Binomial Tree, Cox-Ross-Rubinstein, Method

The Cox-Ross-Rubinstein Binomial Tree method is an instance of the Binomial Options Pricing Model (BOPM) , published originally by Cox, Ross and Rubinstein ... VitisQuantitativeFinanceLibrary 2020.1 LibraryOverview Requirements License TrademarkNotice ReleaseNote UserGuide PricingModelsandNumericalMethods Models NumericalMethods MonteCarloSimulation FiniteDifferenceMethods BinomialTree,Cox-Ross-Rubinstein,Method Overview References InternalDesignofTreeLattice Closed-FormSolutionMethods Risk L1ModuleUserGuide L2KernelUserGuide L3OverlayUserGuide BenchmarkResult QualityandPerformance VitisQuantitativeFinanceLibrary » PricingModelsandNumericalMethods» BinomialTree,Cox-Ross-Rubinstein,Method BinomialTree,Cox-Ross-Rubinstein,Method¶ Overview¶ TheCox-Ross-RubinsteinBinomialTreemethodisaninstanceoftheBinomialOptionsPricingModel(BOPM),publishedoriginallybyCox,RossandRubinsteinintheir1979paper“Option Pricing:ASimplifiedApproach”[CRR1979]. Inthismethod,thebinomialtreeisusedtomodelthepropagationofstockpriceintimetowardsasetofpossibilitiesattheExpirationdate,basedonthestockVolatility.For“N”timestepsintowhichthemodelscenariodurationissubdivided,there areN+1possiblestockpricesattheexpirationtime. BasedontheN+1CallorPutOptionvaluesatexpiration,optionvaluesarebackward-propagatedtotheinitialtimeusingstepprobabilitiesandtheinterest-rate,toobtaintheCallor PutOptionprice.ComparingintermediateCall/Putvaluesduringback-propagationtostockpricesallowsAmericanOptionpricestobecalculated. Cox-Ross-RubinsteinshowthatasNtendsto∞,thebinomialEuropeanPut/CallsolutionstendtowardstheBlack-Scholessolutions.(Bothmodelsmakethesameunderlyingassumptions.)InanexamplewhereK=$35.00and N=150,theyshowthedifferenceislessthan$0.01. Inalaterpaper,Leisen&Reimer[LR1995]proposeamethodtoincreasetheconvergencespeedoftheCRRbinomiallatticetoconvergefaster. Thediagramaboveshowsanexampleofabinomialtree,wherethenumberoftimestepsis\(n\).(Notethat\(n\)stepsresultsin\(n+1\)separatepropagated\(S\)valuesafterthen-thstep.) Ateachsteptheinitialstockprice\(S_0\)ispropagatedinanUppathandaDownpathfromeachnode,withUpandDownfactors\(u\)and\(d\).The“Up”probability is\(p\);Downis\(1-p\). Theequationsinthediagramshowthederivation,where\(\sigma\)isthestockvolatility,\(r\)the“risk-freerate”,\(t\)thescenariodurationand\(n\)thenumberoftimesteps.Thedividend yieldintheaboveisassumedtobezeroandnotincludedintheexpressionfor\(p\),butmaybeincludedwhenrequired. (Diagramsource:WikipediaarticleBinomialOptionsPricingModel(BOPM).) References¶ [CRR1979]Cox,J.C.,Ross,S.A.,Rubinstein,M.,“OptionPricing:ASimplifiedApproach”,JournalofFinancialEconomics(1979) [LR1995]Leisen,D.,Reimer,M.,“BinomialModelsforOptionValuation-ExaminingandImprovingConvergence”,RheinischeFriedrich-Wilhelms-Universität,Bonn,(1995). Next Previous