第十一章奇异期权课后习题及答案

11.1 阐述Black-Scholes 股票期权定价模型中对于一年中股票价格概率分布的假设条件。

Black-Scholes 股票期权定价模型假定一年中股票价格概率分布服从正态分布，同样，它假设股票的连续回报率也是服从正态分布的。

11.2 若一股票价格的波动率为每年30%，则在一个交易日内其相应的价格变化的标准差为多少？在本题中σ=0.3，假设一年中有252个交易日，则 12520.004t ==因此0.019 1.9%or ==11.3 阐述风险中性定价原理。

一个期权或者其他金融衍生品都是通过风险中性定价原理来定价的，期权因此在风险中性下和在真实下有一样的价值。

因此我们为了估价期权而假设这个世界是风险中性的，这简化了分析。

在风险中性情况下，所有证券都期望得到无风险利率的回报率。

因此在一个风险中性世界，用于预计远期现金流的最合适的贴现率是无风险利率。

11.4 计算基于无红利支付股票的欧式看跌期权价格，其中执行价格为＄50，现价为＄50，有效期3个月期，无风险年收益率为10%，波动率为每年30%。

在本题中050,50,0.1,0.3,0.25S X r T σ=====10.2417d ==210.0917d d =-=欧式看跌期权价格是0.10.250.10.2550(0.0.0917)50(0.2417)500.4634500.4045 2.37N e N e -⨯-⨯---=⨯-⨯=11.5 若在两个月后预期支付的红利为＄1.50，则习题11.4中计算会有何变化？在本题中我们在使用BS 公式前必须从股票价格中减去红利的贴现值，因此0S 应该是0.16670.1050 1.5048.52S e-⨯=-= 其他变量不变50,0.1,0.3,0.25X r T σ==== 在本题中10.0414d ==210.1086d d =-=-欧式看跌期权价格是0.10.250.10.2550(0.1086)48.52(0.0414)500.543248.520.4045 3.03N e N e -⨯-⨯---=⨯-⨯=11.6 什么是隐含波动率？如何计算？隐含波动率是使一个期权的Black-Scholes 价格等于它的市场价格的波动率，它用互换程序计算。

1．某投资者于2008年5月份以3．2美元／盎司的权利金买人一张执行价格为380美元／盎司的8月黄金看跌期权，以4．3美元／盎司的权利金卖出一张执行价格为380美元／盎司的8月黄金看涨期权，以380．2美元／盎司的价格买进一张8月黄金期货合约，合约到期时黄金期货价格为356美元／盎司，则该投资者的利润为()美元／盎司。

A．0．9B．1．1C．1．3D．1．5【答案】A【解析】投资者买入看跌期权到8月执行期权后盈利：380-356-3．2=20．8(美元／盎司)；投资者卖出的看涨期权在8月份对方不会执行，投资者盈利为权利金4．3美元／盎司；投资者期货合约平仓后亏损：380．2-356=24．2(美元／盎司)；投资者净盈利：20．8+4．3-24．2=0．9(美元／盎司)。

2．2008年9月20日，某投资者以120点的权利金买入一张9月份到期、执行价格为10200点的恒生指数看跌期权，同时又以150点的权利金卖出一张9月份到期、执行价格为10000点的恒生指数看跌期权。

那么该投资者的最大可能盈利(不考虑其他费用)是(230)点。

A．160B．170C．180D．190【答案】B【解析】该投资者进行的是空头看跌期权垂直套利，其最大收益=(高执行价格-低执行价格)-净权利金=(10200-10000)-(150-120)=170(点)。

3．某交易者以260美分／蒲式耳的执行价格卖出10手5月份玉米看涨期权，权利金为16美分／蒲式耳，与此同时买入20手执行价格为270美分／蒲式耳的5月份玉米看涨期权，权利金为9美分／蒲式耳，再卖出10手执行价格为280美分／蒲式耳的5月份玉米看涨期权，权利金为4美分／蒲式耳。

