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明明要加油 · 2023年10月04日

我算出来S0=15.44， 之后就不会了，这是用到了老师讲的哪个知识点呢？

  1. CFA II
  2. Derivative

NO.PZ2023020101000023

问题如下：

Cummins states that long-/short-hedge fund managers seek to identify and exploit any mispricing that may exist between the price of an option and the price of its underlying stock, utilizing a replicating strategy. Cummins asks Spelding to assess the three scenarios outlined in Exhibit 2, based on the information in Exhibit 1 and assuming that the price of a one-year European-style call option is $19.25.

Exhibit 1: Binomial Model Variables and Values

Exhibit 2: Scenarios and Replicating Strategies

With respect to the replicating strategies, which scenario is most likely correct:

选项：

A.

Scenario 1.

B.

Scenario 2.

C.

Scenario 3.

解释：

The $19.25 price of the call option exceeds its value of $15.44, as calculated based on both the no-arbitrage approach and the expectations approach. Accordingly, the replicating strategy per 100 shares is to (1) sell 1 option, (2) buy h shares, and (3) borrow h * (up/down factor price + up/down call payoff).

The call option calculations follow:

• No-arbitrage approach:

Hedge ratio

h=c+−c−S+−S−=35−0135−75=3560=0.5833h=\frac{c^+-c^-}{S^+-S^-}=\frac{35-0}{135-75}=\frac{35}{60}=0.5833

Call Option value

c=hS+PV(−hS−+c−=0.5833∗100+(−75∗0.5833)1.02+0=$15.44c=hS+PV(-hS^-+c^-=0.5833\ast100+\frac{{(-75\ast0}{.5833)}}{1.02}+0=\$15.44

• Expectations approach:

Probability of an up move π=0.45

Call Option value

c=35∗0.45+01+0.2=$15.44c=\frac{35\ast0.45+0}{1+0.2}=\$15.44

我看了之前老师回答的答案，

【根据无套利组合，每卖出1份call，需要long Δ份股票，因此我们long0.58份股票】

无套利组合是啥，这是李老师在哪里讲的知识点？



为什么不能用C+K=P+S？我最开始尝试用这个思考，发现算不出来。


我也用了老师讲的等式，如下图，推出的是卖出看涨期权【也就是题目所求的操作内容】，等于买空股票加买入债券，发现答案没有我的这个选项。请老师指出我错在哪里呢？




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李坏_品职助教 · 2023年10月04日

嗨，从没放弃的小努力你好：


无套利组合方法（参考基础班讲义P101，这个章节是Valuing Stock Option with Binomial Model）指的是：

就是利用股票和融资来复制一个看涨期权。用到的股票数量就是前面算出来的h，也就是Hedge Ratio。

h = 0.58意思就是，我们可以用0.58个股票+PV(-hS- + C-)这么多的融资来复制1个call option。


这么想：call optoin是赋予期权买方一个在未来行使权力买入股票的机会，也就是说股票越涨，call option越值钱。所以要复制call option肯定是要手里有股票。而套利的本质是尽可能使用借来的钱去买资产，所以要用PV(-hS- + C-)这么多的现金（borrow来的现金）去买股票。

所以理论上：buy stock + borrow 现金 = call option。


由于计算出来的call option的报价19.25远大于它的理论价值15.44，所以为了进行套利，应该高价卖出call option，同时利用buy stock + borrow 现金复制一个15.44的低成本option，这样就实现了套利。

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明明要加油 · 2023年10月04日

老师，您看我最开始写在图里面的公式，因为期权的价值远低于市场价格，当前期权被over valued。所以套利的时候，低买高卖，就是卖出期权获得市场价格19.25，再买入一个组合。 我在等式左右两边加了负号，左边是short call option，等于右边是short H份股票并且long bond。和答案不一样，我不知道自己错在哪里？为什么用这个等式思考不行？

李坏_品职助教 · 2023年10月04日

嗨，努力学习的PZer你好：


对的，你理解的正确。

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加油吧，让我们一起遇见更好的自己！

李坏_品职助教 · 2023年10月04日

嗨，从没放弃的小努力你好：


一个call option的多头 = 买入h份股票+borrow现金。


你一开始已经高价卖出一个call option，等于是call option的空头，那么为了构造套利组合，必须用股票和现金来复制call option的多头来实现“低买”的操作，这样才是完整的低买、高卖，才能实现套利。


你直接对等式取符号，那是两边都做空，没法套利了。

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明明要加油 · 2023年10月04日

哦，我在等式两边同时加负号，相当于short call 可以用什么来复制对吧？ 即：short call（卖出看涨期权）等于卖出股票，买入债券（投资债券）。 实际这道题，应该是一正一负，最终手里头寸为0 ，才是套利的本质喽

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