option pricing （巴克莱银行ppt）课件

OP102OptionPricing MikePawley Aims Bytheendofthismoduleyoushould UnderstandthebinomialoptionpricingmodelBeabletocalculateanoptionpriceusingthebinomialmethodUnderstandthebroadconceptsbehindtheBlack ScholesmodelBeabletocalculateanoptionpriceusingtheBlack Scholesmethod PubGame StartingPosition SX WinNothing WinaPrize OptionPremiumDefinition Theprobability weightedintrinsicvaluediscountedbacktotoday Wine Option Theforwardpriceisbasedonthespotpriceadjustedforcostofcarry Themarketisthuswillingtosellwineforwardatapriceof 110percase OneYearWineoption Marketpriceofwine 100 One yearinterestrate 10 Wine ofcourse paysnoyield InitialPriceAssumption Let sassumethatthemarketforthisspecialwineisabitodd weknowwithcertaintythatthepriceoneyearfromtodaywillbeeither 120or 83 Yourclientwouldliketopurchasea1 yearat the moneycalloption strike 110 onthewine Howmuchshouldyouchargeforthecall suchthatyoucanhedgeitwithouttakinganyrisk InitialPriceAssumption Marketrisesto 120 Marketfallsto 83 Marketprice 100 ShortCallRisk CollectPremiumof Marketrisesto 120 Marketfallsto 83 Call 10Wine 0Premium Net 10 Ifwesimplysellthecallandputonnohedge weappeartobehedgedifthepriceofwinefalls Butwewillobviouslylosemoneyifthepricerises Call 0Wine 0Premium Net 0 BasicHedge Theonlywaytohedgethiscallistobuywinetodayandpaythe10 financingcost orbuywineforwardat 110 whichisthesamething Ifwebuyawholecaseofwineforwardwewouldappeartobehedgedifthepriceofwinerises Butwheredowestandifthepricefalls BasicHedge Marketrisesto 120 Marketfallsto 83 Call 10Wine 10Premium Net 0 Call 0Wine 27Premium Net 27 Buy1caseofwindforwardat 110CollectPremiumof Clearly thisisnottherightapproach either Hedge Thesolutiontothisproblemisbasedontakingapartialpositioninwine1 yearforward Whatweneedisapositionthathasequalgainswhetherthepricemovesupordown Ifwecanfindthisposition weknowhowmuchpremiumtocollect Algebragivesusasolutionforhowmuchwinetobuyforward LetWine bethefacevalueofthewinewemustbuyforward Wewillbuythewineandsellthecallstruckat 110 HedgeCalculation1 ThepremiumwecollectforthecallisCanditsvalueoneyearfromnowis C 1 10 Ifthemarketpricerisesto 120 weownWine withagainof 10 120 110 10Wealsoowe 10onthecalltoitsbuyer Wewantthisalltohaveavalueof0 Wecanwriteitasfollows Ifthemarketpricefallsto 83 weownWine withalossof 27 83 110 27Wewantthelongwineandshortcalltoagainhaveavalueof0 HedgeCalculation2 Withtwoequationsandtwounknownvariables wecansolveforthembyvariousmethods Subtractingthetwoequationsisalegitimatemethod CallValue Wemustthereforebuya27 positionintheforwardwine orspot andpaythefinancingcosts tobehedged Wecanalsocalculatethepricewemustchargeforthecall HedgeWorks Wine 27 Premium 6 64 Marketrisesto 120 Marketfallsto 83 Notethat 6 6x1 1 7 30 forwardpriceofpremium Also ifthemarketrisesby 10 thewinepositionrisesby27 x 10 2 70 Ifthemarketfallsby 27to 83thevalueofthewinepositionfallsby27 x 27 7 30 OptionPricingLogic Eitherwaythemarketmoves ournetpositionisworth 0 Wearehedged Thisisthefundamentallogicofalloptionpricingmodels Thevalueoftheoptionisbasedoncreatingaportfoliocomposedofsomeamountoftheunderlyingcommodity orFXorinterestrate plustheoption Sincethereturnfromthisportfolioisnotafunctionofhowthecommoditypricemovesinthefuture itshouldreturnnomorethananinvestmentinrisk freegovernmentbonds RiskNeutralProbabilities Akeypointtonoteisthatwedonothavetohaveaviewabouttheprobabilityofanup ordown move Thecallisworth 6 64nomatterwhatwethinkaboutprobability Thismethodisneutralaboutthelikelihoodofanupordownmove dependingonlyonthevolatilityandinterestrate BuildingaBinomialTree Wehavetomodelhowassetpricesmightdevelopinthefuture Wewilluseabinomialtree Startingwiththecurrentwinepriceof 100 weassumethatthespotratecanmoveupordownoverthenextperiod p probabilitythatthespotratemoves