金融衍生工具研究第2讲期货价格的决定(期货定价).pdf

1Lecture 2:Determination of Futures PricesLecture 2:Determination of Futures Prices期货价格的决定期货价格的决定(期货定价）期货定价）Outline Operation of margins保证金的运行保证金的运行 Futures pricing期货定价期货定价 Futures price and(expected)spot price期期货价格与未来的现货价格货价格与未来的现货价格2Lecture 2:Determination of Futures PricesFutures Contracts期货合约期货合约 Available on a wide range of underlying assets标的物包括多种资产 Exchange traded在交易所中交易 Specifications need to be defined:期货合约规定：What can be delivered,交割的对象 Where it can be delivered,&交割地点 When it can be delivered交割时间 Settled daily-marking to market每日结算：逐日盯市3Lecture 2:Determination of Futures PricesOperation of Margins保证金制度及其运行保证金制度及其运行 To prevent default,futures exchanges have a set of margin requirements.为避免违约，期货交易所有一系列的保证金要求 Upon entering a contract,a client must deposit a certain amount of money or marketable securities(initial margin)in a margin account with his/her broker at the close of trading on the first day he establishes a new position当客户建立期货合约头寸的当天闭市前，必须存入一定数额的货币或可交易证券（初始保证金）到客户在经纪人公司开设的帐户。Different futures contracts have different margin requirements that are set by the exchanges and may fluctuate from time to time.不同的期货合约有不同的保证金要求，由交易所制定，随时间变化而变动。4Lecture 2:Determination of Futures PricesMaintenance Margins维持保证金维持保证金Maintenance margin is simply the minimum amount of money a client must have in his account after a trading day.The amount is always slightly less than the Initial Margin.在交易日结束后客户的帐户中必须拥有的最低数额的保证金，一般稍低于初始保证金。ContractExchangeInitial M.($)Maintenance M.($)S&P500IMM1968815750SoybeansCBOT810600T-Bill(13w)IMM10275T-noteCBOT13501000T-bondCBOT21601600GoldCOMEX20001500MSCI TaiwanSGX26252100CSSGXS$675S$5005Lecture 2:Determination of Futures PricesMarking to Market逐日逐日盯市盯市The balance in the margin account is adjusted to reflect daily settlement Marking to market.保证金帐户的余额每天都会调整，因为期货合约每日进行结算-盯市。Marking to market takes place through margin payments盯市以保证金支付的形式进行。Standard practice一般做法 At the inception of a contract,a client pays initial margin(5-10%of contract value)to a margin account held by his broker.This can be paid in interest bearing securities so there is no interest cost.合约建立时，客户在保证金帐户中存入初始保证金，一般为合约价值的5-10%，帐户在经纪人公司开设。这也可以由附息证券支付，不存在利息成本。If futures settlement prices rise(fall),the long have a paper profit(loss)and the short a paper loss(profit).The broker receives losses from and pays any profits into the parties margin accounts.若期货清算价格上升，长头寸产生了帐面利润，短头寸产生了帐面损失。经纪人从顾客那收到损失金额，而支付利润到顾客的保证金帐户。6Lecture 2:Determination of Futures PricesContinued If a clients margin account falls below the maintenance margin,he will receive a margin call from his broker to restore his margin account to the initial margin level before the start of the next trading day.This extra fund deposited is called the variation margin.当顾客的保证金帐户余额维持保证金是，投资者收到保证金催付通知，要求在很短时间内将保证金帐户内资金不足到初始保证金的水平。If a client fails to provide the variation margin,the broker will close out his position at available price.如果投资者不能提供变动保证金，经纪人将出售该合约来平仓。Margin in excess of initial margin may be withdrawn by profit-making parties超过初始保证金的资金，投资者可以抽走。If market volatility increases,or the exchange is worried about possible manipulation,it may require an increase in margin levels.如市场波动性增加或者交易所担心市场可能被操纵，会提高保证金水平。7Lecture 2:Determination of Futures PricesMarking to Market:An Example一个例子：盯市一个例子：盯市Operation of margins in a standard futures market:期货市场的保证金操作：一例 A long position in two gold futures contracts持有两份黄金期货合约的多头头寸 The initial margin is$2,000/contract,or$4,000 in total.