A delta hedge is a simple type of hedge that is widely used by derivative dealers to reduce or eliminate a portfolio’s exposure to an underlier. The dealer calculates the portfolio’s delta with respect to the underlier and then adds an offsetting position in the underlier to make the portfolio’s delta zero. The offsetting position may take various forms, but a spot, forward or futures position in the underlier is typical. All that is really required is that the position’s delta offset that of the original portfolio.
For example, a precious metals dealer might sell a call option on gold, resulting in a negative gold delta. To mitigate this exposure, he then purchases enough gold futures to offset the short option’s negative delta. Together, the short option and long futures have a combined gold delta of zero. See Exhibit 1.
Exhibit 1: A dealer sells a call option on gold. Its market value is plotted as a function of the spot price of gold in the top graph. A tangent line to that function has negative slope indicating the position has a negative delta. The dealer then purchases enough gold futures to offset that negative delta. The market value of the gold futures is plotted in the second chart. The slope of that graph is the exact opposite of the slope of the tangent line in the top graph, so the two positions have equal but opposite deltas. The market value of the combined position is indicated in the bottom graph. A tangent line has zero slope, indicating the position is delta hedged.
Note that in the third graph of Exhibit 1, exposure to the price of gold has not been entirely eliminated. While the position’s delta is hedged, it still has negative gamma, and likely negative vega as well. Such residual gamma and vega exposures are inevitable when options positions are delta hedged. One solution is delta-gamma hedging, in which options are added to a portfolio to achieve both a zero delta and zero gamma. Not only will this eliminate gamma exposure, but it will largely address vega exposure as well. Because options can be expensive, dealers rarely employ delta-gamma hedging.
Another problem with delta hedging an options position is the fact that the position’s delta will change with movements in the underlier, thereby throwing off the delta hedge. The inevitable solution to this problem is to constantly adjust the delta hedge as the underlier moves. This technique is called dynamic hedging.
A portfolio that has zero delta is said to be delta neutral. This terminology can be misleading because a portfolio can have exposures to multiple underliers. The portfolio may be delta neutral for one underlier but have a positive or negative delta for another.