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呜啦啦 · 2024年09月05日

long call=long stock+卖bond借钱

  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

如果说long call=long stock+卖bond借钱，那现在需要short call为什么仍然是buy stock，而不是short呢

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李坏_品职助教 · 2024年09月05日

嗨，努力学习的PZer你好：


本题的意思是，当前真实的option市场价格是19.25，而虚拟的option的理论价值（No-arbitrage approach）是15.44，所以为了进行套利，需要卖出被高估的真实option（以19.25的价格卖出），同时买入被低估的虚拟option。


买入被低估的虚拟option，也就是要用buy h shares + borrow去构造一个虚拟的option。


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虽然现在很辛苦，但努力过的感觉真的很好，加油！

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