The Binomial-tree Option Calculator - StudyLib

The methods used in the calculations are the well-known Cox-Ross-Rubinstein's binomial model and a few others. The results are compared with the Black-Scholes ... Studylib Documents Flashcards Chromeextension Login Uploaddocument Createflashcards × Login Flashcards Collections Documents Lastactivity Mydocuments Saveddocuments Profile Business Finance FinancialModeling TheBinomial-treeOptionCalculator advertisement TheBinomial-treeOptionCalculator Thisisashortdocumentationofhowtousethephp-programs;   Binomial-treeOptionCalculatorand Binomial-treeIterationAnalyser forusingthebinomialmethodsforcalculationsonoptions.Themethodsusedinthe calculationsarethewell-knownCox-Ross-Rubinstein'sbinomialmodelandafew others.TheresultsarecomparedwiththeBlack-Scholesvalueifthecalculationsare madeforEuropeanoptionsofAmericancalloptions. TheoutputcalledProbabilityistheprobabilitythattheoptionwillhaveapositive valueatthetimeformaturity. Inputdata Firstofallyouhavetofillintheformwiththeneededinputdata.Thisis:       Theunderlyingassetprice, Theoptionstrikeprice, Timetomaturity, Theriskfreeinterestrate, Thevolatilityand Thenumberofbinomialsteps(minimumandmaximumvaluefortheIteration Analysor). Ifyouwanttocalculatetheimpliedvolatilityforagivenoption,youalsohaveto give:  Thesimulatedoptionprice. Whenalltheinputdataisgivenyoumustselect:    Theoptionexercisestyle, Theoptiontypeand Thetypeofbinomialtreetouse. Calculate(withthecalculator) Now,youcanstartthecalculationbypressingCalculate.Withtheselectedmetodthe followingoutputisgiven:        Optionprice Delta Gamma Theta Vegaand Rho Probabilitytoreachstrikepriceatmaturity. IftheoptionisofEuropeantypeoranAmericancalloption,thevaluesarecompared withthevaluesgivenbyBlack-Scholes. TheprobabilityiscalculatedbytheBlack-Scholesformulasinceweonlycalculatethe probabilitytoreachthestrikeattheenddate. IfyouwanttoSavethecalculateddata,youhavetogivethefilenamesforeachofthe valuesyouliketosave.Ifyousavetheresultsyoucanusetheseresultstocompare differentoptionsingraphs.Thisisexplainedbelow.Youcanusetheradiobuttonsto selectifyouwanttosavetheresultasfunctionoftimeorasfunctionofthe underlyingassetprice. Duringthecalculationyoucanalsosavethebinomialtrees.Thetreesaresavedon filesifthecheckboxfor“Savethetreeonfile”arechecked.Seebelowforcomments onthetrees. ThebuttonClear,clearsalltheinputdataandletyoustartfromthebeginning. NumericalMethods Agreatadvantagesofthisprogram,isthatyoueasilycanchangethetypeoftreeto useinthecalculations.Thisgivesyouthepossibilitytoinvestigatethedifferent resultsgivenbythemodels.Youcanalsocreateplotstoexamineandcomparehow thetree-modelsconverge.Thegraphicalpossibilitiesaredescribedbelow. Thetrees Youcanselectbetweenthefollowingtrees:         CRR–Cox,RossRubenstein[1979]tree, CRR2–Cox,RossRubensteintree,(amodifiedCRRmodel) JR –JarrowRudd[1983]tree, TIAN–Tian[1993]tree, TRG–Trigeorgis[1991]tree, LR –LeisenReimer[1995]tree, LRRE–TheLeisenReimertreewithRichardsonextrapolation, TRI–Trinomialtreeand NEW!!Ihavealsoaddedsomenewcalculationstrategies.Theyarecalled,e.g. CRR++andCRR++REetc,andtheyneedanexplanation.Inthe++strategiesI replacethevalueoftheoptioninsomeofthenodes,closesttomaturitywiththevalue givenbyBlack-Scholes.Thisstrategywillremovemuchoftheoscillationsinthe convergencewhenthenumberofnodesincreases.With++REIalsouseRichardson extrapolationonthe“++”result.Thisispossiblebecausewehaveremovedthe oscillations. SincethisisNOTadocumentwiththeaimtogiveafulldescriptionofthebinomial method,Ireferthereadertotheliterature.Igivesomeofmyreferencesintheendof thisdocument.But,Iwillgiveashotdescriptionoftheparametersusedinthetrees. Alltreesarebuiltfromfourparameters(u,d,pandq).Theparametersuanddtell howmuchtheunderlyingwillgoupordownineachdiscretetime,andthe parameterspandqistheprobabilitiesforthepricetogoupanddownrespectively. Thereforep+q=1. CRRmodel t  ue    d1/ue p t ertd ud CRR-2model 2  a2b21a2b214a2  u 2a  d1/u aert  2 2 2t 1 bae   p JRmodel ertd ud   12  rtt ue2  12 rtt  2 d  e  p ertd ud TIANmodel MV  u2  dMV  2 V1V22V3   V1V22V3   Mert  