初始保证金一合约$2000或者总数为$4000。The maintenance margin is$1,500,or$3,000 total.维持保证金一合约$1500或者总数为$3000。The contract is entered on June 3 at$400 and closed out June 26 at$392.3.6月3日购买，价格为$400，6月26日平仓，价格为$392.38Lecture 2:Determination of Futures PricesMarking to Market:An Example一个例子：盯市一个例子：盯市DailyCumulative MarginFuturesGainGainAccountMarginPrice(Loss)(Loss)BalanceCallDay(US$)(US$)(US$)(US$)(US$)400.004,0005-Jun 397.00(600)(600)3,4000.13-Jun 393.30(420)(1,340)2,6601,340.19-Jun 387.00(1,140)(2,600)2,7401,260.26-Jun 392.30260(1,540)5,0600+=4,0003,000+=4,000 S0erT,:An investor adopts the strategy:Borrow S0dollars at an interest rate r for T years;buy one ounce of gold;short a forward contract on one ounce of gold.若F0 S0erT，投资者采用以下策略：以利率r借T年的S0，买一盎司黄金，卖空一盎司黄金的远期合约At time T one ounce of gold is sold for F0.An amount S0erT,is required to repay the loan at this time and the investor makes a profit of F0-S0erT.在T时刻，一盎司的黄金售出价为F0，S0erT 需用于支付借款，投资者的收益为F0-S0erT29Lecture 2:Determination of Futures Prices If F0 S0erT:An investor who own one ounce of gold can:sell the gold for S0;invest the proceeds at interest rate r for time T;take a long position in a forward contract on one once of gold.若F0CB:one may then short A and long B to make an initial profit of CA-CBinstantly and risklessly.若它们的成本不同，如CACB:则可以卖空A买入B马上获得无风险收益CA-CBThe Arbitrage principle requires 套利原则要求 No restrictions on short selling.无卖空限制 No transaction costs 无交易成本 No bid-ask spread 无买入价和卖出价之间的差别 Borrowing rate=lending rate 借款利率等于贷款利率31Lecture 2:Determination of Futures PricesCash and Carry Arbitrage Forward on A Non-dividend Paying Asset没有现金收益投资型资产没有现金收益投资型资产 Consider two strategies for owning one unit of a stock(paying no dividend)at T:先考虑T时刻拥有股票的策略（不支付股利）Strategy A:buy 1 unit of the stock forward for F0,invest e-rT*F0in risk free assets and use the proceeds to pay for purchase at T.Cost=F0e-rT.策略A：买入价格为F0的股票远期合约，借入e-rT*F0并投资于无风险资产，将投资所得来支付T时刻的支出。Strategy B:Buy 1 unit of the stock at S0and hold.Cost=S0.策略B：现以S0买入股票并持有，成本为S0 Both strategies give the same terminal value ST.They must cost the same.Hence S0=F0e-rTor F0=S0erT.Note:Strategy B is a replicating strategy:it gives the same returns as the derivatives strategy without involving the derivatives.两个策略的期末价值为ST，因此S0=F0e-rTor F0=S0erT。策略B为复制策略：它未使用衍生品最终获得了衍生品相同的收益。32Lecture 2:Determination of Futures PricesExample A share costs$1.00 today with no dividend payable over the coming year.Risk free rate is 4%.What is the futures price of the share in 1 years time?Two strategies for owning the share for one year:Strategy A:buy 1 share forward for F0;deposit e-rTF0in risk free asset;use the proceed to complete the contract.Cost=e-rTF0.Strategy B:buy 1 share today.Cost S0=1.Thus F0=erTS0=e0.04*1=$1.04.33Lecture 2:Determination of Futures PricesWhen an Investment Asset Provides a Known Dollar Income支付确定现金收益投资型资产的远期合约支付确定现金收益投资型资产的远期合约F0=(S0 I)erTwhere I is the present value of the incomeAn example:Known dividend D payable at 0 t S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures当F0S0e(r-q)T，套利者买入指数组合中的股票，卖出期货合约。When F0S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index当When F0S0e(r-q)T，套利者买入期货合约，卖出指数组合中的股票。40Lecture 2:Determination of Futures PricesIndex Arbitrage指数套利指数套利 Index arbitrage involves simultaneous trades in futures and many different stocks指数套利涉及同时买入很多股票以及期货交易。