Delta-Gamma Hedging

What Is Delta-Gamma Hedging?

Delta-gamma hedging is an options strategy that combines both delta and gamma hedges to mitigate the risk of changes in the underlying asset and in the delta itself.

In options trading, delta refers to a change in the price of an option contract per change in the price of the underlying asset. Gamma refers to the rate of change of delta. When fully hedged in this manner, a position is both delta neutral and gamma neutral.

Understanding Delta-Gamma Hedging

Both delta and gamma help to gauge movement in an option’s price relative to how in the money (ITM) or out of the money (OTM) the option is. Traders hedge delta to limit the risk of small price movements in the underlying security, and hedge gamma to protect themselves from the remaining exposure created through the use of a delta hedge. In other words, hedging gamma should have the effect of protecting the trader's position from movement in the option's delta.

Delta moves between -1 and +1. Call options have deltas between 0 and 1, while put options have deltas between 0 and -1. When delta changes, gamma is approximately the difference between the two delta values. Further OTM options have deltas that tend toward zero. Further ITM options have deltas that tend toward 1 (call) or -1 (put).

A delta-gamma hedge is often one that is market-neutral (i.e., zero delta and zero gamma); however, a delta-gamma hedge can, in theory, adopt any static level of delta and/or gamma. Options positions that are delta-gamma hedged are still exposed to changes in value, due to shifts in volatility, interest rates, and time decay.

Defining Individual Hedges

Delta hedging aims to reduce, or hedge, the risk associated with price movements in the underlying asset by taking offsetting long and short positions. For example, a long call position may be delta-hedged by shorting the underlying stock. This strategy is based on the change in premium, or price of the option, caused by a change in the price of the underlying security.

Delta itself measures the theoretical change in premium for each $1 change in the price of the underlying. Gamma hedging attempts to reduce, or eliminate, the risk created by changes in an option's delta.

Gamma itself refers to the rate of change of an option's delta with respect to the change in price of the underlying asset. Essentially, gamma is the rate of change of the price of an option.

A trader who is trying to be delta-hedged or delta-neutral is usually making a trade that volatility will rise or fall in the future. Gamma hedging is added to a delta-hedged strategy to try and protect a trader from larger changes in the portfolio than expected, or time value erosion.

Using a Delta-Gamma Hedge

With delta hedging alone, a position has protection from small changes in the underlying asset. However, large changes will change the hedge (change delta), leaving the position vulnerable. By adding a gamma hedge, the delta hedge remains intact.

Using a gamma hedge in conjunction with a delta hedge requires an investor to create new hedges when the underlying asset’s delta changes. The number of underlying shares that are bought or sold under a delta-gamma hedge depends on whether the underlying asset price is increasing or decreasing, and by how much.

Large hedges that involve buying or selling significant quantities of shares and options may have the effect of changing the price of the underlying asset on the market, requiring the investor to constantly and dynamically create hedges for a portfolio to take into account greater fluctuations in prices.

Gamma hedging essentially involves constantly readjusting the delta hedge as delta changes (i.e., making the position gamma-neutral).

Example of Delta-Gamma Hedging Using the Underlying Stock

Assume a trader is long one call of a stock, and the option has a delta of 0.6. That means that for each $1 the stock price moves up or down, the option premium will increase or decrease by $0.60, respectively. To hedge the delta, the trader needs to short 60 shares of stock (one contract x 100 shares x 0.6 delta). Being short 60 shares neutralizes the effect of the positive 0.6 delta.

As the price of the stock changes, so will the delta. At-the-money (ATM) options have a delta near 0.5. The deeper ITM an option goes, the closer delta gets to one. The deeper OTM an option goes, the closer it gets to zero.

Assume that the gamma on this position is 0.2. That means that for each dollar change in the stock, the delta changes by 0.2. To offset the change in delta (gamma), the prior delta hedge needs to be adjusted.

If delta increases by 0.2, then delta is now 0.8. That means the trader needs 80 short shares to offset delta. They already shorted 60, so they need to short 20 more. Conversely, if delta decreased by 0.2, the delta is now 0.4, so the trader only needs 40 shares short. They have 60, so they can buy 20 shares back.