As gamma increases, delta goes from 0 to about 0.50 for long call options; and from 0 to about minus-0.50 for long put options.

Delta (red) looks like a logistic growth function. Its rate of change (gamma) is highest in the center before inflecting and the rate of change becomes lower, though still positive.

This information is reflected in the blue curve. It shows that when an option is ATM and begins to go ITM, an option begins to act more like the underlying asset. Delta’s rate of change thereby decreases, which causes gamma to decrease.

Gamma is lowest for deltas of 0.0 and 1.0 for call options and lowest for delta of 0.0 and minus-1.0 for put options.

(The inverse is true for short call and put options.)

^{ }

**Exceptions**

It is generally understood that “long gamma” involves being long options and “short gamma” involves being short options.

The exception to this is when there is negative convexity in the payout function.

Convexity is normally positive when being long options.

This is because you pay a fixed premium for the option, and if you get a favorable move, delta increases non-linearly (positive gamma). That gives options a type of leverage or a type of convexity in the potential payoff structure.

Negative convexity is rare. But it does exist in some circumstances and in some parts of the financial markets.

When convexity is negative that means the shape of a bond’s yield curve is concave.

For example, most mortgage-backed securities (MBS) – i.e., bonds that are tied to mortgage lending and subsequent servicing of that – exhibit negative convexity.

This is because of refinancing risk.

In most mortgages, a homeowner takes out a fixed-rate loan. If interest rates fall below the rate they have on their mortgage, many will try to refinance their mortgage.

This helps lock in lower monthly payments and is favorable for the borrower.

But for the investors in mortgage-backed securities, it’s riskier because they are holding onto a bond that’s likely to be paid off faster than what they initially believed.

When interest rates fall, the prices of these mortgage-backed securities rise less than other bonds of similar maturities.

This is due to the fact the amount of time that’s expected to elapse between when the security was issued and when it matures has decreased.

Consequently, most mortgage bonds exhibit negative convexity. Derivatives tied to them may also pose a similar type of relationship due to the inherent concavity in the underlying.

**How Traders Use Gamma**

Traders know that gamma is highest approximately ATM and gets lower the further an option is OTM or ITM.

Since gamma is essentially the convexity of an option’s value, traders will think of it from the perspective of risk/reward.

Shorting OTM options can lead to losses in a non-linear way if left uncovered, so traders need to either avoid such exposures or be adept at hedging them.

This is especially important when option expiration is close. Options values erode with time (theta).

OTM options near expiration can change significantly in value even if there’s an otherwise normal move in the underlying.

For some traders it might be compelling at the end of a week, just before options expiration in many securities, to try and squeeze out some extra P/L by shorting options very near expiration. But it can backfire when a small move in the underlying can lead to a loss of multiples of the premium.

Delta hedging works on small price movements. But because of the convexity of an option’s value, it doesn’t do a good job of protecting exposures as it pertains to larger swings in movements of the underlying.

This is where gamma hedging comes in.

An effective delta hedge for a portfolio works on only a small range of price movements in the underlying, so a trader may also try to neutralize the portfolio’s gamma.

This helps ensure that the portfolio is hedged across a wider variety of price movements in the underlying should they occur rapidly. This could be due to a data release – e.g., macroeconomic data, earnings – or anything that changes the discounted set of expectations that drives the price.

**Delta Hedging vs. Gamma Hedging**

Delta is the first derivative of the profit and loss curve of the underlying. Gamma is the second derivative of the profit and loss curve.

Hedging involves offsetting adverse exposure in these regards by using a combination of the underlying asset, derivatives, and other exposures to get the desired delta or gamma, whether that’s zero, positive, or negative, depending on what the trader is trying to do.

**Delta Hedge Example**

A basic delta hedge could be shorting call options and going long a certain number of shares of the underlying based on the delta of the options and the overall exposure from the options.

That looks like the following. You have that linear delta-1 exposure, but cap it off at a certain price under the standard covered call structure. Also, if the underlying decreases in price, some of that loss is offset by the options premium received.