这种卖出蝶式套利的最大可能亏损是()美分／蒲式耳。

A．4B．6C．8D．10【答案】C【解析】该策略是一种买入蝶式套利(看涨期权)的操作，它需要支付一个初始的净权利金为2美分／蒲式耳(16-9-9+4)。

第一章1.1请解释远期多头与远期空头的区别。

答：远期多头指交易者协定将来以某一确定价格购入某种资产；远期空头指交易者协定将来以某一确定价格售出某种资产。

1.2请详细解释套期保值、投机与套利的区别。

答：套期保值指交易者采取一定的措施补偿资产的风险暴露；投机不对风险暴露进行补偿，是一种“赌博行为”；套利是采取两种或更多方式锁定利润。

1.3请解释签订购买远期价格为$50的远期合同与持有执行价格为$50的看涨期权的区别。

答：第一种情况下交易者有义务以50$购买某项资产（交易者没有选择），第二种情况下有权利以50$购买某项资产（交易者可以不执行该权利）。

1.4一位投资者出售了一个棉花期货合约，期货价格为每磅50美分，每个合约交易量为50，000磅。

请问期货合约结束时，当合约到期时棉花价格分别为（a）每磅48.20美分；（b）每磅51.30美分时，这位投资者的收益或损失为多少？答：(a)合约到期时棉花价格为每磅$0.4820时，交易者收入：（$0.5000-$0.4820）×50，000＝$900;(b)合约到期时棉花价格为每磅$0.5130时，交易者损失:($0.5130-$0.5000) ×50，000=$6501.5假设你出售了一个看跌期权，以$120执行价格出售100股IBM的股票，有效期为3个月。

IBM股票的当前价格为$121。

你是怎么考虑的？你的收益或损失如何？答：当股票价格低于$120时，该期权将不被执行。

当股票价格高于$120美元时，该期权买主执行该期权，我将损失100(st-x)。

1.6你认为某种股票的价格将要上升。

现在该股票价格为$29,3个月期的执行价格为$30的看跌期权的价格为$2.90.你有$5,800资金可以投资。

现有两种策略：直接购买股票或投资于期权，请问各自潜在的收益或损失为多少？答：股票价格低于$29时，购买股票和期权都将损失，前者损失为$5,800$29×(29-p),后者损失为$5，800;当股票价格为（29，30），购买股票收益为$5,800$29×(p-29),购买期权损失为$5,800;当股票价格高于$30时，购买股票收益为$5,800$29×(p-29)，购买期权收益为$$5,800$29×(p-30)-5,800。

1.试着找出一些本章没有讨论到的奇异期权或者自己设计几类奇异期权。

答：巴拉期权/巴黎期权/重置期权/可参见专著：《奇异期权》张光平(Peter G.Zhang)著这里可以介绍几张近年创新出来的奇异期权。

奇异期权是在常规期权(标准的欧式或美式期权)基础上，通过改变合约条款，满足私人定制收益结构或者路径依赖的场外期权产品。

比如2016年，我国券商推出的结构化收益凭证，同时嵌套了挂钩特定指数的敲入和敲出期权组合，敲入和敲出的观察时间不一致，敲入为每天观察日，敲出为每月特定一个观察日。

另外，当标的指数走势表现出趋势后，还设计了阶梯障碍的敲出期权。

这类“理财产品”受到高净值人群的追捧。

看似高票息的背后，潜藏的市场风险也非常大。

2.为什么奇异期权主要在场外交易?它们可能在交易所交易吗?答：场内交易的期权通常是标准化的金融期权。

而大多数的奇异期权条款都是定制化，挂钩的标的和收益结构都没有统一的标准，往往是金融机构根据客户的具体需求开发出来的，其灵活性和多样性是常规期权所不能比拟的，因此多只能在场外交易的。

相应地，奇异期权流动性也比较差，定价和保值往往也更加困难，奇异期权对模型设定正确与否的依赖性常常很强，合约中潜在的风险通常比较模糊，很容易导致非预期的损失，无论是用标的资产进行保值还是用相应的期权进行保值，都需要很小心。

当然，也有很少的一些流行的（参与这多了，就可以标准）奇异期权在交易所交易。

3.有一类定义在两个资产S(t)和Ŝ(t)上的奇异期权,在到期日T该期权持有者的收益是min[S(T),Ŝ(t)]。

请问该期权应该如何定价?答：这个属于支付两资产中最优或者最差回报的奇异期权。

大致的定价思路为，在S(t)和Ŝ(t)构成的相图以及45度线S(t)=Ŝ(t)，利用双变量正态分布密度函数，可以表示出期权价格的积分表达式。

定价推导可参见专著：《奇异期权》张光平(Peter G.Zhang)著，第14章，第26章。

第十一章：收益和风险：资本资产定价模型1.系统性风险通常是不可分散的，而非系统性风险是可分散的。

但是，系统风险是可以控制的，这需要很大的降低投资者的期望收益。

2.（1）系统性风险（2）非系统性风险（3）都有，但大多数是系统性风险（4）非系统性风险（5）非系统性风险（6）系统性风险3.否，应在两者之间4.错误，单个资产的方差是对总风险的衡量。

