Business, Legal & Accounting Glossary
An options strategy designed reduce the risk associated with price movements in the underlying security, achieved through offsetting long and short positions.
Delta hedging is the process of setting or keeping the delta of a portfolio of financial instruments zero, or as close to zero as possible – where delta is the sensitivity of the value of a derivative to changes in the price of its underlying instrument. Mathematically, delta is the partial derivative of the portfolio’s fair value with respect to the price of the underlying security; see The Greeks. Being delta neutral (or, instantaneously delta-hedged) means that the instantaneous change in value of the portfolio for an infinitesimal change in the value of the underlying is zero.
Keeping delta at zero is termed a “static delta hedge”; keeping delta close to zero is a “dynamic delta hedge”. Delta constantly changes, thus, once the delta of a portfolio has been made zero by adjusting its holdings (typically in the underlying security for a portfolio of derivatives) it is zero only for that instant; delta neutrality is instantaneous. The term static delta hedge is, therefore, a misnomer and thus (re)setting delta to zero is often preferred. In dynamic delta hedging, the portfolio is readjusted regularly in order to reset the delta to zero. Between readjustments, the portfolio delta will deviate from zero.
In fact, the amount by which a hedge has to be adjusted to stay delta neutral is related to gamma, the second derivative of the portfolio value with respect to the price of the asset in question. For example, if a position is ‘long gamma’, i.e., has a positive gamma, an increase in the asset price will lead to a positive delta, and one will need to sell some of the asset to ‘flatten’ the delta. Similarly, a decrease in asset price will cause one to buy more of the asset. From this it is intuitively clear that a high volatility of the underlying asset will lead to trading profits.
As above, a portfolio has to be adjusted continuously (i.e. infinitely often in any time interval) in order to maintain absolute delta neutrality. This idea plays an important part in the Black-Scholes model of option pricing, and – indeed – the expected cost of keeping a position in one option, the underlying asset (and cash) delta neutral is equal to the initial fair value (Black-Scholes price) of the option; for the underlying logic see the discussion at Rational pricing.
dynamic hedging strategy
A delta hedge is a simple type of hedge that is widely used by derivatives 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.
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This glossary post was last updated: 29th December, 2021 | 0 Views.