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1、1,Chapter 4Futures and Forwards Prices II,2,Three kinds of underlying investment assets:、Investment assets providing no income、Investment assets providing a known cash income、Investment assets providing a known dividend yieldDevelop pricing models for forwards when the underlying asset has cash flow
2、s associated with it during the period of the forward contract.,forward price for an investment asset,3,2.2 forward price for an investment asset providing a Known Cash Income,Intuition 1To see this,consider a security that had a perfectly predicable set of cash flows that will occur between 0 and T
3、.Denote the present value of those cash flows as I(discounting at the risk free rate).From the short partys perspective:buy one unit of the asset and enter into a short forward contract to sell it for F0 at time T.This cost S0 and is certain to lead to a cash inflow of F0 at time T and income with a
4、 present value of I.,4,A formal prove,Assumption:The underlying asset will provide income with a present value of I during the life of a forward contract.Consider the following portfolios(at time 0):Portfolio A:a long forward contract and a cash amount of on hand.Portfolio B:one unit of underlying a
5、sset and a cash amount of I borrowed at risk-free interest rate.,Clearly A and B once again have the same payoffs at time T,and so once again,setting them equivalent:,Provide income with a present value of I,5,Continued,This is the forward price for an investment asset providing a known cash income(
6、present value I).,6,Reexamine The Formula,We could attribute this formula to the notion opportunity cost that the short party faced:to induce the short party to enter into the contract the long party must pay them interest at least in the amount that the short could get if they simply sold the asset
7、 now and invested it at the risk-free rate.,Recall that we demonstrated that a forward contract on a security that pays no dividends will have,at time 0,a delivery price of:,7,Reexamine The Formula,Recall,however,that during the period of the forward contract,if the short party physically holds the
8、underlying asset providing a known cash income,then they will garner any benefits that accrue to the asset during that period.For example,if the forward contract were written on a stock,and the stock paid a dividend,then if the short party physically held the stock on the ex-dividend date,they would
9、 receive the dividend.,8,Reexamine The Formula,The short party still has no risk(ignoring credit risk)in the forward contract,as a result they should still only earn the risk-free rate for being the short party.The benefits that accrue to holding the underlying asset would reduce the amount that the