5.是的，组合标准差会比组合中各种资产的标准差小，但是投资组合的贝塔系数不会小于最小的贝塔值。

6. 可能为0，因为当贝塔值为0 时，贝塔值为0 的风险资产收益=无风险资产的收益，也可能存在负的贝塔值，此时风险资产收益小于无风险资产收益。

7.因为协方差可以衡量一种证券与组合中其他证券方差之间的关系。

8. 如果我们假设，在过去3 年市场并没有停留不变，那么南方公司的股价价格缺乏变化表明该股票要么有一个标准差或贝塔值非常接近零。

德州仪器的股票价格变动大并不意味着该公司的贝塔值高，只能说德州仪器总风险很高。

9. 石油股票价格的大幅度波动并不表明这些股票是一个很差的投资。

如果购买的石油股票是作为一个充分多元化的产品组合的一部分，它仅仅对整个投资组合做出了贡献。

这种贡献是系统风险或β来衡量的。

所以，是不恰当的。

10. The statement is false. If a security has a negative beta, investors would want to hold the asset to reduce the variability of their portfolios. Those assets will have expected returns that are lower than the risk-free rate. To see this, examine the Capital Asset Pricing Model:E(R S) = R f + S[E(R M) – R f] If S < 0, then the E(R S) < R f11. Total value = 95($53) + 120($29) = $8,515The portfolio weight for each stock is:WeightA = 95($53)/$8,515 = .5913 WeightB = 120($29)/$8,515 = .4087 12.Total value = $1,900 + 2,300 = $4,200So, the expected return of this portfolio is:E(R p) = ($1,900/$4,200)(0.10) + ($2,300/$4,200)(0.15) = .1274 or 12.74%13. E(R p) = .40(.11) + .35(.17) + .25(.14) = .1385 or 13.85%14. Here we are given the expected return of the portfolio and the expected return of each asset in the portfolio and are asked to find the weight of each asset. We can use the equation for the expected return of a portfolio to solve this problem. Since the total weight of a portfolio must equal 1 (100%), the weight of Stock Y must be one minus the weight of Stock X. Mathematically speaking, this means:E(R p) = .129 = .16w X + .10(1 –w X)We can now solve this equation for the weight of Stock X as:.129 = .16w X + .10 – .10w X w X = 0.4833So, the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value, or: Investment in X = 0.4833($10,000) = $4,833.33And the dollar amount invested in Stock Y is:Investment in Y = (1 – 0.4833)($10,000) = $5,166.6715. E(R) = .2(–.09) + .5(.11) + .3(.23) = .1060 or 10.60%16. E(RA) = .15(.06) + .65(.07) + .20(.11) = .0765 or 7.65%E(RB) = .15(–.2) + .65(.13) + .20(.33) = .1205 or 12.05%17. E(R A) = .10(–.045) + .25 (.044) + .45(.12) + .20(.207) = .1019 or 10.19%方差=.10(–.045 – .1019)⌒2 + .25(.044 – .1019)⌒2 + .45(.12 – .1019)⌒2 + .20(.207 – .1019)⌒2 = .00535标准差= (.00535)1/2 = .0732 or 7.32%18. E(R p) = .15(.08) + .65(.15) + .20(.24) = .1575 or 15.75%If we own this portfolio, we would expect to get a return of 15.75 percent.19. a.Boom: E(R p) = (.07 + .15 + .33)/3 = .1833 or 18.33%Bust: E(R p) = (.13 + .03 .06)/3 = .0333 or 3.33%E(Rp) = .80(.1833) + .20(.0333) = .1533 or 15.33%b. Boom: E(R p)=.20(.07) +.20(.15) + .60(.33) =.2420 or 24.20%Bust: E(R p) =.20(.13) +.20(.03) + .60( .06) = –.0040 or –0.40%E(R p) = .80(.2420) + .20( .004) = .1928 or 19.28%P的方差= .80(.2420 – .1928)⌒2 + .20( .0040 – .1928)⌒2 = .0096820.a．Boom: E(R p) = .30(.3) + .40(.45) + .30(.33) = .3690 or 36.90%Good: E(R p) = .30(.12) + .40(.10) + .30(.15) = .1210 or 12.10%Poor: E(R p) = .30(.01) + .40(–.15) + .30(–.05) = –.0720 or –7.20%Bust: E(R p) = .30(–.06) + .40(–.30) + .30(–.09) = –.1650 or –16.50%E(R p) = .20(.3690) + .35(.1210) + .30(–.0720) + .15(–.1650) = .0698 or 6.98%b． p⌒2 = .20(.3690 – .0698)⌒2 + .35(.1210 – .0698)⌒2 + .30(–.0720 – .0698)⌒2 + .15(–.1650 – .0698)⌒2 = .03312p的标准差= (.03312)⌒1/2 = .1820 or 18.20%21. β= .25(.75) + .20(1.90) + .15(1.38) + .40(1.16) = 1.2422.