期货、期权及衍生品-第三章
约翰.赫尔,期权期货和其他衍生品(third edition)习题答案
12.1 一个证券组合当前价值为＄1000万，β值为1.0，S&P100目前位于250，解释一个执行价格为240。
标的物为S&P100的看跌期权如何为该组合进行保险？当S&P100跌到480，这个组合的期望价值是10 ×（480/500）=＄9.6million.买看跌期权10，000，000/500=20，000可以防止这个组合下跌到＄9.6million下的损失。
因此总共需要200份合约12.2 “一旦我们知道了支付连续红利股票的期权的定价方法，我们便知道了股票指数期权、货币期权和期货期权的定价”。
请解释这句话。
一个股票指数类似一个连续支付红利的股票12.3 请说明日圆看涨期权与日圆期货看涨期权的不同之处一个日元的看涨期权给了持有者在未来某个时刻以确定的价格购买日圆的权利，一个日圆远期看涨期权给予持有者在未来时刻远期价格超过特定范围按原先价格购买日圆的权利。
如果远期齐权行使，持有者将获得一个日圆远期和约的多头。
12.4请说明货币期权是如何进行套期保值的？12.5 计算3个月期，处于平价状态的欧式看涨股票指数期权的价值。
指数为250。
无风险年利率为10%，指数年波动率为18%，指数的年红利收益率为3%。
一个日元的看涨期权给了持有者在未来某个时刻以确定的价格购买日圆的权利，一个日圆远期看涨期权给予持有者在未来时刻远期价格超过特定范围按原先价格购买日圆的权利。
如果远期齐权行使，持有者将获得一个日圆远期和约的多头。
12.6 有一美式看涨期货期权，期货合约和期权合约同时到期。
在任何情况下期货期权比相应的标的物资产的美式期权更值钱？当远期价格大于即期价格时，美式远期期权在远期和约到期前的价值大于相对应的美式期权/12.7 计算5个月有效期的欧式看跌期货期权的价值。
期货价格为＄19，执行价格为＄20，无风险年利率为12%。
期货价格的年波动率为20%。
本题中12.8 假设交易所构造了一个股票指数。
期权期货及其衍生品
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感谢观看
期权期货及其衍生品
期权期货及其衍生品是金融市场的重要组成部分，它们为投资者提 供了丰富的投资策略和风险管理工具。
期权期货及其衍生品的重要性
风险管理
期权期货及其衍生品可以帮助投资者管理风险，通过买入 或卖出相应的衍生品，投资者可以控制风险敞口，减少潜 在的损失。
投资策略
期权期货及其衍生品为投资者提供了丰富的投资策略，如 套期保值、套利交易、价差交易等，这些策略有助于投资 者实现投资目标。
详细描述
期货是一种标准化的金融合约，买卖双方约定在未来某一特定日期或该日之前按照约定的价格交割标的资产。期 货市场是一个高度组织化和规范化的市场，具有价格发现、套期保值和投机等功能。通过期货交易，参与者可以 规避价格波动风险、锁定未来采购或销售成本、进行套利交易等。
衍生品市场概述
总结词
衍生品市场是金融市场的一个重要组成部分，主要交易各种衍生品合约，如期权、期货、掉期等。衍 生品市场具有高风险、高收益的特点，对经济和金融市场的影响深远。
权。
详细描述
期权是一种金融合约，其价值来源于标的资产的价格变动。持有期权的人有权在未来的 特定时间或之前，以特定价格买入或卖出标的资产。根据权利的不同，期权可以分为看 涨期权和看跌期权。看涨期权赋予持有者买入标的资产的权利，而看跌期权赋予持有者
卖出标的资产的权利。
期货定义与功能
总结词
期货是一种标准化的金融合约，买卖双方约定在未来某一特定日期或该日之前按照约定的价格交割标的资产。期 货市场具有价格发现、套期保值和投机等功能。
06
结论
期权期货及其衍生品的影响
市场波动性
期权期货及其衍生品可以增加市场的 波动性，因为它们为投资者提供了更 多的交易策略和机会，从而增加了市 场的交易量。
期货、期权及衍生品-第三章
向前滚动对冲
2003年11月，美国某公司借入2年期、到期本息为 1000万英镑的债务，为避免英镑升值，该公司决 定用英镑期货进行向前滚动对冲。 IMM每份英镑期货合约的价值为62,500英镑（英 镑合约为3、6、9、12四个季月）
例：假设今天为5月15日，某石油生产商签订了一份合约：
• 8月15日：将期货进行平仓。
结果: （到期时现货价格分别是55美元与65美元）
选择交割的月份
若实物交 割，尽量 选择与对 冲到期日 一致的交 割月份
若不交 割，选 择随后 交割月 份
• 芝加哥交易所玉米期货合约的交割月份包括3、6、 9、12月，当对冲期限分别是6、7、1月时，应该 选择哪种合约？
* * S2 F1 F2 F1 (S2 F2 ) (S2 S2 )
交叉对冲：当被对冲的资产与期货的标的资产不一致时，称
为交叉对冲。
期货的对冲
• 采用什么形式的对冲策略
• 选用哪一品种期货合约
• 采用期货合约的最佳头寸与时间
合约的选择
• 选择期货合约的标的资产 • 选择对冲的标准是使得期货价格与被对冲资 产价格的相关性达到最大。
*
* h QA * N QF
*
（3-3）
（3-2）
• V=QxP=数量x价格
尾随对冲
• 对于上例中：某公司预计一个月之后买进200万加 仑飞机燃料油，NYMEX每份加热油期货合约规模 为4万2，最佳对冲比率是0.778，假设现在期货价 格与现货价格分别是1.99美元与1.94美元
• VA=200万×1.94=388万
总收入S2+F1-F2 =F1+b2=2.2+0.1=2.3 (总支出S2+F1-F2 =F1+b2= 2.2+0.1=2.3）
期权、期货及其他衍生品定价理论教学大纲
期权、期货及其他衍生品定价理论教学大纲制作人吴可任课教师吴可一、课程名称：期权、期货及其他衍生品定价理论Option, Futures, and other Derivatives Pricing theory二、课程编码：三学时与学分学时：32 /2四、先修课程证券投资分析，金融工程学五、课程教学目标本课程为金融工程概论的后续课程。
专门讲授金融衍生产品的定价与套利技术。
要求学生学习和掌握各种金融衍生产品定价与市场套利技术。
包括期权、期货、远期、互换等产品的定价与套利技术；期权定价模型的扩展、数值求解、奇异期权定价，以及在险价值求解。
目的是要求学生通过定价理论与方法的学习，深入理解和掌握金融产品的定价与套利的关系。
为熟练与科学运用金融衍生产品工具有效进行套期保值和套利奠定坚实的理论基础和技术基础。
