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Weike · 2024年02月21日

h=-0.9

  1. CFA I
  2. Derivative

NO.PZ2021061002000071

问题如下：

Suppose the current price (S0) of a non-dividend-paying stock is $50, and a put option on the stock has an exercise price (X) of $54 with six months left to maturity. Now an investor believes that the stock’s price in six months’ time will be either 10% higher or 10% lower.

Which of the following is true about constructing a perfectly hedged portfolio using put options and their underlying stocks?

选项：

A.

Buy one put option and buy 0.9 units of the underlying asset.

B.

Buy one put option and sell 0.9 units of the underlying asset.

C.

Sell one put option and buy 0.9 units of the underlying asset.

解释：

解析：

S1u = 50 * (1+10%) = 55, p1u=Max(0, 54 -55)= 0

S1d = 50 * (1-10%) = 45, p1d=Max(0, 54 -45)= 9

h = p1u - p1d / S1u - S1d = (0-9) / (55-45) = -0.9

注意计算的h是每份期权对应的标的资产的份数。Long stock与long put构成对冲组合，因此A对。

这边的正负号有什么作用，怎么看出是call 还是put

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2 个答案

pzqa35 · 2024年02月22日

嗨，努力学习的PZer你好：


题目中算的是负的，因为这个h就是一个delta，也就是标的资产价格变动1单位，期权价格变动多少。这个负号说明的就是标的资产的价格变动和期权的价格变动是反方向的，这个也符合put option的逻辑，就是它是在资产价格下跌的时候赚钱，也就是标的资产价格下跌，put option价格上涨，所以put option的delta取值范围就是-1到0，所以这里的负号代表的是一种反向变动的关系。所以long put 和long stock本身就是反向变动的关系，就可以互相对冲，建议同学使用前面提到的公式，那个公式的正负号就是long和short的意思，也更好理解。

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pzqa35 · 2024年02月22日

嗨，爱思考的PZer你好：


构建hedged portfolio，就是利用期权和股票的头寸，使得整个资产头寸不会因为标的资产的价格变动而变动，那么对于这个这个题而言就是np*△p+ns*△s=0。那么我们已知np=1，也就是△p+ns*△s=0，那么ns=-△p/△s。把数字带入ns=-（0-9）/（55-45）=0.9.所以此时的组合就是long put+long 0.9 stock。同学直接利用np*△p+ns*△s=0这个公式来解题会更方便一些，同时long put+long stock也就是我们所学的protective put的组合。

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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！

Weike · 2024年02月22日

答案解析里算的是负0.9，怎么看正负号和long put 的关系

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