开发者:上海品职教育科技有限公司 隐私政策详情

应用版本:4.2.11(IOS)|3.2.5(安卓)APP下载

Ivana 🍭 · 2023年05月26日

麻烦解释一下这题

* 问题详情,请 查看题干

NO.PZ202208260100000802

问题如下:

If Kleinert's clients observe that the one-year put option with a €100 exercise price is trading at €2.50, which of the following statements best describes how Kleinert's clients could take advantage of this to earn a risk-free return greater than 0.37% over the year.

选项:

A.Kleinert should purchase the put option and also purchase approximately 0.23 shares per option to match the hedge ratio.

B.Kleinert should purchase the put option and purchase 50% of the underlying shares given the 50-50 chance the stock will fall and the put option exercised.

C.Kleinert should purchase the put option and purchase 47% of the underlying shares to match the risk-neutral probability of put exercise.

解释:

Solution

A is correct.

If the put option can be purchased for less than the no-arbitrage price, then a potential arbitrage opportunity is available. In this case, Kleinert's clients should purchase the underpriced put option and buy h* units of SparCoin's stock. The hedge ratio, h*, is calculated as:


Note that the negative hedge ratio implies that both the put option and underlying are purchased or sold to create a hedge. This initial purchase of the put option and stock will cost:

€2.50 + 0.2276 × €105.25 = €26.45.

Should the stock price decrease, the value of this portfolio will be:


The strategy generates a risk-free return of (€26.83 – €26.45)/€26.45 = 1.44%, which is greater than the 0.37% return on other available risk-free investments.

中文解析

由上面一问我们可以知道,该看跌期权的无套利价格是2.78,现在市场上看跌期权的价格是2.50。根据低买高卖的套利原理,我们应该买入该看跌期权,对应的如果构成hedged portfolio,需要long stock。然后根据公式计算h0.2276份。

此时,一份的put0.2276stock可以构成一个hedged portfolio.

该组合初始价值为:€2.50 + 0.2276 × €105.25 = €26.45

计算当股价下跌的时候,组合新的价值V1€26.83.

此时可以计算得到return1.44%,是大于题干所说的高于0.37%的。

当然,就选出答案来说,只需计算出h即可。

麻烦解释一下这题。。

1 个答案

Lucky_品职助教 · 2023年05月26日

嗨,从没放弃的小努力你好:


这道题考察了期权定价和无套利原理的应用。首先,我们需要根据期权定价模型计算出该看跌期权的无套利价格为2.78欧元,而市场上的价格为2.50欧元,因此存在低买高卖的套利机会。接下来,我们需要构建一个hedged portfolio来进行套利操作,假设该组合由x份看跌期权和y份标的资产组成,则其价值为V(x,y) = max(100 – S, 0) x + yS,其中S为标的资产当前的价格。由于该组合的价值在未来的一年内无论标的资产的价格如何都将保持不变(在到期时必定会被行权),因此我们可以通过调整y的数量来确保该组合在未来一年的任何时候都能够达到无风险收益。

根据无套利原理,该组合的初始价格应等于其未来现值,即:

x + yS = (100 – S) / (1 + r), 其中r为无风险收益率

此外,由于该组合的delta应等于看跌期权价格相对标的资产价格的负斜率,因此我们可以通过计算delta得到y的数量。在这个例子中,我们得到:

delta = dV/dS = y / S = -N(-d1) = -0.2276

其中N()为标准正态分布的累积分布函数,d1为期权定价模型中的重要参数。因此我们可以通过联立上述两个方程来计算出y的数量:

y = -0.2276S

将上式代入无套利条件中,可得:

x = (100 – S) / (1 + r) + 0.2276S

因此,我们需要购买x份看跌期权和0.2276份标的资产(即SparCoin股票)来构建hedged portfolio。最后,通过计算在标的资产价格下跌时hedged portfolio的价值变化,我们可以确定套利操作的预期回报率,并发现其高于其他可用的无风险投资的收益率。选项A是正确的答案。

----------------------------------------------
虽然现在很辛苦,但努力过的感觉真的很好,加油!