六、适用学科专业经济、金融专业及管理学专业七、基本教学内容与学时安排第一章金融工程的基本分析方法（4学时）第一节MM理论及其涵义第二节无套利定价法第三节风险中性定价法第四节状态价格定价技术第五节积木组合分析法第二章远期和期货的定价（4学时）第一节金融远期和期货市场概述第二节远期价格和期货价格的关系第三节远期定价及其应用第三章互换定价方法（4学时）第一节互换市场概述第二节金融互换的种类第三节互换的定价及其应用第四章B-S期权定价模型（4学时）第一节期权市场概述第二节证券价格的变化过程第三节B-S模型推导第四节B-S期权定价公式的实证研究和应用第五节股票指数期权、货币期权、期货期权定价第五章B-S期权定价公式的扩展（4学时）第一节B-S定价公式的缺陷第二节交易成本第三节波动率微笑和波动率期限结构第四节随机波动率第五节不确定的参数第六节跳跃扩展过程第七节崩盘模型第六章期权定价的数值解法（4学时）第一节二叉树期权定价模型第二节蒙特卡洛模拟第三节有限差分法第五章奇异期权定价（4学时）第一节奇异期权概述第二节奇异期权定价第七章套期保值与套利（4学时）第一节套期保值与套利的概念第二节基于衍生工具的套期保值技术第三节基于衍生工具的套利技术第八章在险价值（4学时）第一节在险价值的定义第二节资产组合的在险价值计算第三节衍生工具在险价值计算第四节蒙特卡罗模拟与历史模拟八、教材及参考书：教材：1．郑振龙，金融工程学，厦门大学出版社出版2.John Hull， Option, Futures, and other Derivatives 清华大学出版社参考书：1．郑振龙，金融工程学，厦门大学出版社出版2．孙金龙史永东，现代金融工程中国金融出版社3．陈信华，金融衍生工具，上海财经大学出版社4．洛伦兹·格利茨，金融工程学（修订版），经济科学出版社5. 叶永刚、郑康彬，金融工程概论，武汉大学出版社6．陆家骝，现代金融经济学，东北财经大学出版社九、考核方式书面考试＋小论文，参考平时作业。
期货期权交易管理制度模板
第一章总则第一条为规范公司期货期权交易行为，有效防范和管控市场风险，确保公司资产安全，维护公司及股东利益，依据《中华人民共和国公司法》、《中华人民共和国证券法》、《中华人民共和国期货和衍生品法》等相关法律法规，结合公司实际情况，制定本制度。
第二条本制度所称期货交易，是指以期货合约或者标准化期权合约为交易标的的交易活动。
衍生品交易，是指期货交易以外的，以互换合约、远期合约和非标准化期权合约及其组合为交易标的的交易活动。
第三条本制度适用于公司及其控股子公司（以下简称“公司”）的期货期权交易活动。
第二章组织与管理第四条公司设立期货期权交易管理委员会（以下简称“委员会”），负责制定期货期权交易策略，审批交易权限，监督交易执行，评估交易风险，并定期向公司董事会报告。
第五条委员会成员由公司高级管理人员、财务部、风险控制部等相关部门负责人组成。
第六条公司期货期权交易业务由公司授权的部门负责，具体职责如下：1. 制定期货期权交易计划；2. 审批交易权限；3. 监督交易执行；4. 评估交易风险；5. 定期向委员会报告。
第三章交易规则第七条公司期货期权交易应以套期保值为主要目的，严格遵守以下原则：1. 交易品种与公司主营业务相关；2. 交易规模与公司风险承受能力相匹配；3. 交易期限与公司风险管理需求相适应。
第八条公司期货期权交易应遵循以下规则：1. 严格遵守相关法律法规和交易所规则；2. 交易前应进行充分的市场调研和风险评估；3. 交易过程中应实时监控市场变化，及时调整交易策略；4. 交易后应定期进行交易回顾和总结。
第九条公司期货期权交易权限的审批程序如下：1. 各部门提出期货期权交易申请；2. 经授权的部门审批；3. 报委员会备案。
第四章风险控制第十条公司应建立健全期货期权交易风险控制体系，包括：1. 交易风险评估模型；2. 风险预警机制；3. 风险止损机制；4. 风险报告制度。
第十一条公司应定期对期货期权交易风险进行评估，确保风险在可控范围内。
课件期货期权其他衍生品第三章
客户保证金
客户保证金的收取比例可由期货经纪公司规定，但不得低于 交易所对经纪公司收取的保证金。 经纪公司对客户保证金进行内部管理时也分为结算保证金和 交易保证金，但对客户而言（如经纪公司出具给客户的交易 结算单）结算保证金和交易保证金二者归为一体，只有保证 金一项。 经纪公司对客户的保证金应在交易所的交易保证金基础上加 2%～3%。但一般情况下，特别是交易不太活跃时，经纪公司 对客户收取的保证金与交易所规定的保证金水平一致。
会员交易保证金不足并未能在规定时限内补足； 持仓量超出其限仓规定标准； 因违规受到交易所强行平仓处罚的； 根据交易所的紧急措施应予强行平仓的； 其他需要强行平仓的。
信息披露制度
信息披露制度是指期货交易所按有关规定定期公 布期货交易有关信息的制度。它包括即时、每日、每 周、每月的交易信息。 即时交易信息
成交回报与结算单的确认
书面下单方式下成交回报与确认要方式 自助下单和网上下单方式成交回报与确认
风险管理
风险率
风险率=客户权益÷按经纪公司规定的保证金比例客户所 有头寸（持仓）占有保证金总额×100%。 客户权益=上日资金余额±当日资金存取±当日平仓盈亏 ±实物交割款项－当日交易手续费±当日浮动盈亏。 对于当日资金存取，存入（入金）资金为+，取出资金 （出金）为－，一般客户没有实物交割款项。 当日实际平仓盈亏指当天所有平仓产生盈亏加总。 当日浮动平仓盈亏指所有未平仓合约（持仓）平仓盈亏总 和，这里要注意的是平仓价均采用当日的结算价。
每周期货交易信息
指交易所在每周最后一个交易日结束后公布的期货交易信息。 信息内容主要有：商品合约名称、周开盘价、周最高价、周最低 价、周收盘价、涨跌（周末收盘价与上周末收盘价之差）、持仓量、 周持仓量变化、周结算价、周成交量、周成交额；各上市商品标准仓 单（交割仓库在完成卖方商品的入库商品验收，确认合格后签发给卖 方的商品所有权凭证）数量及与上次发布的增减量，已申请交割数量 及本周进出库数量。
期货期权及其衍生品配套课件(全34章)Ch03.ppt
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2008
7
Short Hedge
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2008
8
Choice of Contract
Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. This is known as cross hedging.
