Binomial options pricing model - Wikipedia

The binomial pricing model traces the evolution of the option's key underlying variables in discrete-time. This is done by means of a binomial lattice (Tree), ... Binomialoptionspricingmodel FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Numericalmethodforthevaluationoffinancialoptions Infinance,thebinomialoptionspricingmodel(BOPM)providesageneralizablenumericalmethodforthevaluationofoptions.Essentially,themodelusesa"discrete-time"(latticebased)modelofthevaryingpriceovertimeoftheunderlyingfinancialinstrument,addressingcaseswheretheclosed-formBlack–Scholesformulaiswanting. ThebinomialmodelwasfirstproposedbyWilliamSharpeinthe1978editionofInvestments(ISBN 013504605X),[1]andformalizedbyCox,RossandRubinsteinin1979[2]andbyRendlemanandBartterinthatsameyear.[3] ForbinomialtreesasappliedtofixedincomeandinterestratederivativesseeLatticemodel(finance)§ Interestratederivatives. Contents 1Useofthemodel 2Method 2.1Step1:Createthebinomialpricetree 2.2Step2:Findoptionvalueateachfinalnode 2.3Step3:Findoptionvalueatearliernodes 3RelationshipwithBlack–Scholes 4Seealso 5References 6Externallinks Useofthemodel[edit] TheBinomialoptionspricingmodelapproachhasbeenwidelyusedsinceitisabletohandleavarietyofconditionsforwhichothermodelscannoteasilybeapplied.ThisislargelybecausetheBOPMisbasedonthedescriptionofanunderlyinginstrumentoveraperiodoftimeratherthanasinglepoint.Asaconsequence,itisusedtovalueAmericanoptionsthatareexercisableatanytimeinagivenintervalaswellasBermudanoptionsthatareexercisableatspecificinstancesoftime.Beingrelativelysimple,themodelisreadilyimplementableincomputersoftware(includingaspreadsheet). AlthoughcomputationallyslowerthantheBlack–Scholesformula,itismoreaccurate,particularlyforlonger-datedoptionsonsecuritieswithdividendpayments.Forthesereasons,variousversionsofthebinomialmodelarewidelyusedbypractitionersintheoptionsmarkets.[citationneeded] Foroptionswithseveralsourcesofuncertainty(e.g.,realoptions)andforoptionswithcomplicatedfeatures(e.g.,Asianoptions),binomialmethodsarelesspracticalduetoseveraldifficulties,andMonteCarlooptionmodelsarecommonlyusedinstead.WhensimulatingasmallnumberoftimestepsMonteCarlosimulationwillbemorecomputationallytime-consumingthanBOPM(cf.MonteCarlomethodsinfinance).However,theworst-caseruntimeofBOPMwillbeO(2n),wherenisthenumberoftimestepsinthesimulation.MonteCarlosimulationswillgenerallyhaveapolynomialtimecomplexity,andwillbefasterforlargenumbersofsimulationsteps.MonteCarlosimulationsarealsolesssusceptibletosamplingerrors,sincebinomialtechniquesusediscretetimeunits.Thisbecomesmoretruethesmallerthediscreteunitsbecome. Method[edit] functionamericanPut(T,S,K,r,sigma,q,n) { 'T...expirationtime 'S...stockprice 'K...strikeprice 'q...dividendyield 'n...heightofthebinomialtree deltaT :=T/n; up :=exp(sigma*sqrt(deltaT)); p0 :=(up*exp(-q*deltaT)-exp(-r*deltaT))/(up^2-1); p1 :=exp(-r*deltaT)-p0; 'initialvaluesattimeT fori :=0ton{ p[i] :=K-S*up^(2*i-n); ifp[i]<0thenp[i] :=0; } 'movetoearliertimes forj :=n-1downto0{ fori :=0toj{ 'binomialvalue p[i] :=p0*p[i+1]+p1*p[i]; 'exercisevalue exercise :=K-S*up^(2*i-j); ifp[i]