βp = 1.0 = 1/3(0) + 1/3(1.85) + 1/3(βX) βX = 1.1523.E(R i) = R f + [E(R M) – R f] ×βiE(R i) = .05 + (.12 – .05)(1.25) = .1375 or 13.75%24. We are given the values for the CAPM except for the of the stock. We need to substitute these values into the CAPM, and solve for the of the stock. One important thing we need to realize is that we are given the market risk premium. The market risk premium is the expected return of the market minus the risk-free rate. We must be careful not to use this value as the expected return of the market. Using the CAPM, we find:E(R i) = .142 = .04 + .07βi 则βi = 1.4625. E(R i) = .105 = .055 + [E(R M) – .055](.73) 则E(R M) = .1235 or 12.35%26. E(R i) = .162 = R f + (.11 – R f)(1.75).162 = R f + .1925 – 1.75R f则R f = .0407 or 4.07%27. a. E(R p) = (.103 + .05)/2 = .0765 or 7.65%b. We need to find the portfolio weights that result in a portfolio with a of 0.50. We know the 贝塔of the risk-free asset is zero. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So:c. We need to find the portfolio weights that result in a portfolio with an expected return of 9 percent. We also know the weight of the risk-free asset is one minus the weight of the stock since the portfolio weights must sum to one, or 100 percent. So:d. Solving for the of the portfolio as we did in part a, we find:18. ßp = w W(1.3) + (1 –w W)(0) = 1.3w WSo, to find the βof the portfolio for any weight of the stock, we simply multiply the weight of the stock times its β.Even though we are solving for the and expected return of a portfolio of one stock and the risk-free asset for different portfolio weights, we are really solving for the SML. Any combination of this stock and the risk-free asset will fall on the SML. For that matter, a portfolio of any stock and the risk-free asset, or any portfolio of stocks, will fall on the SML. We know the slope of the SML line is the market risk premium, so using the CAPM and the information concerning this stock, the market risk premium is:E(R W) = .138 = .05 + MRP(1.30)MRP = .088/1.3 = .0677 or 6.77%So, now we know the CAPM equation for any stock is:E(R p) = .05 + .0677*贝塔p29. E(R Y) = .055 + .068(1.35) = .1468 or 14.68%E(R Z) = .055 + .068(0.85) = .1128 or 11.28%Reward-to-risk ratio Y = (.14 – .055) / 1.35 = .0630Reward-to-risk ratio Z = (.115 – .055) / .85 = .070630. (.14 – R f)/1.35 = (.115 – R f)/0.85We can cross multiply to get: 0.85(.14 – R f) = 1.35(.115 – R f)Solving for the risk-free rate, we find:0.119 – 0.85R f = 0.15525 – 1.35R f R f = .0725 or 7.25%31.32. [E(R A) – R f]/ A = [E(R B) – R f]/ßBRP A/β A = RP B/β B βB/βA = RP B/RP A33. Boom: E(R p) = .4(.20) + .4(.35) + .2(.60) = .3400 or 34.00%Normal: E(R p) = .4(.15) + .4(.12) + .2(.05) = .1180 or 11.80%Bust: E(R p) = .4(.01) + .4(–.25) + .2(–.50) = –.1960 or –19.60%E(R p) = .35(.34) + .40(.118) + .25(–.196) = .1172 or 11.72%σp⌒2= .35(.34 – .1172)2 + .40(.118 – .1172)2 + .25(–.196 – .1172)2 = .04190σp= (.04190)1/2 = .2047 or 20.47%b. RP i = E(R p) – R f = .1172 – .038 = .0792 or 7.92%c．