d}$$ Example:Risk-neutralValuation Thepriceofanexchange-quotedzero-dividendshareis$30.Overthepastyear,thestockhasexhibitedastandarddeviationof17%.Thecontinuouslycompoundedrisk-freerateis5%perannum.Computethevalueofa1-yearEuropeancalloptionwithastrikepriceof$30usingaone-periodbinomialmodel: Theup-anddown-movefactorsare: $$\begin{align*}u&={e}^{0.17\times\sqrt{1}}=1.1853\\ d&=\frac{1}{1.1853}=0.8437\end{align*}$$ Therisk-neutralprobabilitiesofanup-anddown-moveare: $$\begin{align*}{\pi}_{u}&=\frac{\left({e}^{0.05\times1}\right)-D}{U-D}=\frac{1.0513-0.8437}{1.1853-0.8437}=0.61\\  {\pi}_{d}&=1-0.61=0.39\end{align*}$$ Exhibit1:BinomialTree–Stock $$ \begin{array} \hline {}&{}&1.1853\times$30=$35.60\\ $30&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\ {}&{}&0.8437\times$30=$25.30\\ \end{array}$$ Exhibit2:BinomialTree–Option $$ \begin{array} \hline {}&{}&\max\left(0,$35.6-$30\right)=$5.6\\ {c}_{0}&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\ {}&{}&\max\left(0,$25.3-$30\right)=$0\\ \end{array}$$ Theexpectedvalueoftheoptioninoneyearisgivenby: $${c}_{u}\times{\pi}_{u}+{c}_{d}\times{\pi}_{d}=$5.6\times0.61+$0\times0.39=$3.42$$ Theexpectedvalueoftheoptionatpresentisgivenby: $${c}_{0}=$3.42{e}^{\left(-0.05\times1\right)}=$3.25$$ Two-stepBinomialmodel Inthetwo-periodmodel,thetreeisexpandedtocreateroomforagreaternumberofpotentialoutcomes.Exhibit3belowpresentsthetwo-periodstockpricetree: $$ \begin{array} \hline {}&{}&{}&{}&{S}_{uu}\\ {}&{}&{S}_{u}&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\ {S}_{0}&{\begin{matrix}\\\begin{matrix}\begin{matrix}\quad\quad\quad\Huge\diagup\\\end{matrix}\\\quad\quad\quad\Huge\diagdown\end{matrix}\\\end{matrix}}&{}&{}&{S}_{ud}\quador\quad{S}_{du}\\ {}&{}&{S}_{d}&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\ {}&{}&{}&{}&{S}_{dd}\\ \end{array}$$ Thetwo-stepmodelusesthesameformulaeusedintheone-stepversiontocalculatethevalueofanoption.However,here,wereplace\(t\)with\(\Deltat\),whichisthelengthofone-step.Ifwehavesay,anoptionthatmaturesinoneyearperiod,thenforatwo-stepbinomialmodel,\(\Deltat=1/2=0.5\) Thus,thevalueofanoptionisgivenby: $$f=e^{-r\Deltat}\left(\pif_u+(1-\pi)f_d\right)$$ and  $${\pi}=\frac{{e}^{r\Deltat}-d}{u-d}$$ Example:Two-StepBinomial Thepriceofanexchange-quotedzero-dividendshareis$30.Overthepastyear,thestockhasexhibitedastandarddeviationof17%.Thecontinuouslycompoundedrisk-freerateis5%perannum.Computethevalueofa1-yearEuropeancalloptionwithastrikepriceof$30usingatwo-stepbinomialmodel Theup-anddown-movefactorsare: $$\begin{align*}u&={e}^{0.17\times\sqrt{0.5}}=1.1277\\ d&=\frac{1}{1.1277}=0.8867\end{align*}$$ Sothat, $$ \begin{array}\hline {}&{}&{}&{}&{S}_{uu}=38.15\\ {}&{}&{S}_{u}=33.83&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\{S}_{0}=30&{\begin{matrix}\\\begin{matrix}\begin{matrix}\quad\quad\quad\Huge\diagup\\\end{matrix}\\\quad\quad\quad\Huge\diagdown\end{matrix}\\\end{matrix}}&{}&{}&{S}_{ud}=30\quador\quad{S}_{du}=30\\ {}&{}&{S}_{d}=26.60&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\{}&{}&{}&{}&{S}_{dd}=23.59\\ \end{array}$$ Theoptionvaluesare $$\begin{array}{l|l|l} {S}_{uu}=$38.15&{f}_{uu}=\max\left(⁡$38.15-$30,0\right)&{f}_{uu}=$8.15\\\hline {S}_{ud}=$30&{f}_{ud}=\max\left(⁡$30-$30,0\right)&{f}_{ud}=$0\\\hline {S}_{du}=$30&{f}_{du}=\max\left(⁡$30-$30,0\right)&{f}_{du}=$0\\\hline {S}_{dd}=$23.59&{f}_{dd}=\max\left(⁡$23.59-$30,0\right)&{f}_{dd}=$0\\ \end{array}$$ Therisk-neutralprobabilityis: $$\pi=\frac{\left({e}^{0.05\times0.5}\right)-d}{u-d}=\frac{1.0253-0.8867}{1.1277-0.8867}=0.58$$ Thus, $$f_u=e^{-r\Deltat}\left(\pif_{uu}+(1-\pi)f_{ud}\right)=e^{-0.05\times0.5}\left(0.58\times8.15+0.42\times0\right)=4.6103$$ and $$f_d=e^{-r\Deltat}\left(\pif_{ud}+(1-\pi)f_{dd}\right)=e^{-0.05\times0.5}\left(0.58\times0+0.42\times0\right)=0$$ Thus,thevalueoftheoptionis  $$f=e^{-r\Deltat}\left(\pif_u+(1-\pi)f_d\right)=e^{-0.05\times0.5}\left(0.58\times4.6102+0.42\times0\right)=2.6079$$ Note:Thevalueofaputcanbecalculatedoncethevalueofthecallhasbeendetermined,usingtheput-callparityrelationship. $$\text{CallPrice}+\text{PVofStrikePrice}=\text{PutPrice}+\text{StockPrice}$$ IncreasingtheNumberofStepsintheBinomialModel Binomialmodelswithoneortwostepsareunrealisticallysimple.Assumingonlyoneortwostepswouldyieldaveryroughapproximationoftheoptionprice.Inpractice,thelifeofanoptionisdividedinto30ormoretimesteps.Ineachstep,thereisabinomialstockpricemovement. Asthenumberoftimestepsisincreased,thebinomialtreemodelmakesthesameassumptionsaboutstockpricebehaviorastheBlack–Scholes–Mertonmodel.WhenthebinomialtreeisusedtopriceaEuropeanoption,thepriceconvergestotheBlack–Scholes–Mertonpriceasthenumberoftimestepsisincreased. Delta Thedelta,Δ,ofastockoption,istheratioofthechangeinthepriceofthestockoptiontothechangeinthepriceoftheunderlyingstock.Itisthenumberofunitsofthestockaninvestor/tradershouldholdforeachoptionshortedinordertocreatearisklessportfolio.Thisprocessiscalleddelta-hedging. Thedeltaofacalloptionisalwaysbetween0and1becauseastheunderlyingassetincreasesinprice,calloptionsincreaseinprice. Thedeltaofaputoption,ontheotherhand,isalwaysbetween-1and0becauseastheunderlyingsecurityincreases,thevalueofputoptionsdecrease. Forinstance,supposethatwhenthepriceofastockchangefrom$20to$22,thecalloptionpricechangesfrom$1to$2.Wecancalculatethevalueofdeltaofthecallas: $$\frac{2-1}{22-20}=0.5$$ Thismeansthatiftheunderlyingstockincreasesinpriceby$1pershare,theoptiononitwillriseby$0.5pershare,allelsebeingequal. Supposethataninvestorislongonecalloptiononthestockabove(withadeltaof0.5,or50sinceoptionshaveamultiplierof100).Theinvestorcoulddeltahedgethecalloptionbyshorting50sharesoftheunderlyingstock.Conversely,iftheinvestorislongone put onthestock(withadeltaof-0.5,or-50),theywouldmaintainadeltaneutralpositionbypurchasing50sharesoftheunderlyingstock. Generally, $$\Delta=\frac{f_u-f_d}{S_u-S_d}$$ HowVolatilityisCapturedintheBinomialModel Asthestandarddeviationincreases,sodoesthedivide(dispersion)betweenstockpricesinupanddownstates(\({S}_{u}\)and\({S}_{d}\),respectively).Supposetherewasnodeviationatall.Wouldwehaveabinomialtreeinthefirstplace?Theanswerisno. Withzerostandarddeviation,(\({S}_{u}\)wouldbeequalto\({S}_{d}\),andinsteadofatree,wewouldhaveastraightline.Butprovidedthere’ssomedeviation,thegapbetweenstockpricesintheupstateandstockpricesinthedownstateincreasinglywidensasthedeviationincreases. Tocapturevolatility,therefore,itwouldbeparamounttoevaluatestockpricesateachtimeperiodpresentinthetree. HowtheModifiedBinomialModelcanbeAlteredtoPriceOptionsonNon-zeroDividendStocks,StockIndices,Currencies,andFutures Givenastockthatpaysacontinuousdividendyield\(q\),thefollowingformulacanbeusedtopricetheresultingoption: $$\begin{align*}\text{Probabilityofanupmove}&={\pi}_{u}=\frac{{e}^{\left(r-q\right)t}-d}{u-d}\\\text{Probabilityofadownmove}=1-{\pi}_{u}\end{align*}$$ \(u\)=sizeoftheupmovefactor=\({e}^{\sigma\sqrt{t}}\),and \(d\)=sizeofthedownmovefactor=\({e}^{-\sigma\sqrt{t}}=\frac{1}{{e}^{\sigma\sqrt{t}}}=\frac{1}{u}\) Note:Thesizesoftheupmovefactoranddownmovefactorarethesameasinthezero-dividendmodel. Sometimesitmayalsobenecessarytopriceoptionsconstructedwithastockindexastheunderlying,forinstance,anoptionontheS&P500index.Suchanoptionwouldbevaluedinamannersimilartothatofthedividend-payingstock.It’sassumedthatthestocksformingpartoftheindexpayadividendyieldequalto\(q\). Thebinomialmodelcanalsobemodifiedtoincorporatetheuniquecharacteristicsofoptionsonfutures.Ofnoteisthefactthatfuturescontractsarelargelyconsideredcost-freetoinitiate,andthereforeinarisk-neutralenvironment,theyarezero-growthinstruments.Theonlyformulathatchangesisthatoftheprobabilityofanupmove,where: $${\pi}_{u}=\frac{1-d}{u-d}$$ Whendealingwithoptionsoncurrencies,aplausibleassumptionisthatthereturnearnedonaforeigncurrencyassetisequaltotheforeignrisk-freerateofinterest.Assuch,theprobabilityofanupmoveisgivenby: $${\pi}_{u}=\frac{{e}^{\left({r}_{DC}-{r}_{FC}\right)t}-d}{u-d}$$ AmericanOptions TovalueanAmericanoption,wecheckforearlyexerciseateachnode.Ifthevalueoftheoptionisgreaterwhenexercised,weassignthatvaluetothenode.Ifthat’snotthecase,weassignthevalueoftheoptionunexercised.Wethenworkbackwardthroughthetreeasusual. TheBinomialModelWhenTimeisContinuous