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Sofia nice · 2025年06月20日

老师能不能推演下解题思路

  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+0.02=$15.44c=\frac{35\ast0.45+0}{1+0.02}=\$15.44

1、为啥直接用一时刻行权开始算，不从2时刻推演吗？

2、sell option可以赚19元多，已经够买0.58份股票，还有多余钱，为啥还要借钱？

3、用预期法算期权价值，那么还必要用无套利求H，看方向吗？还是有其他办法？

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1 个答案

李坏_品职助教 · 2025年06月20日

嗨，从没放弃的小努力你好：


  1. 题目说了assuming that the price of a one-year European-style call option is $19.25。 这个期权的期限就是1年的，我们去复制他的时候肯定也是一年的期限。
  2. 借钱是因为，在进行套利的时候，sell市场上的option得到19块钱，这是第一步，而第二步是利用h份股票，以及PV这么多的融资，来复制一个虚拟的call option。这个融资是复制call的过程中不可或缺的一个环节。 当然了，你卖掉option得到的19.25块钱也可以用来1年后去还债。
  3. 你用预期法求出c = 15.44 这个价格小于市场报价的19.25. 这个只结果可以告诉你，你需要复制一份call option。复制call option的时候必然是要用到Long stock，用什么钱去long stock呢？是用借来的钱，所以是borrow。

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