A put option has an N(-d1) of 0.5746. You wish to construct a strategy that mimicks the payoff of a long straddle that shares the same strike as the put above, as well as an appropriate underlying position. If your strategy contains 120 put options, determine how many shares you need to buy or sell to achieve this goal. If you need to buy shares, please input a positive (whole) number. If you need to sell shares, please input a negative (whole) number.
Question
A put option has an N(-d1) of 0.5746.
You wish to construct a strategy that mimicks the payoff of a long straddle that shares the same strike as the put above, as well as an appropriate underlying position.
If your strategy contains 120 put options, determine how many shares you need to buy or sell to achieve this goal.
If you need to buy shares, please input a positive (whole) number. If you need to sell shares, please input a negative (whole) number.
Solution
To construct a strategy that mimics the payoff of a long straddle, you would need to long both a call and a put option at the same strike price. However, in this case, you only have put options. Therefore, you need to create a synthetic long call option using the put options and the underlying asset.
The formula to create a synthetic long call is: Long Call = Long Put + Long Underlying.
Given that N(-d1) is 0.5746, this is the delta of the put option. The delta of the put option ranges from -1 to 0. Since we are longing the put, the delta becomes positive, i.e., 1 - 0.5746 = 0.4254.
This means for every put option you long, you need to long 0.4254 shares of the underlying to create a synthetic long call.
If your strategy contains 120 put options, the number of shares you need to buy is 120 * 0.4254 = 51.048 shares.
Since we can't buy a fraction of a share, we round it to the nearest whole number. Therefore, you need to buy 51 shares to achieve this goal.
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