Thebinomialmodelisessentiallyadiscrete-timemodelwhereweevaluateoptionvaluesatdiscretetimes,say,intervalsofoneyear,intervalsofsixmonths,intervalsofthreemonths,etc. However,ifweweretoshrinkthelengthoftimeintervalstoarbitrarilysmallvalues,we’dendupwithacontinuous-timemodelwherethepricecanmoveatnon-discretetimes.Thebinomialmodelconvergestothecontinuous-timemodelwhentimeperiodsaremadearbitrarilysmall. Questions Question1 Supposewehavea6-monthEuropeancalloptionwith\(K=$23\).Supposethestockpriceiscurrently$22andintwo-timestepsofthreemonths,thestockcangoupordownby10%.Theupmovefactoris1.1whilethedownmovefactoris0.9.Therisk-freerateofinterestis12%. Computethevalueofthecalltoday. $2 $1.54 $1.45 $0 ThecorrectanswerisC. $$ \begin{array} {}&{}&{}&{S}_{uu}=$26.62\\ {}&{S}_{u}=$24.2&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\ {S}_{0}=$22\begin{matrix}&{}&\\&\Huge\diagup&\\&\Huge\diagdown&\\&{}&\end{matrix}&{}&\begin{matrix}{}\\\\{}\end{matrix}&\begin{matrix}{S}_{ud}=$21.78\\\\{S}_{du}=$21.78\end{matrix}\\ {}&{S}_{d}=$19.8&{\Huge\begin{matrix}\diagup\\\diagdown\end{matrix}}&{}\\ {}&{}&{}&{S}_{dd}=$17.82\\ \end{array}$$ \({S}_{u}=22\times1.1=24.2,\) \({S}_{uu}=22\times1.1\times1.1\) Othervaluesatothernodesarecalculatedusingtherelevantup/downfactors. $$\begin{align*}{\pi}_{u}&=\frac{{e}^{rt}-d}{u-d}=\frac{{e}^{0.12\times0.25}-0.9}{1.1-0.9}=0.6523,\\{\pi}_{d}&=1-0.6523=0.3477\end{align*}$$ Let\(f\)representthevalueofthecall: $$ \begin{array}{l|l|l} {S}_{uu}=26.62&{f}_{uu}=\max\left(⁡$26.62-$23,0\right)&{f}_{uu}=$3.62\\\hline {S}_{ud}=$21.78&{f}_{ud}=\max\left(⁡$21.78-$23,0\right)&{f}_{ud}=$0\\\hline {S}_{du}=$21.78&{f}_{du}=\max\left(⁡$21.78-$23,0\right)&{f}_{du}=$0\\\hline {S}_{dd}=$17.82&{f}_{dd}=\max\left(⁡$17.82-$23,0\right)&{f}_{dd}=$0\\ \end{array} $$ Theexpectedvalueofthecallsixmonthsfromnowisgivenby: $$\begin{align*}&0.6523\times0.6523\times$3.62+0.6523\times0.3477\times$0\\&+0.3477\times0.6523\times$0+0.3477\times0.3477\times$0\\&=$1.54\end{align*}$$ $$\text{Valueofthecalltoday}=\(\frac{$1.54}{{e}^{0.12\times0.5}}=$1.45$$ Question2 A1-year$50strikeEuropeancalloptionexistson\(ABC\)stockcurrentlytradingat$49.\(ABC\)paysacontinuousdividendof3%andthecurrentcontinuouslycompoundedrisk-freerateis4%.Assuminganannualstandarddeviationof3%,computethevalueofthecalltoday. $0.31 $0.30 $0.47 $0 ThecorrectanswerisB. $$\begin{align*}u&={e}^{\sigma\sqrt{t}}={e}^{0.03\times1}=1.03\\d&=\frac{1}{1.03}=0.97\end{align*}$$ Notethatthestockisdividend-paying,andthereforetheformulafortheprobabilityofanupmoveisgivenby: Probabilityofanupmove=\({\pi}_{u}=\frac{{e}^{\left(r-q\right)t}-d}{u-d}=\frac{{e}^{\left(0.04-0.03\right)1}-0.97}{1.03-0.97}=0.67\) Probabilityofadownmove=\(1-0.67=0.33\) Let\(S\)representthepriceofthestockand\(f\)representthevalueofthecall.Thisisaone-stepbinomialprocess. $$ \begin{array}{l|l} {S}_{u}=$49\times1.03=$50.47&{f}_{u}=\max\left($50.47-$50,0\right)=$0.47\\\hline {S}_{d}=$49\times0.97=$47.53&{f}_{d}=\max\left($47.53-$50,0\right)=$0\\ \end{array} $$ $$\begin{align*}\text{Valueofthecalloptiononeyearfromtoday}&=\left($0.47\times0.67+$0\times0.33\right)=$0.31\\\text{Valueofthecalltoday}&=\frac{$0.31}{{e}^{0.04}}=$0.30\end{align*}$$ × Share Copy Addedtoclipboard × Featured SwapsPrinciplesforSoundStressTesting–PracticesandSupervisionCountryRisk:Determinants,Measures,andImplicationsViewMore StudywithUs CFA®ExamandFRM®ExamPrepPlatformofferedbyAnalystPrep StudyPlatform LearnwithUs Subscribetoournewsletterandkeepupwiththelatestandgreatesttipsforsuccess OnlineTutoring Ourvideosfeatureprofessionaleducatorspresentingin-depthexplanationsofalltopicsintroducedinthecurriculum. VideoLessons DanielGlyn2021-03-24IhavefinishedmyFRM1thankstoAnalystPrep.AndnowusingAnalystPrepformyFRM2preparation.ProfessorForjanisbrilliant.Hegivessuchgoodexplanationsandanalogies.Andmorethananythingmakeslearningfun.AbigthankyoutoAnalystprepandProfessorForjan.5starsalltheway!michaelwalshe2021-03-18ProfessorJames'videosareexcellentforunderstandingtheunderlyingtheoriesbehindfinancialengineering/financialanalysis.TheAnalystPrepvideoswerebetterthananyoftheothersthatIsearchedthroughonYouTubeforprovidingaclearexplanationofsomeconcepts,suchasPortfoliotheory,CAPM,andArbitragePricingtheory.WatchingtheseclearedupmanyoftheunclaritiesIhadinmyhead.Highlyrecommended.NykaSmith2021-02-18EveryconceptisverywellexplainedbyNilayArun.kudostoyouman!BadrMoubile2021-02-13Veryhelpfull!AgustinOlcese2021-01-27Excellentexplantions,veryclear!JaakJay2021-01-14Awesomecontent,kudostoProf.JamesFrojansindhushreereddy2021-01-07CrispandshortpptofFrmchaptersandgreatexplanationwithexamples.Trustpilotratingscore:4.7of5,basedon61reviews. PreviousPostValuationandRiskManagement NextPostOptionsMarkets RelatedPosts part-1valuation-and-risk-management Aug02,2019 CalculatingandApplyingVaR Aftercompletingthisreading,youshouldbeableto:Explainandgiveexamples...ReadMore foundations-of-risk-managementpart-1 Jun29,2019 ApplyingtheCAPMtoPerformanceMeasu... Aftercompletingthisreading,youshouldbeableto:Calculate,compare,andevaluate...ReadMore financial-markets-and-productspart-1 Sep20,2019 OptionsMarkets Aftercompletingthisreading,youshouldbeableto:Describethevarioustypes,...ReadMore part-1quantitative-analysis Sep29,2019 CommonUnivariateRandomVariables Aftercompletingthisreading,youshouldbeableto:Distinguishthekeyproperties...ReadMore LeaveaCommentCancelreplyYoumustbeloggedintopostacomment. X StartstudyingforFRMorSOAexamsrightaway! Registerforfree