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2008
6
Long Hedge
We define
F1 : Initial Futures Price F2 : Final Futures Price S2 : Final Asset Price If you hedge the future purchase of an asset by entering into a long futures contract then
期货从业资格考试《期货基础知识》知识点汇总
期货从业资格考试《期货基础知识》知识点汇总第一章期货市场概述第一节期货市场的产生和发展1、期货交易的起源于欧洲19世纪中期产生于美国芝加哥，1848年芝加哥期货交易所（CBOT）现代期货市场：标准化合约、保证金制度、对冲机制、统一结算芝加哥商业交易所（CME）:最大的肉类和畜类期货交易中心2007年CBOT和CME合并为芝加哥商业交易所集团（CME Group）:最大期货交易所伦敦金属交易所（LME）:最大的有色金属期货交易中心。
2、期货交易与现货交易、远期交易的关系3、期货交易的基本特征合约标准化、杠杆机制（保证金5%-10%，比例越低作用越大）、双向交易、对冲机制、当日无负债结算制度（下一日开市前）、交易集中化（会员制）4、期货市场的发展商品期货农产品期货金属期货主要品种为有色金属（除铁铬锰外的金属），诞生于英国，纽约商业交易所（NYMEX）拥有成交量最大的黄金期货合约能源期货纽约商业交易所（NYMEX）最具影响力金融期货外汇期货CME设立国际货币市场分部（IMM）利率期货:CBOT上市国民抵押协会债券（GNMA）合约股指期货:美国堪萨斯期货交易所（KCBT）开发价值线综合指数期货合约其他品种天气期货:CME推出，包括温度期货、降雪量期货、霜冻期货、飓风期货指数期货5、期货交易在衍生品交易中的地位交易量：北美:亚太:欧洲衍生品：股指类期货期权:利率类:外汇类衍生品远期:远期外汇合约受欢迎期货期权:选择权，分为看涨期权和看跌期权互换:利率互换和外汇互换第二节期货市场的功能与作用1、期货市场的规避风险、价格发现功能规避风险借助套期保值建立盈亏冲抵机制，锁定成本、稳定收益同种商品的期货价格和现货价格走势一致期货价格和现货价格随期货合约到期日的来临呈现趋同性投机者、套利者的参与是套期保值实现的条件价格发现:预期性、连续性、公开性、权威性2、期货市场的作用锁定生产成本，实现预期利润利用期货价格信号安排生产经营活动提供分散、转移价格风险的工具，有助于稳定国民经济为政府宏观政策的制定提供参考依据有助于现货市场的完善和发展有助于增强国际价格行程中的话语权第三节中国期货市场的发展历程1、中国期货市场的建立1990年10月12日中国郑州粮食批发市场:第一个商品期货市场1991年6月10日深圳有色金属交易所，1992年上海金属交易所1992年9月广东万通期货经纪公司:第一家期货经纪公司1992年底中国国际期货经纪公司2、中国期货市场的治理整顿2000年12月29日中国期货业协会:期货行业自律组织清理后，上海期货交易所、郑州商品交易所、大连商品交易所，品种12种1995年2月国债期货“3.27”事件和5月“3.19”事件:“市场禁止进入制度”3、中国期货市场的规范发展2002年，证监会：《期货从业人员管理办法》、《期货经纪公司高级管理人员任职资格管理办法》、《期货交易所管理办法》、《期货经纪公司管理办法》:奠定基础2003年，最高法院：《最高人民法院关于审理期货纠纷案件若干问题的规定》:完善2003年，《期货从业人员执业行为准则》:第一部行业性行为规范2004年，《国务院关于推进资本市场改革开放和稳定发展的若干意见》:期货经纪公司建设成为金融企业2004年，证监会：《期货经纪公司治理准则》:明确公司制度规定2007年，国务院修订《期货交易管理条例》适用范围由商品期货交易扩大到商品、金融期货和期权合约，实行分级结算制度“期货经纪公司”改为“期货公司”，实行许可制度，由国务院监管机构颁发证期货公司还可申请经营境外期货经纪、期货投资咨询及其他业务2006年5月18日，中国期货保证金监控中心成立，由上海期货交易所、大连商品交易所、郑州商品交易所出资兴办，为非营利性公司制法人。
(NEW)赫尔《期权、期货及其他衍生产品》教材精讲讲义
(NEW)赫尔《期权、期货及其他衍生产品》教材精讲讲义简介赫尔的《期权、期货及其他衍生产品》是一本经典的金融学教材，被广泛用于大学金融学课程的教学。
本文档将对该教材进行精讲，涵盖主要内容和关键概念，旨在帮助读者深入理解和掌握期权、期货及其他衍生产品领域的知识。
本文档采用Markdown格式，方便阅读和使用。
第一章：期权市场简介1.1 期权的定义和特点期权是一种金融衍生工具，它赋予持有者在未来某个时间以特定价格买入或卖出某一标的资产的权利。
期权的特点包括灵活性、杠杆作用、风险限定和多样性等。
1.2 期权市场的组织和参与者期权市场包括交易所市场和场外市场。
交易所市场由交易所组织和管理，参与者包括期权合约买方、卖方、证券公司和交易所监管机构等。
1.3 期权定价模型期权定价模型是评估期权价格的数学模型，常用的模型包括布莱克-斯科尔斯模型和基于风险中性定价的模型。
第二章：期权定价理论2.1 基本期权定价理论基本期权定价理论包括不含股息的欧式期权定价、含股息的欧式期权定价以及美式期权定价等。
2.2 期权市场交易策略期权市场交易策略包括买入期权、卖出期权、期权组合以及期权套利等。
2.3 隐含波动率与期权定价隐含波动率是指根据期权市场价格反推出的波动率水平，它对期权价格的波动具有重要影响。
第三章：期权交易策略3.1 期权买入策略期权买入策略包括买入认购期权、买入认沽期权和买入期权组合等，旨在获得价差和方向性收益。
3.2 期权卖出策略期权卖出策略包括卖出认购期权、卖出认沽期权和卖出期权组合等，旨在获取权利金收入和时间价值消耗。
3.3 期权组合策略期权组合策略包括多头组合和空头组合，以及各种组合的调整和套利策略。
第四章：期货市场简介4.1 期货合约的基本特点期货合约是一种标准化的合约，约定了在未来某个时间以特定价格交割特定数量的标的资产。
4.2 期货交易所和市场参与者期货交易所是组织和管理期货市场的机构，市场参与者包括期货合约买方、卖方、交易所监管机构和期货经纪人等。
HullOFOD eSolutionsCh 第九版期权期货及其他衍生品课后答案
CHAPTER 3Hedging Strategies Using FuturesPractice QuestionsProblem 3.1.Under what circumstances are (a) a short hedge and (b) a long hedge appropriate?A short hedge is appropriate when a company owns an asset and expects to sell that asset in the future. It can also be used when the company does not currently own the asset but expects to do so at some time in the future. A long hedge is appropriate when a company knows it will have to purchase an asset in the future. It can also be used to offset the risk from an existing short position.Problem 3.2.Explain what is meant by basis risk when futures contracts are used for hedging.Basis risk arises from the hedger’s uncertainty as to the difference between the spot price and futures price at the expiration of the hedge.Problem 3.3.Explain what is meant by a perfect hedge. Does a perfect hedge always lead to a better outcome than an imperfect hedge? Explain your answer.A perfect hedge is one that completely eliminates the hedger’s risk. A p erfect hedge does not always lead to a better outcome than an imperfect hedge. It just leads to a more certain outcome. Consider a company that hedges its exposure to the price of an asset. Suppose the asset’s price movements prove to be favorable to the company. A perfect hedge totally neutralizes the company’s gain from these favorable price movements. An imperfect hedge, which only partially neutralizes the gains, might well give a better outcome.Problem 3.4.Under what circumstances does a minimum-variance hedge portfolio lead to no hedging at all?A minimum variance hedge leads to no hedging when the coefficient of correlation between the futures price changes and changes in the price of the asset being hedged is zero.Problem 3.5.Give three reasons why the treasurer of a company might not hedge the company’s exposure to a particular risk.(a) If the company’s competitors are not hedging, the treasurer might feel that the company will experience less risk if it does not hedge. (See Table 3.1.) (b) The shareholders might not want the company to hedge because the risks are hedged within their portfolios. (c) If there is a loss on the hedge and a gain from the company’s exposure to the underlying asset, the treasurer might feel that he or she will have difficulty justifying the hedging to other executives within the organization.Problem 3.6.Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65, the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and the coefficient of correlation between the two changes is 0.8. What is the optimal hedge ratio for a three-month contract? What does it mean?The optimal hedge ratio is 065080642081..⨯=.. This means that the s ize of the futures position should be 64.2% of the size of the company’s exposure in a three-month hedge.Problem 3.7.A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts on a stock index to hedge its risk. The index futures is currently standing at 1080, and each contract is for delivery of $250 times the index. What is the hedge that minimizes risk? What should the company do if it wants to reduce the beta of the portfolio to 0.6?The formula for the number of contracts that should be shorted gives 20000000128891080250,,.⨯=.⨯ Rounding to the nearest whole number, 89 contracts should be shorted. To reduce the beta to 0.6, half of this position, or a short position in 44 contracts, is required.Problem 3.8.In the corn futures contract, the following delivery months are available: March, May, July, September, and December. State the contract that should be used for hedging when the expiration of the hedge is in a) June, b) July, and c) JanuaryA good rule of thumb is to choose a futures contract that has a delivery month as close aspossible to, but later than, the month containing the expiration of the hedge. The contracts that should be used are therefore(a) July(b)September(c)MarchProblem 3.9.Does a perfect hedge always succeed in locking in the current spot price of an asset for a future transaction? Explain your answer.No. Consider, for example, the use of a forward contract to hedge a known cash inflow in a foreign currency. The forward contract locks in the forward exchange rate — which is in general different from the spot exchange rate.Problem 3.10.Explain why a short hedger’s position improves when the basis strengthens unexpectedly and worsens when the basis weakens unexpectedly.The basis is the amount by which the spot price exceeds the futures price. A short hedger is long the asset and short futures contracts. The value of his or her position therefore improves as the basis increases. Similarly, it worsens as the basis decreases.Problem 3.11.Imagine you are the treasurer of a Japanese company exporting electronic equipment to the United States. Discuss how you would design a foreign exchange hedging strategy and the arguments you would use to sell the strategy to your fellow executives.The simple answer to this question is that the treasurer should1.Estimate the company’s future cash flows in Japanese yen and U.S. dollars2.Enter into forward and futures contracts to lock in the exchange rate for the U.S. dollarcash flows.However, this is not the whole story. As the gold jewelry example in Table 3.1 shows, the company should examine whether the magnitudes of the foreign cash flows depend on the exchange rate. For example, will the company be able to raise the price of its product in U.S. dollars if the yen appreciates? If the company can do so, its foreign exchange exposure may be quite low. The key estimates required are those showing the overall effect on the company’s profitability of changes in the exchange rate at various times in the future. Once these estimates have been produced the company can choose between using futures and options to hedge its risk. The results of the analysis should be presented carefully to other executives. It should be explained that a hedge does not ensure that profits will be higher. It means that profit will be more certain. When futures/forwards are used both the downside and upside are eliminated. With options a premium is paid to eliminate only the downside.Problem 3.12.Suppose that in Example 3.2 of Section 3.3 the company decides to use a hedge ratio of 0.8. How does the decision affect the way in which the hedge is implemented and the result?If the hedge ratio is 0.8, the company takes a long position in 16 December oil futures contracts on June 8 when the futures price is $88.00. It closes out its position on November 10. The spot price and futures price at this time are $90.00 and $89.10. The gain on the futures position is(89.10 − 88.00) × 16,000 = 17,600The effective cost of the oil is therefore20,000 × 90 – 17,600 = 1,782, 400or $89.12 per barrel. (This compares with $88.90 per barrel when the company is fully hedged.)Problem 3.13.“If the minimum -variance hedge ratio is calculated as 1.0, the hedge must be perfect." Is this statement true? Explain your answer.The statement is not true. The minimum variance hedge ratio is S Fσρσ It is 1.0 when 05=.ρ and 2S F =σσ. Since 10<.ρ the hedge is clearly not perfect.Problem 3.14.