InvestmentsLecture13Pricing:FuturesPossibleFutureProfitsofFutureContractLongPositionShortPositionSpeculationonDisneyYoubelievethatDisneywillperformverywellinthenextcoupleofmonths,butyoudonothaveanycashWhatcanyoudo?SpeculationonDisneyStrategy1:BuyaoneyearDisneyforwardcontractStrategy2:BuyDisneystocktodayusingfundsborrowedforoneyearMarketPricesCurrentspotDisneyprice:S

=$26.35Forwardpriceforacontractexpiringinoneyear:F

=$26.351-yearinterestrate:r

=2%CashFlowsofStrategy1:

ForwardContractActionTodayInoneyearBuyForwardContract0ST–F=ST–26.35CashFlowsofStrategy2:

SyntheticForwardContractActionTodayIn3monthsBuyDisney–26.35STBorrow26.35at2%26.35–26.48NetCashFlows0ST

–26.48SummaryBothstrategiesallowthespeculatortoparticipateinthepricechangesofDisneyPayoffsafteroneyear:Strategy1:ST–Ft,T=ST–$26.35Strategy2:ST–Ster(T–t)

=ST–$26.48Strategy1hasalwayshigherpayoffsthanStrategy2(F<Ser(T-t))Canthisbeanequilibrium?ArbitragestrategyShorttheassetInvesttheproceedsforaperiodT-tatrTakealongpositionintheforwardforFWhatifF>Ser(T-t)?ForwardPricingInequilibrium,thepriceofaforwardcontractonafinancialassetwithnoincomeequals:F=Ser(T-t)=S(1+r)T-tF

=26.35

*1.005

=$26.48PricingFinancialFutures:

IndexFutures–NoDividendsBydefinition,afuturescontractrequiresnocashoutlayatinitiation.Atdelivery,thepayoffis:ST–Ftoalongposition;F–STtoashortposition;PayoffDiagramPayoffdiagramforindexfuturescontract:ReplicatingtheFuturesPayoffToreplicatethepayoff:SotheFuturesPricemustsatisfy:PricingFinancialFutures:

IndexFutures–WithDividendsAgain,bydefinition,afuturescontractrequiresnocashoutlayatinitiation.Atdelivery,thepayoffis:ST–Ftoalongposition;F–STtoashortposition;However,ifyouinvestintheindexyouwouldreceivedividends(D=yS0)inadditiontothecapitalgainincorporatedintoST.PayoffDiagramforIndexFuturesContractPayoffDiagramforStockandBondReplicatingtheFuturesPayoffToreplicatethepayoff:SotheFuturesPricemustsatisfy:Summary:IndexFuturesPricingIndexoptionwithoutdividends:F=S0(1+rf)Indexoptionwithdividendyieldofy%:F=S0(1+rf

–y)ExampleAssumethattheTSE35currentlyhasavalueof200.Thedividendyieldontheunderlyingstocksintheindexisexpectedtobe4%overthenextsixmonths.New-issuesix-monthTreasurybillsnowsellforasix-monthyieldof6%WhatisthetheoreticalvalueofasixmonthfuturescontractontheTSE35.Supposethefuturespricewasinstead212.Describeanarbitrageopportunitythatisavailable.CommodityFuturesWhenweconsidercommodityfutures,thehedgingstrategybecomesmoretrickysincetoimplementthehedgingstrategyyouhavetoholdthecommodity.Storagecostsbecomeimportant.Benefitsofownershipofthephysicalgoodbecomeimportant.ConvenienceYieldThebenefitofowningthephysicalcommodityisoftenreferredtoasconvenienceyield.Theeffectofconvenienceyieldonthefuturespriceissimilartotheeffectofdividendsonindexfuturesprices.PayofffromFuturesandfromHoldingCommodityReplicatingtheFuturesPayoffToreplicatethepayoff:SotheFuturesPricemustsatisfy:StorageCostsStoragecostsarelikenegativedividendsandaffectthepricingrelationshipintheobviousway:ExampleConsiderafuturescontractonmangoesthatcallsforthedeliveryof2000poundsofthefruitthreemonthsfromnow.Thespotpriceofmangoesis$2perpound.Thethree-monthrisk-freerateis2%.Itcosts$.10/poundtostoremangoesforthreemonths.Whatshouldbethepriceofthisfuturescontract?BasisRiskThedifferencebetweenthecurrentspotpriceonanasset(theasset’spriceforimmediatedelivery)andthecorrespondingfuturesprice(thepricestatedinthefuturescontracts)isknownasthebasisforthefutures:Basis=Currentspotprice–FuturespriceTheriskthatthebasiswillnarroworwiden,causinggainsorlossestotheseinvestors,isknownas“basisrisk”.ConvergenceofPricesTimePricesFuturesPriceSpotPriceForwardCurvesTherelationshipbetweenthetime-to-maturityoffuturescontractsandthefuturespriceiscalledtheforwardcurve.Dependingontheunderlyingasset,thesecurvesmaybeupwardsloping(incontango)ordownwardsloping(inbackwardation)ForwardCurvesIndexfutures:ForwardCurvesCommodityfutures:Pricing:OptionsOptionPricingModelsTwoapproaches:TrackingPortfolio.Solveexplicitlyforreplicatingportfolio.Risk-neutralprobabilities.Adjustprobabilitiessothatexpectedcashflowsgiveoptionprices.Twosettings:Binomial(discretetime)Black-Scholes(continuoustime)34TheBinomialValuationModel:

TheTrackingPortfolioApproach

Considerthefollowingcalloption: S0=$100, K=$125, rf=8Supposeatyear-endstockcantakeonlytwovalues: ST=$200or$50Thenthepayoffforcalloptionis: $75whenstockgoesupand$0whenstockgoesdown.200 7550 0Stock Call100TrackingPortfolioComparetothepayofffromaportfolioof: 1shareofstock+borrowing$46.30at8%Payoffsfromthispositionare: $150whenstockgoesupand$0whenstockgoesdown Cashoutlaytoestablishthisposition=$100-$46.30=$53.70

Payoffofleveredportfolioistwicetheoptionpayoff,therefore: C0=(1/2)$53.70=$26.85Considerthefollowingposition(samedataasbefore) 1shareofstock+2writtencalls Netpayoffs: Stockvalue $50 $200 -Callobligation -0 -$150 Netpayoff $50 $50 PVof$50=$50/1.08=$46.30Hence: S0-2C0=46.30; S0=100andC0=$26.8537Hedgeratio(H):Inthiscase:Therefore: H=0.5shares $100 $25

-Callobligation $75 $0

$25 $25PVof$25at8%=$25/1.08=$23.15CalculatingtheHedgeRatio38Valueofhedgedposition=PVofcertainpayoff 0.5S0-C0=$23.15 $50-C0=$23.15 C0=$26.85Hsharesand1callwrittenalwaysresultsinaperfectlyhedgedportfolioYieldssurepayoffattimeTofGeneralbinomialmodelpricingformula:GeneralPrinciple39Example:ArbitrageProfitsGivenpreviousdataexceptC0=$30Initialcashflows: Write1option $30 Purchase0.5shares -$50 Borrow$20at8% $20 (Repayin1year)Cashflowin1yearateachstockprice: Writtenoption $0 -$75 PurchaseHshares $25 $100 Borrow$20at8%

Repayin1year-$21