chapter4-1无文字金融工程专业

1,Chapter4FuturesandForwardsPrices,2,Ourobjectiveistolinkthepriceofthefuturesorforwardcontracttothepriceoftheunderlyinginstrumentandtoidentifyfactorsthatinfluencetherelationshipbetweentheseprices.,Objectiveofthischapter,3,1、PRELIMINARIES,formanyofthecontractsthataretradedinthemarket,itcanbearguedthattheforwardpriceandfuturespriceofanassetareveryclosetoeachotherwhenthematuritiesofthetwocontractsarethesame.,4,1、PRELIMINARIES,notationWewillusealotofnotation,soletsbeveryclearaboutwhateachsymbolmeans:T=maturitydateoftheforwardcontractt=currenttimeSt=priceofunderlyingattimetST=PriceofunderlyingattimeTK=deliverypriceintheforwardcontractFt=forwardpriceattimetf=valueofalongforwardcontractattimet.r=riskfreerate,5,1、PRELIMINARIES,.ContinuousCompoundingCompoundinterestispaidontheoriginalprincipalandontheaccumulatedpastinterest.Notation:T:Investmentperiod(years)PV:PresentValueofinitialinvestmentFV:FutureValueofinitialinvestmentR:Nominalinterestrateperannumm:Compoundingfrequency,6,Continued,WhentheinterestiscompoundedonceayearforTyears:Whatifinterestispaidmorefrequently?Hereareafewexamplesoftheformula:,m=1,m=4,m=12,7,continued,ConsideranamountPVinvestedforTyearsataninterestrateofRperannum.,Thelimitasmtendstoinfinityisknownascontinuouscompounding,8,示例,例：假设存款金额为100，名义年利率为10%，存款期限为1年，在年度计息的条件下，明年的期末存款余额为：1001.1=110半年计息一次：1001.051.05=110.25每季度计息一次：1001.0254110.38连续复利：100e0.10=110.52,9,实际利率与计息次数,10,continued,ThelimitasmtendstoinfinityisknownascontinuouscompoundingThestandardfutureandpresentvalueformulasare:PVFV：PV*eRTFVFVPV：FV*e-RTPVForexample,ifyourdiscountrateis8%,andyouaregoingtoreceive$200in2years,thepresentvalueofthatcashflowis:,11,continued,Youshouldbeawareofhowtoconvertbetweeninterestratesofdifferentcompoundingfrequency.Forexample,ifryistheannuallycompoundedrateofreturnwhatistheequivalentquarterlycompoundedrateofreturn?Youcansetthisupeasilyenoughbyrealizingthatifyouinvestinbothforagiventimeperiod(say1year),youwindupwiththesameamountofcash,thatis:,12,continued,Ingeneraltoconvertbetweenarater1thathascompoundingfrequencym1andrater2thathascompoundingfrequencym2,youusethisformula:So,ifyouhadasemi-annualrateof6%andwantedtoknowtheequivalentquarterlyrate:,13,continued,Toconvertbetweenadiscreterate,rdwithcompoundingfrequencymdandacontinuouslycompoundedraterc,use:(thisassumest=1),Adepositaccountpays12%perannumwithcontinuouscompounding,butinterestisactuallypaidquarterly.Howmuchinterestwillbepaideachquarterona$10,000deposit?,14,1、PRELIMINARIES,.shortsellingRecallthatwhenyoutakeashortpositioninaninstrument,itmeansthatyouaregivingupcontroloverthatinstrument.Shortselling/融券:sellinganassetthatisnotownedwiththeintentionofbuyingitbacklater.Whenyou“short”astockyouborrowerthestock(getcontrol)andthensellit(losecontrol.),15,shortselling,investor,16,1.