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CC · 2023年08月30日

Value的计算和套利

  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

老师，1）options trade / stock trade/ Financing 分别是什么？

2）如何做这个套利？


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1 个答案

李坏_品职助教 · 2023年09月02日

嗨，爱思考的PZer你好：


这个题告诉我们一个看涨期权(call option)现在卖19.25美金，然后给出了目前的股价，执行价格等一系列信息，问我们怎么进行套利。

这个题目的套利需要通过期权交易（option trade）和股票交易（stock trade）来完成。


我们先算一算这个call 理论上值多少钱，解析算了后发现这个call只值15.44美金，却在市场上卖19.25，那么说明看涨期权被高估了，就可以进行套利。


根据套利的基本原理，我们需要买入被低估的，卖出被高估的。目前这个call肯定是被高估了，所以卖出1份call。又根据无套利组合，每卖出1份call，需要买入h份股票进行对冲。因此对于一张看涨期权，我们买入0.58份股票。


卖出一份看涨期权可以赚19.25, 买入0.58份股票需要花58元，很显然我们钱不够，而是需要借钱（也就是financing，融资）。所以总结下来就是 short call、long stock,同时借钱。


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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

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