As
you can see, the hedging bandwidth moves in line with the absolute
value of the portfolio gamma, and reaches a fixed level as gamma
approaches zero.
Academic models. There is a substantial academic
literature on optimal techniques for delta hedging, and some of it even
deigns to consider whether the techniques under discussion could ever
be deployed in the real world.* I found (Zakamouline 2006) helpful, and chapter 4 of Volatility Trading
by Euan Sinclair offers a thorough and accessible survey of this topic.
I have intentionally avoided endorsing one method over another,
because, as is so often the case in options trading, the appropriate
method for an individual will depend on her book, her experience, her
risk tolerance, the nature of her strategy, etc. The smart way to
approach the topic of delta hedging is not as a search for some one
right answer, but rather for the best fit given your situation. One
issue I haven’t discussed yet is the matter of books with multiple
underlying assets: a trader holding options on Google ( ) and U.S.
Steel ( ) should not treat their deltas equally. It’s not a problem to
hedge at the level of individual products, i.e. buying/selling the
relevant shares of GOOG and X as needed, but for larger books this
quickly becomes less desirable. One solution to this problem is to
beta-weight individual assets to some smaller set, i.e. weighting ( ),
GOOG, and ( )options to the Nasdaq 100 and hedging with index
products.
Inexperienced traders sometimes tend to pay too much attention to
up-front transaction costs and minimize the costs of carrying unhedged
risk. Whatever method you use, the most important thing is that you use
one. Delta hedging is a fairly advanced topic, but it’s something that
every options trader needs to consider.
* I’m calling this the lazy guide to delta hedging
precisely because there are no formulas involved. Unless you’re
managing a very large book and/or institutional money, you needn’t
necessarily bother with the ivory tower approaches. If you do want to
geek out, the bibliographies in the links above offer ample opportunity.