寻找卖空交易的股票.要制定一个卖空交易，交易商A必须首先确认他能够借到的他计划卖空的股票数量。经纪人保持一个称为一箱列表（ABoxList)，列出那些可以做卖空交易的股票清单。经纪人通过以下方式更新这个列表：经纪人自己的股票清单，经纪人客户同意借出的股票，以及其他第三方经纪人的股票。,17,2.执行一个卖空交易.在确认了可以做卖空交易的股票后，交易商A在交易日，或称作“T”，执行一个卖空交易。大部分股票交易市场有为期三天的结算期。就是在交易日之后三天内，或“T3”，完成股票换现金的交易。,18,3.借到卖空交易的股票和完成卖空交易.在T3的早上，交易商A的证券借贷部这一天实际交付做卖空股票的义务。他们查看他们的列表，用列表中的股票来完成卖空交易。,19,4.卖空收益的现金用做借来股票的抵押.当交易结算时，从出售股票收到的现金作为交易商A的借来的股票的抵押。交易商A的经纪人用抵押的现金来投资。,20,5.利息付给卖空的出售人抵押现金投资的利息收入的一部分用来支付管理费和股票借入费。由于管理费用的高成本，结余的利息通常只会付给那些大宗卖空交易的卖者。,21,6.通过卖空卖者支付的股息（如果要求的话）.如果卖空的股票支付股息的话，借出股票的人有收取股息的资格，因为他并没有卖掉这些股票，而仅仅是借出而已。,22,在未来的某个时候，或者是当交易商A决定回购他的卖空股票，或者股票借贷人要收回股票.在第一种情况下，卖空交易就结束了。在第二者情况下，经纪人会设法寻找另一个借贷方,借贷将转移到新的交易人或经纪人。交易商A的卖空交易将不受影响。如果无法找到替代的借贷方的话，经纪人可能通知交易商A借贷已经被收回，交易商A必须立刻平仓他的卖空。,23,24,卖空的特点,特点被卖空证券是经纪商从其他客户账户中借用出来的。在平仓之前，证券所发生的现金流入，比如红利或者利息等，都由卖空者代为支付。卖空所得资金由经纪商进行保管，同时卖空者还需向经纪商缴纳一定比例的保证金。当卖空者平仓时，经纪商要在市场上买回被卖空证券。当被借证券的原所有者需要时，经纪人会设法寻找另一个借贷方,否则卖空者必须立即平仓，从市场上买回被卖空证券。,25,1、PRELIMINARIES,.ArbitrageandtheLawofOnepriceAllcurrentmethodsofpricingderivativeassetsutilizethenotionofarbitrage.Arbitrageisatypeoftransactioninwhichaninvestorseekstoprofitwhenthesamegoodsellsfortwodifferentprices.Iftwoinvestmentopportunitiesofferequivalentoutcomes,theymusthaveequivalentprices.theLawofOnePrice,26,continued,Assetpricesareobtainedfromconditionsthatprecludearbitrageopportunities.Wesaythatthepriceofasecurityisa“fair”level,orthatthesecurityiscorrectlypriced,iftherearenoarbitrageopportunities.Determiningarbitrage-freepricesisatthecenterofvaluingderivativeassets.,27,1、PRELIMINARIES,.InvestmentAssetsvsConsumptionAssetsInvestmentassets:assetsheldbysignificantnumbersofpeoplepurelyforinvestmentpurposes(Examples:bond,gold)Consumptionassets:assetsheldprimarilyforconsumptionpurposes(Examples:copper,oil),arbitrageargumentscanbeusedtoobtainexactforwardpricesincaseofinvestmentassets,canonlygiveanupperboundtoforwardprices,28,Inorderforthepricingmethodpresentedinthislecturetowork,weneedfortheretobealeastsomeparticipantsinthemarketforwhichthefollowingaretrue,oratleastareclosetobeingtrue.Therearenotransactionscosts.Alltradingprofits(netoflosses)aresubjecttothesametaxrate.Themarketparticipantscanborroworlendatthesamerisk-freeinterestrate.Marketparticipantscantakeadvantageofarbitrageopportunitieswhentheyoccur.,Assumptions,29,2、Theconceptofpriceversusvalue,Thepriceinaforwardcontractisknownasthedeliveryprice(K).Theforwardprice(F)foraparticularforwardcontractataparticulartimeisthedeliveryprice(k)thatwouldapplyifthecontractwereenteredintoatthattime(F=Kattimezero).Keepsomethingsinmind:KandFarenotthesamething.Attimezerotheyare,butKissetinthecontractandFmovesovertime.,30,远期价格是指使远期合约签订时价值为零的交割价格。