Approximate expected real return = .1172 – .035 = .0822 or 8.22%1 + E(R i) = (1 + h)[1 + e(r i)]1.1172 = (1.0350)[1 + e(r i)]e(r i) = (1.1172/1.035) – 1 = .0794 or 7.94%Approximate expected real risk premium = .0792 – .035 = .0442 or 4.42%Exact expected real risk premium = (1.0792/1.035) – 1 = .0427 or 4.27%34. w A = $180,000 / $1,000,000 = .18 w B = $290,000/$1,000,000 = .29βp = 1.0 = w A(.75) + w B(1.30) + w C(1.45) + w Rf(0) w C = .33655172Invest in Stock C = .33655172($1,000,000) = $336,551.721 = w A + w B + w C + w Rf 1 = .18 + .29 + .33655172 + w Rf w Rf = .19344828Invest in risk-free asset = .19344828($1,000,000) = $193,448.28w X(.172) + w Y(.0875) + (1 –w X –w Y)(.055)35. E(R p) = .1070 =βp = .8 = w X(1.8) + w Y(0.50) + (1 –w X – w Y)(0)w X = –0.11111 w Y = 2.00000 w Rf = –0.88889Investment in stock X = –0.11111($100,000) = –$11,111.1136. E(R A) = .33(.082) + .33(.095) + .33(.063) = .0800 or 8.00%E(R B) = .33(–.065) + .33(.124) + .33(.185) = .0813 or 8.13%股票A：方差=.33(.082 – .0800)⌒2 + .33(.095 – .0800)⌒2 + .33(.063 – .0800)⌒2 = .00017 标准差=(.00017)⌒1/2 = .0131 or 1.31%股票B：方差=.33(–.065 – .0813)⌒2 + .33(.124 – .0813)⌒2 + .33(.185 – .0813)⌒2 = .01133 标准差= (.01133)1/2 = .1064 or 1064%Cov(A,B) = .33(.092 – .0800)(–.065 – .0813) + .33(.095 – .0800)(.124 – .0813) + .33(.063– .0800)(.185 – .0813) = –.000472ρA,B = Cov(A,B) / （标准差A 标准差B）= –.000472 / (.0131)(.1064) = –.337337. E(R A) = .30(–.020) + .50(.138) + .20(.218) = .1066 or 10.66%E(R B) = .30(.034) + .50(.062) + .20(.092) = .0596 or 5.96%A的方差 =.30(–.020 – .1066)⌒2 + .50(.138 – .1066)⌒2 + .20(.218 – .1066)⌒2 = .00778 2 A的标准差= (.00778)⌒1/2 = .0882 or 8.82%B的方差=.30(.034 – .0596)⌒2 + .50(.062 – .0596)⌒2 + .20(.092 – .0596)⌒2 = .00041B的标准差= (.00041)⌒1/2 = .0202 or 2.02%Cov(A,B) = .30(–.020 – .1066)(.034 – .0596) + .50(.138 – .1066)(.062 – .0596)+ .20(.218 – .1066)(.092 – .0596) = .001732ρA,B = Cov(A,B) / A的标准差*B的标准差= .001732 / (.0882)(.0202) = .970138. a. E(R P) = w F E(R F) + w G E(R G)E(R P) = .30(.10) + .70(.17) = .1490 or 14.90%b. The variance of a portfolio of two assets can be expressed as:标准差= (.18675)⌒1/2 = .4322 or 43.22%39. a. The expected return of the portfolio is the sum of the weight of each asset times the expected return of each asset, so:E(R P) = w A E(R A) + w B E(R B) = .45(.13) + .55(.19) = .1630 or 16.30%c. As Stock A and Stock B become less correlated, or more negatively correlated, the standard deviation of the portfolio decreases.40.(iv) The market has a correlation of 1 with itself.(v) The beta of the market is 1.(vi) The risk-free asset has zero standard deviation.(vii) The risk-free asset has zero correlation with the market portfolio.(viii) The beta of the risk-free asset is 0.b. Firm A: E(R A) = R f + βA[E(R M) – R f]E(R A) = 0.05 + 0.85(0.12 – 0.05) = .1095 or 10.95%Firm B: E(R B) = R f +βB[E(R M) – R f]E(R B) = 0.05 + 1.5(0.12 – 0.05) = .1550 or 15.50%Firm C: E(R C) = R f + βC[E(R M) – R f]E(R C) = 0.05 + 1.23(0.12 – 0.05) = .1358 or 13.58%According to the CAPM, the expected return on Firm C‘s stock should be 13.58 percent. However, the expected return on Firm C‘s stock given in the table is 17 percent. Therefor e, Firm C‘s stock is underpriced, and you should buy it.43. First, we need to find the standard deviation of the market and the portfolio, which are:M的标准差= (.0429)⌒1/2 = .2071 or 20.71%Z 的标准差= (.1783)⌒1/2 = .4223 or 42.23%Now we can use the equation for beta to find the beta of the portfolio, which is:βZ = (相关系数Z,M)*(标准差Z) / 标准差M = (.39)(.4223) / .2071 = .80Now, we can use the CAPM to find the expected return of the portfolio, which is:E(R Z) = R f + βZ[E(R M) – R f] = .048 + .80(.114 – .048) = .1005 or 10.05%。

