Delta-Neutral Hedge

The questions asks: what is the most effective way to initiate a delta-neutral hedge on a long stock position? The answer given is: “add put options to the portfolio as the delta approaches zero.”

I answered that you should add put options as the delta approaches -1. My reasoning is that, in order to affect a delta-neutral hedge, we must purchase # of Put Options = (1/delta) x (# of Shares in Long Position). It follows, then that as the delta of the put option approaches |1|, the number of options that must be purchased to affect the hedge decreases, because the hedge ratio is decreasing. Why, then, does the answer suggest that it is better to add put options when delta approaches zero; implying that it is better to add puts when more options are needed to hedge the position, and when the relative price movement between the options and the underlying stock is less correlated, thus weakening the efficacy of the hedge?

The rationale from the book says: “the number of puts to purchase depends on the hedge ratio, which depends on the option’s delta. Because the delta of the put option is negative, as the option delta moves closer to -1, the number of options necessary to maintain the hedge falls.” This rationale seems to support my answer, but it goes on to list the aforementioned answer. Am I missing something (more than likely), or is the book wrong (and no, it doesn’t ask which method is least effective).

Thanks in advance.

As delta approaches zero your original hedge is less effective than it was when you opened it. To maintain that hedge you’d need to add more.

if you’re originally long 100 shares and have a delta of .5 you’d need 200 puts to hedge. If your delta moves to .2 you’d now need 500 puts or an additional 300 puts to maintain the neutral hedge. If your delta was moving to -1 from an original .5 you’d now be over hedged. So if you’re adding put options as delta approaches -1 you’re hedging when you don’t need to. That’s the way I understood it but I got this question wrong originally too.