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跨式策略

Straddle Strat

专题
Finance / 金融
难度
L3

第 1 小问

题目详情

价差定义：

  1. 跨式（Straddle）= 同一执行价、同一到期日上的 +1 份看涨和 +1 份看跌
  2. 看涨（看跌）价差 = 同一到期日上，+1 份更实值的看涨（看跌）并 -1 份更虚值的看涨（看跌）

假设股票 ABC 的价格为 $62，利率和股息均为 0，我们分析的是一个月后到期的期权。

如果执行价 60 的跨式价值为 $23，那么执行价 60 的看涨和看跌分别值多少？

Spread Definitions:

  1. Straddle = +1 call and +1 put on the same strike with the same time to expiration
  2. Call(Put) = +1 call(put) that is more in the money and -1 call(put) that is less in the money with the same time expiration.

Suppose the price of stock ABC is $62, interest rates and dividends are 0, and we are analyzing options expiring in one month.

If the straddle on 60 strike is worth $23, what is the value of the 60 call and 60 put?

解析

跨式期权的价值就是相同执行价、相同到期日的看涨期权和看跌期权价格之和。由于这里有一个执行价为 60 的跨式，因此可以写成：

Straddle on 60=Call on 60+Put on 60 23=Call on 60+Put on 60 Call on 60=23−Put on 60\begin{equation*} \textrm{Straddle on 60} = \textrm{Call on 60} + \textrm{Put on 60} \end{equation*} \\\ \\ 23 = \textrm{Call on 60} + \textrm{Put on 60} \\\ \\ \textrm{Call on 60} = 23 - \textrm{Put on 60}

要确定执行价 60 的看涨期权价值，先需要确定执行价 60 的看跌期权价值。这可以通过看涨看跌平价公式求出，该公式联系了相同执行价、相同到期日的看涨和看跌期权价值。

Call Value−Put Value=Stock Price−Strike Price×e−r⋅T\begin{equation*} \textrm{Call Value} - \textrm{Put Value} = \textrm{Stock Price} - \textrm{Strike Price} \times e^{-r \cdot T} \end{equation*}

其中 rr 是无风险利率，TT 是到期时间。由于题目给定 r=0r=0，该公式可简化为：

Call Value−Put Value=Stock Price−Strike Price Call Value−Put Value=62−60 Call Value−Put Value=2\begin{equation*} \textrm{Call Value} - \textrm{Put Value} = \textrm{Stock Price} - \textrm{Strike Price} \end{equation*} \\\ \\ \textrm{Call Value} - \textrm{Put Value} = 62 - 60 \\\ \\ \textrm{Call Value} - \textrm{Put Value} = 2

现在有两个方程：

  1. Call Value=23−Put Value\textrm{Call Value} = 23 - \textrm{Put Value}
  2. Call Value−Put Value=2\textrm{Call Value} - \textrm{Put Value} = 2

解这个线性方程组可得，看涨期权价值为 $12.50，看跌期权价值为 $10.50。

Original Explanation

The value of a straddle option is the sum of the prices of the call option and the put option with the same strike price and expiration date. Since we have a straddle on a 60 we can say:

Straddle on 60=Call on 60+Put on 60 23=Call on 60+Put on 60 Call on 60=23−Put on 60\begin{equation*} \textrm{Straddle on 60} = \textrm{Call on 60} + \textrm{Put on 60} \end{equation*} \\\ \\ 23 = \textrm{Call on 60} + \textrm{Put on 60} \\\ \\ \textrm{Call on 60} = 23 - \textrm{Put on 60}

To determine the call on 60 value, we need to first determine the value of the 60 put option. This can be found using the put-call parity formula which relates the values of the call and put options with the same strike price and expiration date.

Call Value−Put Value=Stock Price−Strike Price×e−r⋅T\begin{equation*} \textrm{Call Value} - \textrm{Put Value} = \textrm{Stock Price} - \textrm{Strike Price} \times e^{-r \cdot T} \end{equation*}

Here rr is the risk free rate and TT is the time to expiration. Since r=0r=0 is given in the problem statement we can simplify this formula to:

Call Value−Put Value=Stock Price−Strike Price Call Value−Put Value=62−60 Call Value−Put Value=2\begin{equation*} \textrm{Call Value} - \textrm{Put Value} = \textrm{Stock Price} - \textrm{Strike Price} \end{equation*} \\\ \\ \textrm{Call Value} - \textrm{Put Value} = 62 - 60 \\\ \\ \textrm{Call Value} - \textrm{Put Value} = 2

Now we are left with two formulas:

  1. Call Value=23−Put Value\textrm{Call Value} = 23 - \textrm{Put Value}
  2. Call Value−Put Value=2\textrm{Call Value} - \textrm{Put Value} = 2

Solving these linear equations we get that the value of the call option is $12.50 and the value of the put option is $10.50

第 2 小问

题目详情

价差定义：

  1. 跨式（Straddle）= 同一执行价、同一到期日上的 +1 份看涨和 +1 份看跌
  2. 看涨（看跌）价差 = 同一到期日上，+1 份更实值的看涨（看跌）并 -1 份更虚值的看涨（看跌）

假设股票 ABC 的价格为 $62，利率和股息均为 0，我们分析的是一个月后到期的期权。

延续上面的例子，若 60-65 看涨价差的价值为 $2.2，那么执行价 65 的看涨期权价值是多少？

Spread Definitions:

  1. Straddle = +1 call and +1 put on the same strike with the same time to expiration
  2. Call(Put) = +1 call(put) that is more in the money and -1 call(put) that is less in the money with the same time expiration.

Suppose the price of stock ABC is $62, interest rates and dividends are 0, and we are analyzing options expiring in one month.

Continuing this example, suppose the value of the 60-65 call spread is $2.2, what is the value of the 65 call?

解析

看涨价差的价值就是执行价 65 的看涨期权价值与执行价 60 的看涨期权价值之差。前面我们已经求得执行价 60 的看涨期权价值为 $12.50。因此有

Call Spread 60-65=Call Option 65−Call Option 60 2.20=Call Option 65−12.50 Call Option 65=14.70\begin{equation*} \textrm{Call Spread 60-65} = \textrm{Call Option 65} - \textrm{Call Option 60} \end{equation*}\\\ \\ 2.20 = \textrm{Call Option 65} - 12.50\\\ \\ \boxed{\textrm{Call Option 65} = 14.70}

Original Explanation

The value of the call spread is simply the difference between the value of a 65 call and the value of a 60 call. Previously we found that the value of the 60 call is $12.50. This leaves us with the following formula

$$ \begin{equation*} \textrm{Call Spread 60-65} = \textrm{Call Option 65} - \textrm{Call Option 60} \end{equation*}\\

\ 2.20 = \textrm{Call Option 65} - 12.50\\

\ \boxed{\textrm{Call Option 65} = 14.70}