up overthenextperiod 1 p theprobabilitythatthespotratemoves down overthenextperiod Pricescanmoveup u byacertainamount Pricescanmovedown d byacertainamount Probability WeightedForwardPrice Notethattheprobabilitiesofupanddownmovescarrynosubjectivecontent Theyaremerelycalculatedvalues whichforcethetwopossiblepricesonthetreetogivebackthemarketforwardprice Thesenon subjectiveprobabilitiesareriskneutralprobabilities Theyhavenoviewabouttheriskofeitheranupordownmove Theyareneutralaboutwhichwaythemarketmoves p 1 p u anddmustproduceresults suchthattheprobability weightedpriceequalsthemarketforwardprice BinomialTreeFormulae Data Usingtheabovemarketrateswewillpricea1 yearatthemoney 110strike winecalloption YoucanseeallofthecalculationsinthespreadsheetOP102BinomialPricingWineExample1 Spotwine 100Maturity1yearLengthofeachInterval0 25yearInterestRate10 compoundedquarterly 3 monthPVfactor0 97645Wineyield0 Volatility18 TreeCalculations UpandDownMovesinTree BinomialTree PeriodToday3months6months9months12monthsForwardPrice102 41104 88107 41110 00PriceTree143 33294130 9964119 72119 72174109 42109 4174100 00100 0010091 3991 3931283 5383 52702176 3379569 767633 SpotPriceBinomialTree ProbabilityTree PeriodToday3months6months9months12months0 1396420 2284350 37370 35517070 61130 4357571 00000 47520 3387580 38870 277080 15110 14360170 0587280 0228276TotalProbability1 00001 00001 00001 00001 0000 ProbabilityTree TerminalValueProbabilityOptionValue QuickCalculation 33 33 0 1396 4 65479 7217 0 3551 3 45290 0000 0 3387 0 00000 0000 0 1436 0 00000 0000 0 0228 0 00008 1076Total7 3705PVofTotal EuropeanCallTree PeriodToday3months6months9months12months33 332923 586516 28149 721736311 03325 8029617 37053 463802 067600 0000000 CallOptionPriceTree EuropeanStyle max 119 7217 110 0 9 7217max 100 110 0 0Foreachofthepriorquarters cellsaredeterminedbytakingtheweightedaverageofupmovementsanddownmovementsandthenpresent valuingtheresultback3months EuropeanPutTree PeriodToday3months6months9months12months001 44104 0273 7954837 37058 3451013 08616 0168321 35426 47297931 07240 232 PutOptionPriceTree EuropeanStyle AmericanOptions ForAmericanoptions ateachnodeinthetree thequestionbecomes Whichisworthmore 1 TheEuropeanoptionvaluation or2 Theintrinsicvalueoftheoptionifexercisedatthatpoint Thegreaterofthetwonumbersisplacedinthenodeandtheoptionvaluationprocesscontinues AmericanOptions Therighttoexerciseacallpriortoexpiryonanassetwhichpaysnodividendisworthless unlesstheassetbeginspayingadividend Thereisnomarketsituationwhenwewouldratherholdazero yieldassetthanholdthecallandleaveourmoneyinaninterest bearingdeposit Letusthereforechangetheexampletoanassetwhichpaysayieldof5 Theforwardpriceisnow 104 762 sowewillsetthestriketo 104 762sowestillhaveanATMstrike YoucanseeallofthecalculationsinspreadsheetOP102BinomialPricingWineExample2 AmericanCallTree PeriodToday3months6months9months12months38 57127 11317 90114 96011 3567 9237 0084 19702 2230 000 CallOptionPriceTree AmericanStyle 7 923 MAX AssetPrice Strike EuropeanOptionValue AmericanPutTree PeriodToday3months6months9months12months000 95103 6762 1276667 9207 104 761904813 36913 3687921 23521 23488428 42434 994 PutOptionPriceTree AmericanStyle CurrencyOptions WherePVfd domestic base PVfactorPVff foreign currency PVfactor Exercises Nowisyourchancetotestyourunderstandingoftheprevioussections TrytheexercisesinspreadsheetOP102BinomialExercises TheanswersareinOP102BinomialAnswers BlackScholes PVfd Exp dtPVfr Exp rtr interestrated assetyieldt timetoexpiryln naturallogarithmfunctionvol log normalvolatilityoftheassetpriceN thecumulativenormaldistributionfunction BlackScholes N d2 Probabilityofendingupinthemoney FxN d1 the mostlikelyvalue oftheassetontheN d2 exercisedate inthecasethattheoptionisexercised redtriangleinfollowing