2t Ve p ertd ud TRGmodel IntheTRGmodelthelogarithmoftheprice(insteadofthepriceitself)isusedto buildthetree.Therefore,whenthetreeisbuildweaddu=dxandsubtractbyd=dx. 2  12 2  2 dxtrt  2   11 12dx  p22r2t    LRmodel Thismodeldiffersfromtheothermethodsinacommonsense.Themodelconverges muchfasterthantheotheronesanddonotoscillate.Thereasonforthefast convergenceisthechoiceofparameters.WiththeLR-parametersthestrikepriceis centredinthetreeandtheprobabilitiesaregivenbyDeMoivre-Laplacetheorem, whichisanapproximationofthenormaldistribution.ForthereaderIsuggestthe articlebyLeisenandReimer(seereferences).Butashortexplanationisgivenhere: aert 1  ln(S/K)r2Tt 2  d1 Tt d2d1Tt pB(d2,N) pB(d2Tt,N) WhereBistheinverseofthebinomialdistributionandNthenumberofrefinements. WeusethePeizer-Prattmethod[case:j+½=n–(j+½),n=2j+1]: 1 2   2  111 1 z p exp n 1 244 6   n    3    Thenwehave  p ua p   da1p  1p TRImodel IntheTRImodelwecalculatethreenewnodesineachtime-step.Thismeansthe pricecangoupwithu,downwithdorstaythesame,withtheprobabilitiespu,pdand pm:     2 2 1tr0.5t pu 2 r0.52t 2t t pm1 2 2tr0.52t 2t     2 2 1tr0.5t pd 2 r0.52t 2 t t 2 2 t  ue    d1/ue t Graphics Themostimportantfunctionsintheprogramarethegraphicalpossibilities.Belowthe input/outputformthereareanumberofbuttonsthatwillgiveyouanumberofgraphs.       PriceCurves Delta Gamma Theta Vega Rho Eachofthesebuttonscreatesthreedifferentgraphs;thegivenvariableasfunctionof theunderlyingvalue,asfunctionoftimeandasfunctionofthevolatility.Ifthe checkbox“Printinputdataintheplots”ischecked,aboxwiththeinputvaluesis presentedintheupperrightcorner. WiththeAvista–buttonyoucanalsoplottheimpliedstockpriceasfunctionoftime thatkeepstheoptionvalueconstant.Thisisaveryimportantplotbecausethetimevalueofanoption“eats”theoptionvalueduringtheoptionlifetime.Thiscurve showshowmuchtheunderlyingstockprice,havetoincreasetocompensateforthe lossofthetimevalue. WiththeIterationAnalysoryoucanplothowtheoptionpricefordifferentbinominal methodconverges.IfyouothisforanEuropeanoptionoranAmericancalloption, theconvergenceiscomparedwiththevaluegivenbytheBlack-Scholesmethod. MoreGraphics Ifyougivefilenamesfortheoutputvariables,thedataissavedontheserver.Witha programonhttp://janroman.dhis.org/DataGraph.phpyoucangivethesamefilename andplotthedata.Withthisoptionyoucancomparegraphsfore.g.,differentstrikes ordifferenttimestomaturity. Aprintableresult WiththebuttonPrintableyougetaprintableversionofthecalculationresult.This canbeprintedorcopiedandpastedintoadocumentetc. References [1] [2] [3] [4] [5] [6] [7] [8] Cox,J.,Ross,S.A.,RubensteinM.(1979):“OptionPricing:Asimplified Approach”,JournalofFinancialEconomics7,1979,pp.145-166. HullJ.“OptionFuturesandotherDerivatives”Prentice-Hall,NewJersey. Jarrow,R.RuddA.(1983):“OptionPricing”.Homewood,Illinois1983, pp.183-188. Pratt,J.W.(1968):“ANormalApproximationforBinomial,F,Beta,and OtherCommon,RelatedTailProbabilities,II”,TheJournaloftheAmerican StatisticalAssociation,Bd.63.1968,pp.1457-1483. Tian,Y.(1993):“AModifiedLatticeApproachtoOptionPricing”,Journal ofFuturesMarkets,Vol13,No.5,pp.564-577. Trigeorgis,Lenos(1991):“ALog-transformedBinomialNumericalAnalysis forValuingComplexMulti-OptionInvestments”,JournalofFinancialand QuantitativeAnalysis26,No.3,September1991,pp.309-326. Leisen,D.,Reimer,M.(1996):”BinomialModelsforOptionValuation– ExamineandImprovingConvergence”.AppliedMathematicalFinance,vol.3 1996,pp.319-346. Leisen,D.,(1998):“PricingtheAmericanputoption:Adetailedconvergence analysisforbinomialmodels”.JournalofEconomicDynamicsandControl. 22(1998),pp.1419-1444. Relateddocuments AFurtherAnalysisofConvergenceRateandPatternoftheBinomialModels Download advertisement Addthisdocumenttocollection(s) Youcanaddthisdocumenttoyourstudycollection(s) Signin Availableonlytoauthorizedusers Title Description (optional) Visibleto Everyone Justme Createcollection Addthisdocumenttosaved Youcanaddthisdocumenttoyoursavedlist Signin Availableonlytoauthorizedusers   SuggestushowtoimproveStudyLib (Forcomplaints,use anotherform ) Youre-mail Inputitifyouwanttoreceiveanswer Rateus 1 2 3 4 5 Cancel Send