当前位置:文档之家› 《金融工程》作业一、二、三

《金融工程》作业一、二、三

《金融工程》作业一、二、三
《金融工程》作业一、二、三

作业一

1、一只股票现在价格是100元。有连续两个时间步,每个步长6个月,每个单步二叉树预

期上涨10%,或下跌10%,无风险利率8%(连续复利),运用无套利原则求执行价格为100元的看涨期权的价值。

2、假设市场上股票价格S=20元,执行价格X=18元,r=10%,T=1年。如果市场报价欧式看涨

期权的价格是3元,试问存在无风险的套利机会吗如果有,如何套利

3、股票当前的价格是100元,以该价格作为执行价格的看涨期权和看跌期权的价格分别是

3元和7元。如果买入看涨期权、卖出看跌期权,再购入到期日价值为100 的无风险债券,则我们就复制了该股票的价值特征(可以叫做合成股票)。试问无风险债券的投资成本是多少如果偏离了这个价格,市场会发生怎样的套利行为

作业解答:

1、按照本章的符号,u=,d=,r=,所以p=( ×。这里p是风险中性概率。期权的价值是:

(0.70412×21+2×××0+×0) =。

2、本题中看涨期权的价值应该是S-Xe-rT==。显然题中的期权价格小于此数,会引发套利活动。套利者可以购买看涨期权并卖空股票,现金流是20-3=17。17以10%投资一年,成为17 ==。到期后如果股票价格高于18,套利者以18元的价格执行期权,并将股票的空头平仓,则可获利=元。若股票价格低于18元(比如17元),套利者可以购买股票并将股票空头平仓,盈利是=元。

3、无风险证券的投资成本因该是100-7+3=96元,否则,市场就会出现以下套利活动。

第一,若投资成本低于96元(比如是93元),则合成股票的成本只有97元(7-3+93),相对于股票投资少了3元。套利者以97元买入合成股票,以100元卖空标的股票,获得无风险收益3元。

第二,若投资成本高于96元(比如是98元),则合成股票的成本是102元,高于股票投资成本2元。套利者可以买入股票同时卖出合成股票,可以带来2元的无风险利润。

作业二:

1.A公司和B公司如果要在金融市场上借入5年期本金为2000万美元的贷款,需支付的年利率分别为:

A公司需要的是浮动利率贷款,B公司需要的是固定利率贷款。请设计一个利率互换,其中银行作为中介获得的报酬是%的利差,而且要求互换对双方具有同样的吸引力。

2.X公司希望以固定利率借入美元,而Y公司希望以固定利率借入日元,而且本金用即期汇率计算价值很接近。市场对这两个公司的报价如下:

请设计一个货币互换,银行作为中介获得的报酬是50个基点,而且要求互换对双方具有同样的吸引力,汇率风险由银行承担。

3. 一份本金为10亿美元的利率互换还有10月的期限。这笔互换规定以6个月的LIBOR 利率交换12%的年利率(每半年计一次复利)。市场上对交换6个月的LIBOR利率的所有期限的利率的平均报价为10%(连续复利)。两个月前6个月的LIBOR利率为%。请问上述互换对支付浮动利率的那一方价值为多少对支付固定利率的那一方价值为多少

作业答案:

1. A公司在固定利率贷款市场上有明显的比较优势,但A公司想借的是浮动利率贷款。而B公司在浮动利率贷款市场上有明显的比较优势,但A公司想借的是固定利率贷款。这为互换交易发挥作用提供了基础。两个公司在固定利率贷款上的年利差是%,在浮动利率贷款上的年利差是。如果双方合作,互换交易每年的总收益将是%-%=%。因为银行要获得%的报酬,所以A公司和B公司每人将获得%的收益。这意味着A公司和B公司将分别以LIBOR-%和13%的利率借入贷款。合适的协议安排如图所示。

% %

12% LIBOR+%

LIBOR LIBOR

2. X 公司在日元市场上有比较优势但想借入美元,Y 公司在美元市场上有比较优势但想借入日元。这为互换交易发挥作用提供了基础。两个公司在日元贷款上的利差为%,在美元贷款上的利差为%,因此双方在互换合作中的年总收益为%-%=%。因为银行要求收取%的中介费,这样X 公司和Y 公司将分别获得%的合作收益。互换后X 公司实际上以%-%=%的利率借入美元,而Y 实际上以%-%=%借入日元。合适的协议安排如图所示。所有的汇率风险由银行承担。

日元5% 日元%

日元5% A 金融中介 B 美元10%

美元% 美元10%

3. 根据题目提供的条件可知,LIBOR 的收益率曲线的期限结构是平的,都是10%(半年计一次复利)。互换合约中隐含的固定利率债券的价值为

0.33330.10.83330.16106103.33e e -?-?+=百万美元

互换合约中隐含的浮动利率债券的价值为

()0.33330.1100 4.8101.36e -?+=百万美元

因此,互换对支付浮动利率的一方的价值为103.33101.36 1.97-=百万美元,对支付固定利率的一方的价值为-百万美元。

加元5% 加元%

加元5% A 金融中介 B 美元LIBOR+1%

美元LIBOR+% 美元LIBOR+1%

本金的支付方向在互换开始时与箭头指示相反,在互换终止时与箭头指示相同。金融中介在此期间承担了外汇风险,但可以用外汇远期合约抵补。

作业三:

1.假设某不付红利股票价格遵循几何布朗运动,其预期年收益率16%,年波动率30%,该股票当天收盘价为50元,求:

第二天收盘时的预期价格,第二天收盘时股价的标准差,在量信度为95%情况下,该股票第二天收盘时的价格范围。

2.某股票市价为70元,年波动率为32%,该股票预计3个月和6个月后将分别支付1元股息,市场无风险利率为10%。现考虑该股票的美式看涨期权,其协议价格为65元,有效期8个月。请证明在上述两个除息日提前执行该期权都不是最优的,并请计算该期权价格。

3.某股票目前价格为40元,假设该股票1个月后的价格要么为42元、要么38元。连续复利无风险年利率为8%。请问1个月期的协议价格等于39元欧式看涨期权价格等于多少 习题答案:

1、 由于),(~t t S

S ???σμφ 在本题中,S =50,=,=,t=1/365=.因此, S/50

,因此,第二天预期股价为元,标准差为元,在95%的置信水平上第2天股价会落在至+,即元至元之间。

2、 D 1=D 2=1,t 1=,T=,r=,X=65

221()0.10.1667()0.10.25[1]65(1) 1.07

[1]65(1) 1.60r T t r t t X e e X e

e ---?---?-=-=-=-=

可见, 221()2()1[1]

[1]r T t r t t D X e D X e ----<-<-

显然,该美式期权是不应提早执行的。

红利的现值为:

0.250.10.500.1 1.9265e e -?-?+=

该期权可以用欧式期权定价公式定价: S =70-=,X =65,T=,r=,σ=

21210.320.6667

0.320.66670.3013

d d d =?=-= N(d 1)=,N(d 2)=

因此,看涨期权价格为:

0.10.6667

?-??=

e-?

68.07350.7131650.618410.94

3、构造一个组合,由一份该看涨期权空头和Δ股股票构成。如果股票价格升到42元,该组

合价值就是42Δ-3。如果股票价格跌到38Δ元,该组合价值就等于38Δ。令:

42Δ-3=38Δ

得:Δ=元。也就是说,如果该组合中股票得股数等于,则无论1个月后股票价格是升到42元还是跌到38元,该组合的价值到时都等于元。因此,该组合的现值应该等于:

元。

这意味着:

-c+40Δ=

c=40×元。

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