郑振龙《金融工程》第2版课后习题第十六章奇异期权1．奇异期权的主要类型有哪些?答：奇异期权的基本类型包括分拆与组合、弱式路径依赖、强式路径依赖、时间依赖、维数和阶数。

（1）分拆与组合：最基本的奇异期权是对常规期权和其他一些金融资产的分拆和组合，以得到所需要的回报。

（2）路径依赖：期权的价值会受到标的变量所遵循路径的影响，它又分为弱式路径依赖和强式路径依赖两种。

强式路径依赖奇异期权必须增加独立路径依赖变量；弱式路径依赖奇异期权则无需增加这样的变量。

（3）时间依赖：在期权特征中加入时间依赖，使期权的一些变量随时间而变化。

（4）多维期权：该类期权存在多个独立变量。

（5）高阶期权：该类期权的损益和价值取决于另一个（些）期权的价值。

2．分别为弱式路径依赖期权、强式路径依赖期权、多维期权、高阶期权举出几例。

答：（1）弱式路径依赖弱式路径依赖期权的价值会受到路径变量的影响，但无需增加独立路径依赖变量。

美式期权、障碍期权都是弱式路径依赖期权。

（2）强式路径依赖强式路径依赖期权的损益除了取决于标的资产的目前价格和时间外，还取决于资产价格路径的一些特征，需增加独立路径依赖变量。

亚式期权、回溯期权都是强式路径依赖期权。

（3）多维期权该类期权存在多个独立变量。

彩虹期权、资产交换期权都是多维期权。

（4）高阶期权高阶期权的损益和价值取决于另一个（些）期权的价值。

复合期权、选择者期权都是高阶期权。

3．分析障碍期权的性质。

答：障碍期权的回报以及它们的价值要受到资产到期前遵循路径的影响，因而属于路径依赖期权。

但是，障碍期权的路径依赖性质是较弱的。

因为该期权只需知道障碍是否被触发，而不需要关于路径的其他任何信息，关于路径的信息也不会成为其定价模型中的一个新增独立变量。

如果障碍水平没有被触发，障碍期权到期时的损益情况仍然和常规期权是相同的。

因此，障碍期权是属于弱式路径依赖。

4．基于某个资产价格的欧式向下敲出期权的价值与基于该资产期货价格的欧式向下敲出期权价值相等吗(该期货合约到期日与期权到期日相同)?答：不相等，因为两者被敲出的可能性大小不同。

奇异期权期权市场是世界上最具有活力和变化的市场之一，盈利和避险的需要不断推动新工具的产生。

本章我们将介绍其中一些常见的新型期权本章我们将介绍其中一些常见的新型期权，，分析其定价和保值机制分析其定价和保值机制。

这些思路和方法将有助于我们理解市场中不断创新的期权工具助于我们理解市场中不断创新的期权工具。

概述到目前为止，我们所涉及的主要是标准的欧式或美式期权，比这些常规期权更复杂的衍生证券常常被叫做奇异期权（Exotic Options），比如执行价格不是一个确定的数，而是一段时间内的平均资产价格的期权，或是在期权有效期内如果资产价格超过一定界限，期权就作废，等等。

大多数的奇异期权都是在场外交易的，往往是金融机构根据客户的具体需求开发出来的，其灵活性和多样性是常规期权所不能比拟的。

但是相应地，奇异期权的定价和保值往往也更加困难，奇异期权对模型设定正确与否的依赖性常常很强，合约中潜在的风险通常比较模糊，很容易导致非预期的损失，无论是用标的资产进行保值还是用相应的期权进行保值（在后面我们将会看到，这种保值方法被称为静态保值），都需要很小心。

由于奇异期权的多样性，要对它们进行完全的描述是不可能的，我们只能介绍一些常见的奇异期权，阐述相关的定价和保值技术，为读者提供一个借鉴，当遇到性质相同的问题时，可以加以利用。