“If there is no basis risk, the minimum variance hedge ratio is always 1.0." Is this statement true? Explain your answer.The statement is true. Using the notation in the text, if the hedge ratio is 1.0, the hedger locks in a price of 12F b +. Since both 1F and 2b are known this has a variance of zero and must be the best hedge.Problem 3.15“For an asset where futures prices for contracts on the asset are usually less than spot prices, long hedges are likely to be particularly attractive." Explain this statement.A company that knows it will purchase a commodity in the future is able to lock in a price close to the futures price. This is likely to be particularly attractive when the futures price is less than the spot price.Problem 3.16.The standard deviation of monthly changes in the spot price of live cattle is (in cents per pound)1.2. The standard deviation of monthly changes in the futures price of live cattle for the closestcontract is 1.4. The correlation between the futures price changes and the spot price changes is 0.7. It is now October 15. A beef producer is committed to purchasing 200,000 pounds of live cattle on November 15. The producer wants to use the December live-cattle futures contracts to hedge its risk. Each contract is for the delivery of 40,000 pounds of cattle. What strategy should the beef producer follow?The optimal hedge ratio is 12070614..⨯=.. The beef producer requires a long position in 20000006120000⨯.=, lbs of cattle. The beef producer should therefore take a long position in 3 December contracts closing out the position on November 15.Problem 3.17.A corn farmer argues “I do not use futures co ntracts for hedging. My real risk is not the price of corn. It is that my whole crop gets wiped out by the weather.”Discuss this viewpoint. Should the farmer estimate his or her expected production of corn and hedge to try to lock in a price for expected production?If weather creates a significant uncertainty about the volume of corn that will be harvested, the farmer should not enter into short forward contracts to hedge the price risk on his or her expected production. The reason is as follows. Suppose that the weather is bad and the farmer’sproduction is lower than expected. Other farmers are likely to have been affected similarly. Corn production overall will be low and as a consequence the price of corn will be relatively high. The farmer’s problems arising from the bad harvest will be made worse by losses on the short futures position. This problem emphasizes the importance of looking at the big picture when hedging. The farmer is correct to question whether hedging price risk while ignoring other risks is a good strategy.Problem 3.18.On July 1, an investor holds 50,000 shares of a certain stock. The market price is $30 per share. The investor is interested in hedging against movements in the market over the next month and decides to use the September Mini S&P 500 futures contract. The index is currently 1,500 and one contract is for delivery of $50 times the index. The beta of the stock is 1.3. What strategy should the investor follow? Under what circumstances will it be profitable?A short position in 50000301326501500,⨯.⨯=⨯,contracts is required. It will be profitable if the stock outperforms the market in the sense that its return is greater than that predicted by the capital asset pricing model.Problem 3.19.Suppose that in Table 3.5 the company decides to use a hedge ratio of 1.5. How does the decision affect the way the hedge is implemented and the result?If the company uses a hedge ratio of 1.5 in Table 3.5 it would at each stage short 150 contracts. The gain from the futures contracts would be.1=⨯50.2$5570.1per barrel and the company would be $0.85 per barrel better off than with a hedge ratio of 1.Problem 3.20.A futures contract is used for hedging. Explain why the daily settlement of the contract can give rise to cash flow problems.Suppose that you enter into a short futures contract to hedge the sale of an asset in six months. If the price of the asset rises sharply during the six months, the futures price will also rise and you may get margin calls. The margin calls will lead to cash outflows. Eventually the cash outflows will be offset by the extra amount you get when you sell the asset, but there is a mismatch in the timing of the cash outflows and inflows. Your cash outflows occur earlier than your cash inflows.A similar situation could arise if you used a long position in a futures contract to hedge the purchase of an asset at a future time and the asset’s price fe ll sharply. An extreme example of what we are talking about here is provided by Metallgesellschaft (see Business Snapshot 3.2).Problem 3.21.An airline executive has argued: “There is no point in our using oil futures. There is just as much chance that the price of oil in the future will be less than the futures price as there is that it will be greater than this price.” Discuss the executive’s viewpoint.It may well be true that there is just as much chance that the price of oil in the future will be above the futures price as that it will be below the futures price. This means that the use of a futures contract for speculation would be like betting on whether a coin comes up heads or tails. But it might make sense for the airline to use futures for hedging rather than speculation. The futures contract then has the effect of reducing risks. It can be argued that an airline should not expose its shareholders to risks associated with the future price of oil when there are contracts available to hedge the risks.Problem 3.22.Suppose the one-year gold lease rate is 1.5% and the one-year risk-free rate is 5.0%. Both rates are compounded annually. Use the discussion in Business Snapshot 3.1 to calculate the maximum one-year forward price Goldman Sachs should quote for gold when the spot price is $1,200.Goldman Sachs can borrow 1 ounce of gold and sell it for $1200. It invests the $1,200 at 5% so that it becomes $1,260 at the end of the year. It must pay the lease rate of 1.5% on $1,200. This is $18 and leaves it with $1,242. It follows that if it agrees to buy the gold for less than $1,242 in one year it will make a profit.Problem 3.23.The expected return on the S&P 500 is 12% and the risk-free rate is 5%. What is the expected return on the investment with a beta of (a) 0.2, (b) 0.5, and (c) 1.4?a)00502(012005)0064.