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Avalon · 2024年10月13日

麻烦助教帮忙看下这题

  1. CFA I
  2. Derivative

* 问题详情，请 查看题干

NO.PZ202208260100000802

问题如下：

Privatbank Kleinert KGaA, a private wealth manager in Munich, has a number of clients with large holdings in the German fintech firm SparCoin AG. Kleinert’s analyst is concerned about a drop in SparCoin’s share price in the next year and is recommending to clients that they consider purchasing a one-year put with an exercise price of €100. SparCoin’s spot price (S0) is €105.25, and it pays no dividends. The risk-free rate is 0.37%.


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h为0.2276份。

此时，一份的put和0.2276份stock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

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1 个答案

李坏_品职助教 · 2024年10月13日

嗨，努力学习的PZer你好：


117.88是第一阶段之后，股票上涨的价格S1u。由于题目的期权是1年的，所以S1u也就是期末的股价上涨价格。


此处需要借用上一题的部分条件：

这里给出了：股价有50%的概率上涨12%，那也就是S1u = 105.25 * 1.12 = 117.88.

股价也有50%的概率会跌10%，那么S1d = 105.25 * (1-10%) = 94.73




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

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