本节的主要内容是：对奇异期权的主要类型进行大致的区分，以帮助读者更好地理解奇异期权。

这些类型包括：分拆与组合；弱路径依赖；强路径依赖；时间依赖、维数和阶数。

必须注意的是，因为奇异期权变化很多，本节内容并不能包括奇异期权的所有特点。

一、分拆与组合最基本的奇异期权是对常规期权和其他一些金融资产的分拆和组合，从而得到我们所需要的回报。

这一方法是金融工程的核心之一。

分拆和组合的思想还可以用在为奇异期权定价上。

通过对奇异期权到期时回报的数学整理，常常可以把期权分成常规期权、简单期权和其他金融资产的组合，从而大大简化期权定价过程。

SOLUTIONS TO TEXT PROBLEMS:Quick Quizzes1. Public goods are goods that are neither excludable nor rival. Examples include national defense,knowledge, and uncongested nontoll roads. Common resources are goods that are rival but not excludable. Examples include fish in the ocean, the environment, and congested nontoll roads. 2. The free-rider problem occurs when people receive the benefits of a good but avoid paying for it.The free-rider problem induces the government to provide public goods because if the government uses tax revenue to provide the good, everyone pays for it and everyone enjoys its benefits. The government should decide whether to provide a public good by comparing the good’s costs to its benefits; if the benefits exceed the costs, society is better off.3. Governments try to limit the use of common resources because one person’s use of the resourcediminishes others’ use of it, so there is a negative externality which leads people to use common resources excessively.Questions for Review1. An excludable good is one that people can be prevented from using. A rival good is one for whichone person's use of it diminishes another person's enjoyment of it. Pizza is both excludable, sincea pizza producer can prevent someone from ea ting it who doesn't pay for it, and rival, since whenone person eats it, no one else can eat it.2. A public good is a good that is neither excludable nor rival. An example is national defense, whichprotects the entire nation. No one can be prevented from enjoying the benefits of it, so it is not excludable, and an additional person who benefits from it does not diminish the value of it to others, so it is not rival. The private market will not supply the good, since no one would pay for itbecause they cannot be excluded from enjoying it if they don't pay for it.3. Cost-benefit analysis is a study that compares the costs and benefits to society of providing a publicgood. It is important because the government needs to know which public goods people value most highly and which have benefits that exceed the costs of supplying them. It is hard to dobecause quantifying the benefits is difficult to do from a questionnaire and because respondents have little incentive to tell the truth.4. A common resource is a good that is rival but not excludable. An example is fish in the ocean. Ifsomeone catches a fish, that leaves fewer fish for everyone else, so it's a rival good. But theocean is so vast, you cannot charge people for the right to fish, or prevent them from fishing, so it is not excludable. Thus, without government intervention, people will use the good too much,since they don't account for the costs they impose on others when they use the good.Problems and Applicat ions1. a. The externalities associated with public goods are positive. Since the benefits from thepublic good received by one person don't reduce the benefits received by anyone else, thesocial value of public goods is substantially greater than the private value. Examplesinclude national defense, knowledge, uncongested non-toll roads, and uncongested parks.Since public goods aren't excludable, the free-market quantity is zero, so it is less than the215efficient quantity.b. The externalities associated with common resources are generally negative. Sincecommon resources are rival but not excludable (so not priced) the use of the commonresources by one person reduces the amount available for others. Since commonresources are not priced, people tend to overuse them their private cost of using theresources is less than the social cost. Examples include fish in the ocean, theenvironment, congested non-toll roads, the Town Commons, and congested parks.2. a. (1) Police protection is a natural monopoly, since it is excludable (the police may ignoresome neighborhoods) and not rival (unless the police force is overworked, they're availablewhenever a crime arises). You could make an argument that police protection is rival, ifthe police are too busy to respond to all crimes, so that one person's use of the policereduces the amount available for others; in that case, police protection is a private good.(2) Snow plowing is most likely a common resource. Once a street is plowed, it isn'texcludable. But it is rival, especially right after a big snowfall, since plowing one streetmeans not plowing another street.(3) Education is a private good (with a positive externality). It is excludable, sincesomeone who doesn't pay can be prevented from taking classes. It is rival, since thepresence of an additional student in a class reduces the benefits to others.(4) Rural roads are public goods. They aren't excludable and they aren't rival sincethey're uncongested.