+.⨯.-.=. or 6.4%b)00505(012005)0085.+.⨯.-.=. or 8.5%c)00514(012005)0148.+.⨯.-.=. or 14.8%Further QuestionsProblem 3.24.It is now June. A company knows that it will sell 5,000 barrels of crude oil in September.It uses the October CME Group futures contract to hedge the price it will receive. Each contract is on 1,000 barrels of ‘‘light sweet crude.’’ What position should it take? What price risks is it still exposed to after taking the position?It should short five contracts. It has basis risk. It is exposed to the difference between the October futures price and the spot price of light sweet crude at the time it closes out its position in September. It is also possibly exposed to the difference between the spot price of light sweet crude and the spot price of the type of oil it is selling.Problem 3.25.Sixty futures contracts are used to hedge an exposure to the price of silver. Each futures contract is on 5,000 ounces of silver. At the time the hedge is closed out, the basis is $0.20per ounce. What is the effect of the bas is on the hedger’s financial position if (a) the traderis hedging the purchase of silver and (b) the trader is hedging the sale of silver?The excess of the spot over the futures at the time the hedge is closed out is $0.20 per ounce. If the trader is hedging the purchase of silver, the price paid is the futures price plus the basis. Thetrader therefore loses 60×5,000×$0.20=$60,000. If the trader is hedging the sales of silver, the price received is the futures price plus the basis. The trader therefore gains $60,000.Problem 3.26.A trader owns 55,000 units of a particular asset and decides to hedge the value of her position with futures contracts on another related asset. Each futures contract is on 5,000 units. The spot price of the asset that is owned is $28 and the standard deviation of the change in this price over the life of the hedge is estimated to be $0.43. The futures price of the related asset is $27 and the standard deviation of the change in this over the life of the hedge is $0.40. The coefficient of correlation between the spot price change and futures price change is 0.95.(a) What is the minimum variance hedge ratio?(b) Should the hedger take a long or short futures position?(c) What is the optimal number of futures contracts with no tailing of the hedge?(d) What is the optimal number of futures contracts with tailing of the hedge?(a) The minimum variance hedge ratio is 0.95×0.43/0.40=1.02125.(b) The hedger should take a short position.(c) The optimal number of contracts with no tailing is 1.02125×55,000/5,000=11.23 (or 11 when rounded to the nearest whole number)(d) The optimal number of contracts with tailing is 1.012125×(55,000×28)/(5,000×27)=11.65 (or 12 when rounded to the nearest whole number).Problem 3.27.A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6correlation with gasoline futures price changes. The company will lose $1 million for each 1 cent increase in the price per gallon of the new fuel over the next three months. The new fuel's price change has a standard deviation that is 50% greater than price changes in gasoline futures prices. If gasoline futures are used to hedge the exposure what should the hedge ratio be? What is the company's exposure measured in gallons of the new fuel? What position measured in gallons should the company take in gasoline futures? How many gasoline futures contracts should be traded? Each contract is on 42,000 gallons.Equation (3.1) shows that the hedge ratio should be 0.6 × 1.5 = 0.9. The company has anexposure to the price of 100 million gallons of the new fuel. It should therefore take a position of 90 million gallons in gasoline futures. Each futures contract is on 42,000 gallons. The number of contracts required is therefore9.2142000,42000,000,90 or, rounding to the nearest whole number, 2143.Problem 3.28.A portfolio manager has maintained an actively managed portfolio with a beta of 0.2. During the last year the risk-free rate was 5% and equities performed very badly providing a return of−30%. The portfolio manage produced a return of −10% and claims that in the circumstances it was good. Discuss this claim.When the expected return on the market is −30% the expected return on a portfolio with a beta of 0.2 is0.05 + 0.2 × (−0.30 − 0.05) = −0.02or –2%. The actual return of –10% is worse than the expected return. The portfolio manager done 8% worse than a simple strategy of forming a portfolio that is 20% invested in an equity index and 80% invested in risk-free investments. (The manager has achieved an alpha of –8%!)Problem 3.29. (Excel file)The following table gives data on monthly changes in the spot price and the futures price for a certain commodity. Use the data to calculate a minimum variance hedge ratio.Denote i x and i y by the i -th observation on the change in the futures price and the change in the spot price respectively.096130i i x y =.