10、 long would have to pay the short to induce them to enter into the contract.,F0=K=(S-I)erT,9,Example:Assume a 10 month forward on a dividend-paying stock.Current price is$50.Assume r=8%,and dividends of$0.75 in 3,6,and 9 months.I=.75e-.08(.25)+.75e-.08(.5)+.75e-.08(.75)=2.162T-t=10/12=.83333 years:F
11、=(50-2.162)e.08*.83333=51.136,2.2 forward price for an investment asset providing a Known Cash Income,10,Second example:Assume a 3 month forward on a dividend-paying stock with current price of$100.Assume r=4%,and the stock will pay a dividend of$2.00 in 1 month.I=2.00-.04(1/12)=1.9933T-t=3/12=.25 y
12、ears:F=(100-1.9933)e.04*.25=$98.99We can use this example to demonstrate the arbitrage opportunity that enforces this rule.,2.2 forward price for an investment asset providing a Known Cash Income,11,To see this,consider what would be the case if F0(S0-I)erTLets say that we saw F=101,how would the ar
13、bitrageur exploit the opportunity?At time 0:Short the forward contract(i.e.agree to deliver the stock in three months for$101).Borrow$100 today at the risk-free rate and buy the stock.At time 1 month:Reinvest the dividend at the risk-free rate.At time 3 months they do 4 things:Deliver the stock into
14、 the forward contract and receive$101.Receive$2.103(2e.04(2/12)from the reinvested dividends.Repay$101.005(100e.04(3/12)for the$100 you borrowed at time 0.Net 3 cash:+101+2.013 101.005=$2.008,2.2 forward price for an investment asset providing a Known Cash Income,12,We can show this on a timeline:,2
15、.2 forward price for an investment asset providing a Known Cash Income,0,1,3,Actions,Cash Positions,13,We can show this on a timeline:,2.2 forward price for an investment asset providing a Known Cash Income,0,1,3,Actions,Cash Positions,Short futures contractBorrow$100 at 4%Buy stock for$100,0+$100-$
16、100$0,14,We can show this on a timeline:,2.2 forward price for an investment asset providing a Known Cash Income,0,1,3,Actions,Cash Positions,Short futures contractBorrow$100 at 4%Buy stock for$100,0+$100-$100$0,Receive$2 dividendReinvest at 4%,+$2-$2$0,15,We can show this on a timeline:,2.2 forward
17、 price for an investment asset providing a Known Cash Income,0,1,3,Actions,Cash Positions,Short futures contractBorrow$100 at 4%Buy stock for$100,0+$100-$100$0,Receive$2 dividendReinvest at 4%,Receive$101 from futures deliveryRepay loan$101.005(100e.04(3/12)Receive reinvested dividends 2.013(2e.04(2
18、/12),+$101.000-$101.005+$2.013+2.008,16,2.3 forward price for an investment asset providing a Known Dividend Yield,Some securities such as stock indices and currencies essentially have a continuous dividend yield instead of a discrete dividend.Thus you can think of the asset as paying a continuous d
19、ividend at rate q,based on the value of the security.该投资性资产按照q的百分比率，持续不断地产生股利支付，而该股利支付被立即用于连续复利的再投资即该资产会按照q的连续复利率不断增值。,17,2.3 forward price for an investment asset providing a Known Dividend Yield,Again consider two portfolios at time zero:Portfolio A:One long forward contract and cash equal to Ke-r