  • 1

    回答
  • 0

    关注
  • 550

    浏览
相关问题

NO.PZ202208260100000802问题如下If Kleinert's clients observe ththe one-yeput option with a €100 exercise priis trang €2.50, whiof the following statements best scribes how Kleinert's clients coultake aantage of this to earn a risk-free return greater th0.37% over the year.A.Kleinert shoulpurchase the put option analso purchase approximately 0.23 shares per option to matthe hee ratio.B.Kleinert shoulpurchase the put option anpurchase 50% of the unrlying shares given the 50-50 chanthe stowill fall anthe put option exerciseC.Kleinert shoulpurchase the put option anpurchase 47% of the unrlying shares to matthe risk-neutrprobability of put exercise. Solution A is correct. If the put option cpurchasefor less ththe no-arbitrage price, then a potentiarbitrage opportunity is available. In this case, Kleinert's clients shoulpurchase the unrpriceput option anbuy h* units of SparCoin's stock. The hee ratio, h*, is calculateas:Note ththe negative hee ratio implies thboth the put option anunrlying are purchaseor solto create a hee. This initipurchase of the put option anstowill cost:€2.50 + 0.2276 × €105.25 = €26.45.Shoulthe stopricrease, the value of this portfolio will be:The strategy generates a risk-free return of (€26.83 – €26.45)/€26.45 = 1.44%, whiis greater ththe 0.37% return on other available risk-free investments. 中文解析由上面一问我们可以知道,该看跌期权的无套利价格是2.78,现在市场上看跌期权的价格是2.50。根据低买高卖的套利原理,我们应该买入该看跌期权,对应的如果构成heeortfolio,需要long stock。然后根据公式计算h为0.2276份。此时,一份的put和0.2276份stock可以构成一个heeportfolio.该组合初始价值为€2.50 + 0.2276 × €105.25 = €26.45计算当股价下跌的时候,组合新的价值V1为€26.83.此时可以计算得到return为1.44%,是大于题干所说的高于0.37%的。当然,就选出答案来说,只需计算出h即可。 解析里面说原来期权价格是2.5,是怎么来的?

2024-10-15 07:04 1 · 回答

NO.PZ202208260100000802问题如下 Privatbank Kleinert KGaa private wealth manager in Munich, ha number of clients with large holngs in the Germfintefirm SparCoin AG. Kleinert’s analyst is concerneabout a op in SparCoin’s share priin the next yeanis recommenng to clients ththey consir purchasing a one-yeput with exercise priof €100. SparCoin’s spot pri(S0) is €105.25, anit pays no vin. The risk-free rate is 0.37%. If Kleinert's clients observe ththe one-yeput option with a €100 exercise priis trang €2.50, whiof the following statements best scribes how Kleinert's clients coultake aantage of this to earn a risk-free return greater th0.37% over the year.A.Kleinert shoulpurchase the put option analso purchase approximately 0.23 shares per option to matthe hee ratio.B.Kleinert shoulpurchase the put option anpurchase 50% of the unrlying shares given the 50-50 chanthe stowill fall anthe put option exerciseC.Kleinert shoulpurchase the put option anpurchase 47% of the unrlying shares to matthe risk-neutrprobability of put exercise. Solution A is correct. If the put option cpurchasefor less ththe no-arbitrage price, then a potentiarbitrage opportunity is available. In this case, Kleinert's clients shoulpurchase the unrpriceput option anbuy h* units of SparCoin's stock. The hee ratio, h*, is calculateas:Note ththe negative hee ratio implies thboth the put option anunrlying are purchaseor solto create a hee. This initipurchase of the put option anstowill cost:€2.50 + 0.2276 × €105.25 = €26.45.Shoulthe stopricrease, the value of this portfolio will be:The strategy generates a risk-free return of (€26.83 – €26.45)/€26.45 = 1.44%, whiis greater ththe 0.37% return on other available risk-free investments. 中文解析由上面一问我们可以知道,该看跌期权的无套利价格是2.78,现在市场上看跌期权的价格是2.50。根据低买高卖的套利原理,我们应该买入该看跌期权,对应的如果构成heeortfolio,需要long stock。然后根据公式计算h为0.2276份。此时,一份的put和0.2276份stock可以构成一个heeportfolio.该组合初始价值为€2.50 + 0.2276 × €105.25 = €26.45计算当股价下跌的时候,组合新的价值V1为€26.83.此时可以计算得到return为1.44%,是大于题干所说的高于0.37%的。当然,就选出答案来说,只需计算出h即可。 题目解析里面117.88和94.73是怎么算出来的,我在题干没找到相关信息

