The Three Alternatives
A perfectly neutral straddle would have zero total deltas. This means that the call should have 0.50 deltas and the put -0.50. This ideal situation rarely presents itself: the call’s delta may be close to 0.50 but more than likely will be 0.47 or 0.53. How can one go long a neutral (or “real”) straddle?
The first possibility is to purchase an unequal number of calls and puts. For example if the 75 calls have a delta of 0.55 and the 75 puts a delta of -0.45, a neutral straddle can be constructed by buying 11 of the puts and 9 of the calls. The delta of this position will be (11 X -0.45) + (9 X 0.55), or zero. Voilà, a perfectly neutral straddle.
The second possibility is to combine the purchase of puts with a long stock position. Using the same options as above, you could purchase 20 puts and 900 shares of the underlying stock. The total deltas would be (20 X -0.45) + (9 X 1.00), again, zero.
The third possibility is to combine the purchase of calls with a short position in the underlying stock. The calls have positive deltas, and shorting a stock creates a position with negative delta, since a short stock will result in a loss if its price increases.
In theory, one should be indifferent when asked to choose amongst these 3 alternatives. In practice, the long stock, long put straddle may be the more practical one.