Delta One desks focus on 'delta hedging', a very common investment-banking activity. Usually it refers to the way a bank hedges its long and short exposures across a portfolio of investments that may include assets such as shares, as well as derivatives such as futures and options.
To make a portfolio price neutral', you need an overall delta' of zero. That means for every 5% your long positions gain (say stocks you own), your short positions (created using, for instance, short futures or put options) also lose 5% and vice versa, so your overall gain or loss is zero.
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