=.∑∑222447423594i i x y =.=.∑∑2352i i x y =.∑ An estimate of F σ is05116=. An estimate of S σ is04933=.An estimate of ρ is0981=.The minimum variance hedge ratio is049330981094605116SF.=.⨯=..σρσProblem 3.30.It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the December futures contract on a stock index to change beta of the portfolio to 0.5 during the period July 16 to November 16. The index is currently1,000, and each contract is on $250 times the index.a)What position should the company take?b)Suppose that the company changes its mind and decides to increase the beta of theportfolio from 1.2 to 1.5. What position in futures contracts should it take?a)The company should short(1205)1000000001000250.-.⨯,,⨯or 280 contracts.b)The company should take a long position in(1512)1000000001000250.-.⨯,,⨯or 120 contracts.Problem 3.31. (Excel file)A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager is concerned about the performance of the market over the next two months and plans to use three-month futures contracts on the S&P 500 to hedge the risk. The current level of the index is 1250, one contract is on 250 times the index, the risk-free rate is 6% per annum, and the dividend yield on the index is 3% per annum. The current 3 month futures price is 1259.a)What position should the fund manager take to eliminate all exposure to the market overthe next two months?b)Calculate the effect of your strategy on the fund manager’s returns if the level of themarket in two months is 1,000, 1,100, 1,200, 1,300, and 1,400. Assume that the one-month futures price is 0.25% higher than the index level at this time.a) The number of contracts the fund manager should short is 50000000087138201259250,,.⨯=.⨯ Rounding to the nearest whole number, 138 contracts should be shorted.b) The following table shows that the impact of the strategy. To illustrate the calculations in the table consider the first column. If the index in two months is 1,000, the futures price is 1000×1.0025. The gain on the short futures position is therefore(1259100250)2501388849250$-.⨯⨯=,,The return on the index is 3212⨯/=0.5% in the form of dividend and250125020%-/=- in the form of capital gains. The total return on the index istherefore 195%-.. The risk-free rate is 1% per two months. The return is therefore205%-. in excess of the risk-free rate. From the capital asset pricing model we expect the return on the portfolio to be 0872*******%%.⨯-.=-. in excess of the risk-free rate. The portfolio return is therefore 16835%-.. The loss on the portfolio is01683550000000.⨯,, or $8,417,500. When this is combined with the gain on the futures the total gain is $431,750.Problem 3.32.It is now October 2014. A company anticipates that it will purchase 1 million pounds of copper in each of February 2015, August 2015, February 2016, and August 2016. The company has decided to use the futures contracts traded in the COMEX division of the CME Group to hedge its risk. One contract is for the delivery of 25,000 pounds of copper. The initial margin is $2,000per contract a nd the maintenance margin is $1,500 per contract. The company’s policy is to hedge 80% of its exposure. Contracts with maturities up to 13 months into the future are considered to have sufficient liquidity to meet the company’s needs. Devise a hedging stra tegy for the company. (Do not make the “tailing” adjustment described in Section 3.4.)Assume the market prices (in cents per pound) today and at future dates are as follows. What is the impact of the strategy you propose on the price the company pays for copper? What is the initial margin requirement in October 2014? Is the company subject to any margin calls?To hedge the February 2015 purchase the company should take a long position in March 2015 contracts for the delivery of 800,000 pounds of copper. The total number of contracts required is 8000002500032,/,=. Similarly a long position in 32 September 2015 contracts is required to hedge the August 2015 purchase. For the February 2016 purchase the company could take a long position in 32 September 2015 contracts and roll them into March 2016 contracts during August 2015. (As an alternative, the company could hedge the February 2016 purchase by taking a long position in 32 March 2015 contracts and rolling them into March 2016 contracts.) For the August 2016 purchase the company could take a long position in 32 September 2015 and roll them into September 2016 contracts during August 2015.The strategy is therefore as followsOct. 2014: Enter into long position in 96 Sept. 2015 contractsEnter into a long position in 32 Mar. 2015 contractsFeb 2015: Close out 32 Mar. 2015 contractsAug 2015: Close out 96 Sept. 2015 contractsEnter into long position in 32 Mar. 2016 contractsEnter into long position in 32 Sept. 2016 contractsFeb 2016: Close out 32 Mar. 2016 contractsAug 2016: Close out 32 Sept. 2016 contractsWith the market prices shown the company pays.