Delta Hedge

Explained:

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 some 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 dealer might sell a call option on gold, resulting in a negative gold delta of 5,200 ounces. To mitigate this exposure, he then purchases 5,200 ounces of gold spot. Together, the short option and long gold have a combined gold delta of zero. See Exhibit 1.

Delta Hedging a Short Call Option
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 has been fit to that function. Its negative slope indicates that the position has a negative delta. The dealer then purchases enough spot gold to offset that negative delta. The market value of the spot gold 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 that 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. Because of this, the position will suffer a small loss from either a rise or decline in the price of gold.

Such residual gamma exposure is typical 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. 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 readjust 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.