Very often a computer is used to generate the trades program trading通常用电脑程序来交易 Occasionally(e.g.,on Black Monday)simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0and S0does not hold.Arbitrage is often blamed for destabilizing the market.有时同时交易不可能，理论上的套利不存在。有人批评套利，认为其会影响市场稳定。41Lecture 2:Determination of Futures PricesCurrency Futures外汇期货合约外汇期货合约 The underlying asset in foreign currency futures is a certain number of units of foreign currency whose spot price is measured in dollars of one unit of the foreign currency.The holder of a foreign currency can earn interest at risk free rate prevailing in the foreign country.42Lecture 2:Determination of Futures Prices A foreign currency is analogous to a security providing a dividend yield The continuous dividend yield is the foreign risk-free interest rate It follows that if rf is the foreign risk-free interest rate Futures and Forwards on Currencies外汇期货合外汇期货合约和远期合约约和远期合约FS er rTf00()43Lecture 2:Determination of Futures PricesFutures on An Asset Paying Storage Costs支付储支付储藏成本的期货合约藏成本的期货合约 For investment assets投资性资产:if U is the present value of all storage costs that will incur before maturity of a futures contract.It follows,F0=(S0+U)erT.U can be viewed as a negative dollar dividend.If the storage costs are proportional to the price of the commodity,they can be regarded as providing a negative dividend yield,u.Then,F0=S0e(r+u)T.44Lecture 2:Determination of Futures PricesFutures on Consumption Assets消费性资产的期消费性资产的期货合约货合约 Arbitrage arguments can be used to obtain exact futures prices for investment assets,but can only give an upper bound to futures prices in the case of consumption assets.If F0(S0+U)erT.An arbitrager can Borrow an amount S0+U at risk free rate and use it to purchase one unit of the commodity and to pay storage costs.Short a futures contract on one unit of the commodity.This strategy leads to a profit of F0-(S0+U)erTat T.45Lecture 2:Determination of Futures PricesFutures on Consumption Assets消费性资产的期消费性资产的期货合约货合约 Since most commodities are held for consumption and holders are reluctant to sell the commodity and buy futures,therefore,when F0(S0+U)erTor F00(netshorthedge),futurespricesisnormallydownward estimate of futures spot price.Z0(net longhedge),futures price is upward biased.As m approachesinfinity,futures price becomes unbiased.60Lecture 2:Determination of Futures Prices课堂问题课堂问题Arbitrage is a.a zero initial wealth trading strategy that has a likelihood of making profits without risk of a lossb.a way of resolving disputesc.a surefire way of making money for arbitragersd.a zero investment trading strategy in which the likelihood of portfolio gain overwhelms the likelihood of loss61Lecture 2:Determination of Futures PricesConsider a forward contract that began earlier.As we move toward maturity,the value of the forward a.must always be positiveb.can be zeroc.must always be negatived.none of the above62Lecture 2:Determination of Futures PricesA forward contract that began earlier matures on July 15.The value of the contract is$2.50,the spot is$100,a zero coupon bond maturing on July 15 is worth$0.98,and the forward price is$100.The arbitrage profit that you can make isa.$2b.$1.50c.$1d.none of the above63Lecture 2:Determination of Futures Prices Forward contracts on electricity are hard to price because the simple cost-of-carry argument breaks down as it is impossible to store large quantities of electricity for long periods of time.True/False and Why?64Lecture 2:Determination of Futures Prices 中国期货交易所的主要交易品种有哪些？本章的期货定价原理对产品设计与创新有何启示？