+.⨯.-.=.3690008(3723036910)37156for copper in February, 2015. It pays.+.⨯.-.=.3650008(3728036480)37140for copper in August 2015. As far as the February 2016 purchase is concerned, it loses 3728036480800.-.=. on the .-.=. on the September 2015 futures and gains 37670364301240 February 2016 futures. The net price paid is therefore.+.⨯.-.⨯.=.377000880008124037348.-.=. on theAs far as the August 2016 purchase is concerned, it loses 3728036480800.-.=. on the September 2016 futures. The September 2015 futures and gains 38820364202400net price paid is therefore388000880008240037520.+.⨯.-.⨯.=.The hedging strategy succeeds in keeping the price paid in the range 371.40 to 375.20.In October 2014 the initial margin requirement on the 128 contracts is 1282000$⨯, or $256,000. There is a margin call when the futures price drops by more than 2 cents. This happens to the March 2015 contract between October 2014 and February 2015, to the September 2015 contract between October 2014 and February 2015, and to the September 2015 contract between February 2015 and August 2015. (Under the plan above the March 2016 contract is not held between February 2015 and August 2015, but if it were there would be a margin call during this period.)。
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交叉对冲：当被对冲的资产与期货的标的资产不一致时，称
为交叉对冲。
期货的对冲
• 采用什么形式的对冲策略
• 选用哪一品种期货合约
• 采用期货合约的最佳头寸与时间
合约的选择
• 选择期货合约的标的资产 • 选择对冲的标准是使得期货价格与被对冲资 产价格的相关性达到最大。
期货价格收敛到现货价格的特性
• 随着期货合约交割月的逼近，期货价格会逐渐 收敛到标的资产的即期价格，在达到交割日期 时，期货价格会等于或非常接近于即期价格 • 举反例说明：假设在交割时间内，期货的价格高 于现货价格，则交易员可以按如下方式获得无风 险套利：
• 卖出一个期货合约（即持有短头寸） • 买入资产 • 进行交割
例：假设今天为5月15日，某石油生产商签订了一份合约：
• 8月15日：将期货进行平仓。
结果: （到期时现货价格分别是55美元与65美元）
选择交割的月份
若实物交 割，尽量 选择与对 冲到期日 一致的交 割月份
若不交 割，选 择随后 交割月 份
• 芝加哥交易所玉米期货合约的交割月份包括3、6、 9、12月，当对冲期限分别是6、7、1月时，应该 选择哪种合约？
8月15
对冲结果
长头寸对冲
• 定义：持有期货长头寸的对冲策略。当已知将来需要买入 一定资产并想在今天锁定价格时，可以采用长头寸对冲。
例：假定今天为1月15日，某铜产品加工商知道在5月15日 需要买入10万磅铜来履行某合约。 • 当前铜的现货价格为每磅340美分 • 当前铜在5月份的期货价格为每磅320美分 对冲策略 • 1月15日：买入4份5月份的铜期货（每份2万5千磅） • 5月15日：对交易进行平仓。 • 结果： （5月份铜的价格为325与305）
长头寸对冲
对冲结果
日 期 现货市场 铜价格 340美分/磅 买入铜 价格325美分/磅 （305） 花费325000 （花费305000） 期货市场 买出铜期货价格 320美分/磅
1月15
5月15
卖出铜期货价格 325美分/磅（305）
对冲结果
盈利5000 （亏损15000）
总花费320000 （总花费320000）
总收入S2+F1-F2 =F1+b2=2.2+0.1=2.3 (总支出S2+F1-F2 =F1+b2= 2.2+0.1=2.3）
中盛粮油基差风险
生产各类精炼大豆油产品，以供内地市场销售。 对冲方式：卖出CBOT豆油期货。
国内成品大豆油从2005年2月的每吨5800元下 跌到了6月的每吨5000元左右。 国际市场上，美国农业部连续6次下调大豆油 产量预测，导致大豆油和大豆油期货价格节节 攀升。
2003.11，买进160份2004.9到期的英镑期货 英镑期货价格：1英镑=1.6500美元
2004.9，卖出2004.9到期的英镑期货 价格： 1英镑= 1.6550美元 2005.6，卖出6月期货英镑期货 价格：1英镑= 1.6600美元 同时，买进160份2005.6到 期的英镑期货 价格： 1英镑= 1.6570美 元 同时，买进160份2005.12 到期的英镑期货 价格： 1英镑= 1.6630美元 在现货市场上买入1000万 英镑用于还本付息 价格： 1英镑= 1.6655美元
采用期货的对冲策略
期货市场的参与者
• 如何利用期货进行对冲？
期货的对冲
• 采用什么形式的对冲策略
• 选用哪一品种期货合约
• 采用期货合约的最佳头寸与时间
短头寸对冲
• 定义：选择期货的短头寸来进行对冲。 • 当对冲者已经拥有了某种资产并期望在将来某时 刻卖出资产，可选择期货短头寸进行对冲。
短头寸对冲
h* 使得对冲者头寸变化的方差达到极小的对冲比率
最优对冲比率
对冲比率：持有期货合约的头寸大小与资产风 险暴露数量大小的比率
最优对冲比率为
S h F
*
• 例：某商品价格每季度变化的标准差是0.65，关于
该商品的期货价格每季度变化的标准差是0.81，两
种价格变化的相关系数为0.8，那么一份3月期的合
长（短）头寸
种类 现货市场 期货市场 目的
短头寸对冲 现货多头或 者未来要卖 出资产 长头寸对冲 现货空头或 者未来要买 入资产
期货空头
防止现货市 场价格下跌 风险 防止现货市 场价格上升 风险
期货多头
反对对冲的观点
• 竞争者角度 • 例：激进珠宝商不采用对冲措施，保守珠宝商买了期货 合约来对冲未来18个月购买黄金的风险。 黄金价格的变化 对珠宝业价 对激进珠宝 格影响 商利润影 响 上涨 上涨 无影响 对保守珠宝商 利润影响 增加
*
* h QA * N QF
*
（3-3）
（3-2）
• V=QxP=数量x价格
尾随对冲
• 对于上例中：某公司预计一个月之后买进200万加 仑飞机燃料油，NYMEX每份加热油期货合约规模 为4万2，最佳对冲比率是0.778，假设现在期货价 格与现货价格分别是1.99美元与1.94美元
• VA=200万×1.94=388万
长头寸对冲
对冲结果
日 期 1月15 现货市场 铜价格 340美分/磅 买入铜 价格325美分/磅 （305） 期货市场 买出铜期货价格 320美分/磅
5月15
卖出铜期货价格 325美分/磅（305）
对冲结果
花费325000 （花费305000）
盈利5000 （亏损15000）
总花费320000 （总花费320000）
对冲成功案例
对冲失败案例
• 为规避油价风险,2008年下半年东航的航油对冲量占到
公司全年预计耗油量的35.9%。
• 该公司航油套期保值合约于2008年12月31日的公允价值 损失约为人民币62亿元。
基差风险
• 完美对冲：完全消除风险的对冲。 • 在实践中，对冲往往不会像前述那么完美： a. 需要对冲价格风险的资产与期货合约的标的资产 可能并不完全一致； b.对冲者可能并不确定资产买入及卖出的时间； c. 对冲者可能需要在期货到期月之前将期货进行平 仓。
东航亏损与中盛粮油的亏损对比？
基差风险
注意：
a. 基差风险会使对冲者的头寸得以改善或恶化
实际价格=F1+b2
P53 3.9 3.10
基差风险
注意：
b. 当对冲者的资产合约与对冲合约的标的资产不同时，这
时基差风险会更大。 定义S*2是期货合约中的资产的现货价格，则通过对冲，
公司确保购买或出售资产的价格为：
在8月15日以市价卖出100万桶石油。 • 当前原油的现货价格：每桶60美元； • 当前 8月份的石油期货价格：每桶59美元； 对冲策略： • 5月15日：卖出1000份8月份的期货合约（每份1000桶）
例：假设今天为5月15日，某石油生产商签订了一份合约：
• 8月15日：将期货进行平仓。
结果: （到期时现货价格分别是55美元与65美元）
短头寸对冲
对冲结果：
日 期 现货市场 原油价格 60$/桶 卖出原油 价格55$/桶 （65） 收入5500万 （收入6500万） 期货市场 卖出原油期货价 格59$/桶 买入原油期货价 格55$/桶 （65） 盈利400万 （亏损600万） 总收入5900万 $ （总收入5900万 $）
5月15
合约的选择 • 期货合约的头寸
QA : 被对冲资产头寸大小 QF : 一份期货合约的规模 N * : 最优期货合约数量
交叉对冲？
QA N QF
交叉对冲
• 接下来的分析，采用如下符号：
S 在对冲期限内，即期价格S的变化
F 在对冲期限内，期货价格F的变化
S， F S和F的标准差 S和F之间的相关系数
b1 S1 F 1 , b2 S2 F 2
b1 0.30, b2 0.10
短（长）头寸对冲结果：
基差风险
日 期
t1
现货市场
期货市场
基差=St-Ft
S1=2.50 S2=2.00
F1=2.20 F2=1.90
0.30 0.1
t2
对冲结果
短头寸 （长头寸）
收入2 (支付2)
盈利0.3 （亏损0.3）
下跌
下跌
无影响
减少
思考
• 某玉米农场的农场主有以下论点：”我不采 用期货来对冲我面临的风险，我面临的真正 风险并不是玉米价格的变化，而是糟糕的气 候可能使我颗粒无收。“ • 讨论：该农场主是否应对玉米预期产量有所 估计然后采用对冲策略来锁定预期价格，如 果对冲，应该采用哪种方式。
对冲成功案例
2004-2008原油每桶的价格从33美元攀升至最高147美元
1992年
1993年后期
1993年底
MRGM公司在 1992年与客户 签订了一份10 年的远期供油 合同
MGRM公司在NY MEX建立了相当于 5500万桶油的 期货合约长头寸 （相当于Castle公 司3年半的产量）
石油输出国组织 （OPEC）未能在 减产问题上达成 协议，油价价格 直线下滑，从每 桶19美元跌至 15美元。
约最佳对冲比率是多少？其含义是什么？
举例
F 0.0313 S 0.0263 0.928
0.0263 h 0.928 0.0313 0.778
最优合约数量
• 应该购买的合约份数
QA : 被对冲资产头寸大小
h QA QF : 一份期货合约的规模 N QF * N : 最优期货合约数量
• 在tn时刻：对合约n-1进行平仓；进入合约n的短头寸； • 在T时刻：对合约n进行平仓。