2024-10-13 20:24 1 · 回答

NO.PZ202208260100000802问题如下 If Kleinert's clients observe ththe one-yeput option with a €100 exercise priis trang €2.50, whiof the following statements best scribes how Kleinert's clients coultake aantage of this to earn a risk-free return greater th0.37% over the year.A.Kleinert shoulpurchase the put option analso purchase approximately 0.23 shares per option to matthe hee ratio.B.Kleinert shoulpurchase the put option anpurchase 50% of the unrlying shares given the 50-50 chanthe stowill fall anthe put option exerciseC.Kleinert shoulpurchase the put option anpurchase 47% of the unrlying shares to matthe risk-neutrprobability of put exercise. Solution A is correct. If the put option cpurchasefor less ththe no-arbitrage price, then a potentiarbitrage opportunity is available. In this case, Kleinert's clients shoulpurchase the unrpriceput option anbuy h* units of SparCoin's stock. The hee ratio, h*, is calculateas:Note ththe negative hee ratio implies thboth the put option anunrlying are purchaseor solto create a hee. This initipurchase of the put option anstowill cost:€2.50 + 0.2276 × €105.25 = €26.45.Shoulthe stopricrease, the value of this portfolio will be:The strategy generates a risk-free return of (€26.83 – €26.45)/€26.45 = 1.44%, whiis greater ththe 0.37% return on other available risk-free investments. 中文解析由上面一问我们可以知道,该看跌期权的无套利价格是2.78,现在市场上看跌期权的价格是2.50。根据低买高卖的套利原理,我们应该买入该看跌期权,对应的如果构成heeortfolio,需要long stock。然后根据公式计算h为0.2276份。此时,一份的put和0.2276份stock可以构成一个heeportfolio.该组合初始价值为€2.50 + 0.2276 × €105.25 = €26.45计算当股价下跌的时候,组合新的价值V1为€26.83.此时可以计算得到return为1.44%,是大于题干所说的高于0.37%的。当然,就选出答案来说,只需计算出h即可。 h是负数为什么要long stock呢

2024-08-16 18:53 1 · 回答

NO.PZ202208260100000802 问题如下 If Kleinert's clients observe ththe one-yeput option with a €100 exercise priis trang €2.50, whiof the following statements best scribes how Kleinert's clients coultake aantage of this to earn a risk-free return greater th0.37% over the year. A.Kleinert shoulpurchase the put option analso purchase approximately 0.23 shares per option to matthe hee ratio. B.Kleinert shoulpurchase the put option anpurchase 50% of the unrlying shares given the 50-50 chanthe stowill fall anthe put option exercise C.Kleinert shoulpurchase the put option anpurchase 47% of the unrlying shares to matthe risk-neutrprobability of put exercise. Solution A is correct. If the put option cpurchasefor less ththe no-arbitrage price, then a potentiarbitrage opportunity is available. In this case, Kleinert's clients shoulpurchase the unrpriceput option anbuy h* units of SparCoin's stock. The hee ratio, h*, is calculateas:Note ththe negative hee ratio implies thboth the put option anunrlying are purchaseor solto create a hee. This initipurchase of the put option anstowill cost:€2.50 + 0.2276 × €105.25 = €26.45.Shoulthe stopricrease, the value of this portfolio will be:The strategy generates a risk-free return of (€26.83 – €26.45)/€26.45 = 1.44%, whiis greater ththe 0.37% return on other available risk-free investments. 中文解析由上面一问我们可以知道,该看跌期权的无套利价格是2.78,现在市场上看跌期权的价格是2.50。根据低买高卖的套利原理,我们应该买入该看跌期权,对应的如果构成heeortfolio,需要long stock。然后根据公式计算h为0.2276份。此时,一份的put和0.2276份stock可以构成一个heeportfolio.该组合初始价值为€2.50 + 0.2276 × €105.25 = €26.45计算当股价下跌的时候,组合新的价值V1为€26.83.此时可以计算得到return为1.44%,是大于题干所说的高于0.37%的。当然,就选出答案来说,只需计算出h即可。 题目求的是有套利机会的时候,满足最后套利收益大于无风险收益时怎么构建这个对冲组合,h的公式是这么构建对冲组合套利收益为零。为什么还可以用h的公式来算

2023-10-03 14:03 1 · 回答