远期价格是理论上的交割价格。关于远期价格的讨论也要分远期合约签订时和签订后两种情形。一份公平合理的远期合约在签订的当天应使交割价格等于远期价格。如果实际交割价格不等于这个理论上的远期价格，该远期合约价值对于多空双方来说就都不为零，实际上隐含了套利空间。在远期合约签订之后，交割价格已经确定，远期合约价值不一定为零，远期价格也就不一定等于交割价格。,30,类似地，在期货合约中，我们定义期货价格（FuturesPrices）为使得期货合约价值为零的理论交割价格。,31,32,continued,Thevalueofaforwardcontractfistheprofitonthiscontract.Forforwardcontracts,nocashispaidoutupfront,sothecontractshavezerovaluewhenfirstwritten.,AttimeT(longposition),Howdowedeterminethevalueoftheforwardattimet0?,但值得注意的是，对于期货合约来说，一般较少谈及“期货合约价值”这个概念。基于期货的交易机制，由于期货每日盯市结算、每日结清浮动盈亏，因此期货合约价值在每日收盘后都归零。,33,34,Buytheoldfuturescontract,Sellthenewone,DeliverthetwocontractsandmakeaprofitofFtK,fnew=0,ft=(Ft-K)e-r(T-t),35,Thevalueofforwardcontract,Thepresentvalueofthisdifferenceisthevalueoftheforward/futurescontract(longposition)attimet,ft,ft=(Ft-K)e-r(T-t),36,3.forwardpriceforaninvestmentasset,Threekindsofinvestmentassets:、Investmentassetsprovidingnoincome、Investmentassetsprovidingaknowncashincome、Investmentassetsprovidingaknowndividendyield,37,2.1forwardpriceforaninvestmentassetprovidingnoincome,IntuitionItisveryeasytogetboggeddownintheexactdetailsofthevariouspricingformulasthatwewilluse,soletsfirstconsidertheeconomicintuitionthatunderliesalloftheseformulas.Considerwhentwopartiesenterintoaforwardcontractonanon-incomeproducingasset.Thelongparty(LP)agreestopaytheshortparty(SP)KdollarsattimeTinexchangefortheunderlyingasset.FromtheviewpointofSP,SPisholdingtheassetontheirbehalfuntilthematuritydate.AssumingnocreditriskthismeansthatitisthesameasifSPboughttheunderlyingassettoday,heldontoitforTyears,andthensolditatpriceK.,38,Intuition,Fromtheshortsperspectivethereisnoriskinthiscontract.Theypaythecurrentmarket(spot)pricefortheasset(S0),theyholditforTyears,andthentheysellitforKdollars.ThiscostS0andiscertaintoleadtoacashinflowofK/F0attimeT.,Sohowwouldtheshortpartydeterminethedeliverypricethattheywouldbewillingtoenterinto?Lookatitthisway,theyhavetopayS0dollarstoday,andtheywanttoearntherisk-freerateonthosedollars,sothedeliverypriceKortheforwardpriceattimezeroF0mustbeF0=K=S0erT!,39,Aformalprove,Considerthefollowingportfolios(attime0):PortfolioA:alongforwardcontractandacashamountofonhand.PortfolioB:oneunitofunderlyingasset.,Aunitofasset,buy,40,Aformalprove,Notice,then,thatbothportfoliosAandBeachgiveyouthesameendresultorvaluethevalueofthesecurityatmaturity.Onconditionthattherearenoarbitrageopportunities,atsomeearliertime,suchastime0,thesetwoportfoliosshouldbeidenticaltoo.Ifnot,ifABorBA,thenyouwouldshorttheexpensiveone