20、T.Portfolio B:e-qT units of the security with all income being reinvested in the security.,Thus at time T you will once again have one unit of the security,which is worth ST.Clearly A and B once again have the same payoffs at time T,and so once again,setting them equivalent:f0+Ke-rT=S0e-qTOr f0=S0e-
21、qT-Ke-rTsetting f0 to 0 and solving for K leads to:K=F0=S0e(r-q)T,18,2.3 forward price for an investment asset providing a Known Dividend Yield,Thus,the formula for determining the forward price is:F0=S0e(r-q)Twhere q is the dividend yield,expressed in annual terms.变量S代表以人民币表示的一单位外汇（美元）的即期价格(1$？￥)美元
22、持有人能获得美国的无风险利率rf的收益（债券或存款）；人民币的持有人也能获得中国的无风险利率r的收益。都以连续复利计息。1、购买 单位的即期外汇2、卖空一单位外汇的一个远期合约（在T时刻将一单位外汇以F的价格卖出）,利率平价关系,A foreign currency can be regarded as an investment asset paying a known yield.The yield is the risk-free rate of interest in the foreign currency.,19,U.S.prime rate is the base rate on
23、 corporate loans posted by at least 70%of the 10 largest U.S.banks,20,2.4 Stock Index Futures,A stock index tracks the changes in the value of a hypothetical portfolio of stocks.,1225.38,21,2.4 Stock Index Futures Contracts,股价指数期货（Stock Index Futures），是指由交易双方签订的，约定在将来某一特定时间交收“一定点数的股价指数”的标准化期货合约，亦即是以
24、股价指数为交易标的的一种期货合约。,22,股指期货交易的特点,特殊的避险功能。外汇期货和利率期货交易仅能回避其各自的非系统性风险，而股指期货的诞生则主要是用于回避或改变股票市场的系统性风险。特殊的结算方式和交易结果。股指期货的标的资产不是某种商品或金融工具，而是抽象的股票指数。若持有到期，以最后交易日的结算价格为准，并于之后的第一个营业日为结算日。在结算日，交易双方只要交付或收取根据结算价与开仓时股指差价所折成的一定金额的货币即可，即采用现金结算的方式，而勿须也无法进行实物交割。同时，也正是由于股指期货交易并未发生实际的股票收付，故交易中也不会发生任何股东权利和义务的转移。现金结算是由指数“点
25、”的变化和每个指数“点”所代表的价值来共同决定的。,指数点数的价值（value of an index point)或者合约乘数（contract multiple）.用z表示。,23,24,most indices can be though of as a security that pays dividends.Usually it is convenient to consider them as paying dividends continuously.Let q be the dividend yield rate,then the futures price is given
26、by:F0=S0e(r-q)T.Example 5.5,25,Index Arbitrage,Once again we can use arbitrage arguments to prove the equation:If F Se(r-q)T,then buy the stocks underlying the index and short futures contracts.If F Se(r-q)T,do the opposite:short the stocks and buy the futures contract.For indices involving many sto
27、cks,index arbitrage is sometimes accomplished by trading a relatively small representative sample of stocks whose movements closely mirror those of the index.,Index arbitrage,26,Hedging an Equity Portfolio,1.多头套期保值与空头套期保值股指期货的标的资产是市场股票指数，因此运用股指期货进行套期保值，管理的是股票市场的系统性风险。例如，当投资者预期在将来特定时刻投资股票，但担心实际购买时大盘整
28、体上扬而蒙受损失，便可通过预先进入股指期货多头的方式消除系统性风险；当投资者看好手中所持有的股票不愿轻易卖出，但担心大盘下跌给自己带来损失，就可以通过股指期货空头对冲系统性风险。,30只以上的股票组合，其风险（方差）有95%以上是市场共有风险。,27,Hedging an Equity Portfolio,2.股指期货合约数量的选择,P:Current value of the portfolioA:Current value of the stocks underlying one futures contractIf the portfolio mirrors the index,the