(5) City streets are common resources when congested. They aren't excludable, sinceanyone can drive on them. But they are rival, since congestion means every additionaldriver slows down the progress of other drivers. When they aren't congested, city streetsare public goods, since they're no longer rival.b. The government may provide goods that aren't public goods, such as education, becauseof the externalities associated with them.3. a. Charlie is a free rider.b. The government could solve the problem by sponsoring the show and paying for it with taxrevenue collected from everyone.c. The private market could also solve the problem by making people watch commercials thatare incorporated into the program. The existence of cable TV makes the good excludable,so it would no longer be a public good.4. a. Since knowledge is a public good, the benefits of basic scientific research are available tomany people. The private firm doesn't take this into account when choosing how muchresearch to undertake; it only takes into account what it will earn.b. The United States has tried to give private firms incentives to provide basic research bysubsidizing it through organizations like the National Institute of Health and the NationalScience Foundation.c. If it's basic research that adds to knowledge, it isn't excludable at all, unless people in othercountries can be prevented somehow from sharing that knowledge. So perhaps U.S.firms get a slight advantage because they hear about technological advances first, butknowledge tends to diffuse rapidly.5. When a person litters along a highway, others bear the negative externality, so the private costsare low. Littering in your own yard imposes costs on you, so it has a higher private cost and is thus rare.6. When the system is congested, each additional rider imposes costs on other riders. For example,when all seats are taken, some people must stand. Or if there isn't any room to stand, somepeople must wait for a train that isn't as crowded. Increasing the fare during rush hourinternalizes this externality.7. On privately owned land, the amount of logging is likely to be efficient. Loggers have incentives todo the right amount of logging, since they care that the trees replenish themselves and the forest can be logged in the future. Publicly owned land, however, is a common resource, and is likely to be overlogged, since loggers won't worry about the future value of the land.Since public lands tend to be overlogged, the government can improve things by restricting thequantity of logging to its efficient level. Selling permits to log, or taxing logging, could be used to reach the appropriate quantity by internalizing the externality. Such restrictions are unnecessary on privately owned lands, since there is no externality.8. a. Overfishing is rational for fishermen since they're using a common resource. They don'tbear the costs of reducing the number of fish available to others, so it's rational for them tooverfish. The free-market quantity of fishing exceeds the efficient amount.b. A solution to the problem could come from regulating the amount of fishing, taxing fishingto internalize the externality, or auctioning off fishing permits. But these solutionswouldn't be easy to implement, since many nations have access to ocea ns, so internationalcooperation would be necessary, and enforcement would be difficult, because the sea is solarge that it is hard to police.c. By giving property rights to countries, the scope of the problem is reduced, since eachcountry has a greater incentive to find a solution. Each country can impose a tax or issuepermits, and monitor a smaller area for compliance.d. Since government agencies (like the Coast Guard in the United States) protect fishermenand rescue them when they need help, the fishermen aren't bearing the full costs of theirfishing. Thus they fish more than they should.e. The statement, "Only when fishermen believe they are assured a long-term and exclusiveright to a fishery are they likely to manage it in the same far-sighted way as good farmersmanage they land," is sensible. If fishermen owned the fishery, they would be sure not tooverfish, because they would bear the costs of overfishing. This is a case in whichproperty rights help prevent the overuse of a common resource.f. Alternatives include regulating the amount of fishing, taxing fishermen, auctioning offfishing permits, or taxing fish sold in stores. All would tend to reduce the amount offishing from the free-market amount toward the efficient amount.9. The private market provides information about the quality or function of goods and services inseveral different ways. First, producers advertise, providing people information about the product and its quality. Second, private firms provide information to consumers with independent reportson quality; an example is the magazine Consumer Reports. The government plays a role as well, by regulating advertising, thus preventing firms from exaggerating claims about their products,regulating certain goods like gasoline and food to be sure they are measured properly and provided without disease, and not allowing dangerous products on the market.10. To be a public good, a good must be neither rival nor excludable. When the Internet isn’tcongested, it is n ot rival, since one person’s use of it does not affect anyone else. However, at times traffic on the Internet is so great that everything slows down at such times, the Internet is rival. Is the Internet excludable? Since anyone operating a Web site can charge a customer for visiting the site by requiring a password, the Internet is excludable. Thus the Internet is notstrictly a public good. Since the Internet is usually not rival, it is more like a natural monopolythan a public good. However, since most people’s Web sites contain information and exclude no one, the majority of the Internet is a public good (when it is not congested).11. Recognizing that there are opportunity costs that are relevant for cost-benefit analysis is the key toanswering this question. A richer community can afford to place a higher value on life and safety.So the richer community is willing to pay more for a traffic light, and that should be considered in cost-benefit analysis.。