29、number of futures contracts is,28,Hedging an Equity Portfolio,2.股指期货的最优套期保值比率 t0时，套期保值者在现货市场为多头，拥有NA个现货单位；在期货市场上卖出N*份期货合约。在t1时，结清持有的头寸，z为期货合约乘数。套期保值的组合价值变化为：,29,Hedging an Equity Portfolio,30,Hedging an Equity Portfolio,如果：（1）RM所表示的市场证券组合收益率与我们用于套期保值的股指期货价格变动一致；（2）套期保值期间，被套期保值的股票组合的系数能很好地代表其真实的系统性风险
30、，则的确是股指期货最优套期保值比率的一个良好近似。,h*,31,Example,6月20日，一个养老基金准备为价值100万英镑的股票组合对冲。对冲的期限为5个月。股票组合的1.5。股票组合的红利收益为q4%，r10%（连续复利）。选择使用12月到期的股指期货。现在的FTSE100为6400。contract multiple10英镑每指数点为了使股票组合规避风险，应该卖出多少份股指期货？,32,Example,Value of portfolio=$1,500,000 of portfolio=1.5Index S0=1500Risk-free rate=10%(per year;contin
31、uous compounding)Dividend yield on index:q=4%(continuous compounding)Contract multiple$250We assume that the index futures contract with 4 months to maturity is used to hedge the value of the portfolio over the next 3 months.Suppose the index turns out to be St=1350 in 3 months,33,为股票指数的收益率。包括资本利得和红
32、利收益两部分,34,3.改变投资组合的系统性风险暴露投资者可以利用股指期货，根据自身的预期和特定的需求改变股票投资组合的系数，从而调整股票组合的系统性风险与预期收益。设定股票组合的原系数为，目标系数为。则需要交易的股指期货份数为 当时，意味着投资者希望提高所承担的系统性风险，获取更高的风险收益，应进入股指期货多头，上式大于零；当时，意味着投资者希望降低所承担的系统性风险，应进入股指期货空头，上式小于零。显然最优套期保值比率是目标的特例。,34,Hedging an Equity Portfolio,35,总结：投资性资产的远期/期货价格,36,2.5 风险中性假设远期和期货价格的决定,无套利均
33、衡分析的过程和结果与市场参与者的风险偏好无关。如果对一个问题的分析过程与投资者的风险偏好无关，则可以将问题放到一个假设的风险中性世界里进行分析，所得到的结果在真实的世界中也应当成立。风险中性假设：在风险中性世界里，所有参与者都是风险中性的，所有的资产不管其风险大小如何，总的预期收益率都等于无风险收益率。所有资产现在的市场均衡价格都是未来预期值用无风险利率折现后的现值。,37,2.5 风险中性假设远期和期货价格的决定,远期合约在到期日的多头价值为：远期合约在0时刻的价值为：在风险中性世界中，无收益资产价格增长率 因此,38,2.5 风险中性假设远期和期货价格的决定,远期合约在到期日的多头价值为：
34、远期合约在0时刻的价值为：在风险中性世界中，红利收益率q的资产价格增长率为，因此,39,本书的讨论焦点：标的资产为投资性资产（截止目前考虑的是投资性金融资产）。所谓投资性资产是指投资者主要出于投资目的而持有的资产，如股票、债券等金融资产和黄金、白银等资产。由于投资性资产的投资决策不受消费等其他目的的影响，投资者所关注的是金融资产中所蕴涵的风险收益特征而非金融产品本身，因此标的资产及其期货之间存在高度的可替代性，只要相对价格水平不合理，就有套利活动。所以，在这样的市场上，只要没有其他的制度制约套利行为，期货的定价就成为一个纯粹的风险收益问题，相应地无套利原则和持有成本模型就成为远期/期货定价的基
35、本原理。,39,40,消费性资产则是指那些投资者主要出于消费目的而持有的资产，如石油、铜、农产品等。对于消费性资产来说，远期定价公式不再适用，而是转化为：原因在于消费性的标的资产具有消费价值，而远期却无法即时消费，消费性的标的资产与其远期之间并不具有完全的可替代性。因此即使在远期价格相对偏低的时候投资者也不会轻易出售现货，购买远期，从而使得单纯基于风险收益考虑的金融无套利原则不再完全有效。,41,消费性资产与便利收益（convenient yields）便利收益衡量了 中，左边小于右边的程度。对于投资性资产而言，便利收益y0。便利收益反映了市场对未来商品可获得性的预期。在远期/期货合约有效期间
36、，标的资产越短缺，实际运营对标的资产的需求越大，资产消费性越强，则便利收益就越高，远期/期货价格距离单纯出于无套利考虑的价位就越远。,42,总结：远期/期货价格,注：c为持有成本（cost of carry），storage cost plus the interest that is paid to finance the asset less the income earned on the asset。,43,无套利条件下，。可以从两个角度分析F和S之间的关系：第一，当标的资产在远期（期货）存续期内没有收益、已知现金收益较小、或已知收益率小于无风险利率时，当前远期（期货）价格应高于标的资
37、产的当前现货价格；当标的资产在远期（期货）存续期内的已知现金收益较大或已知收益率大于无风险利率时，当前远期（期货）价格应小于标的资产的当前现货价格。在远期（期货）到期日，远期（期货）价格将收敛于标的资产的现货价格（这是套利行为决定的）。,2.1同一时刻远期（期货）价格与标的资产现货价格的关系,44,第二，对定价公式进行变换，可得 在现实生活中，大量实证研究表明，在面临新的市场信息冲击时，投资者越来越多地先在远期（期货）市场上进行操作，使得新信息往往先在远期（期货）市场上得到反映，然后才传达至现货市场，从而使得F反过来具有引领S价格变化的信号功能。当前远期（期货）价格对当前现货价格的这种引领作用也被称为远期（期货）的“价格发现”（Price Discovery）功能。,44,45,45,46,根据资本资产定价原理，若标的资产的系统性风险为0，则y=r，F=E(ST)；若标的资产的系统性风险大于零，则yr，FE(ST)。在现实生活中，大多数标的资产的系统性风险都大于零，因此在大多数情况下，F都小于E(ST)。,46,47,期中作业,、对比ETF指数套利与传统指数套利对股指期货定价效率的影响。、认识沪深300股指期货合约，并计算交易一份沪深300股指期货需要交纳的初始保证金数量。以上问题以打印方式，于11 月7 日交作业。,
