Options Greeks: Gamma Explained

Traders often talk about the "greeks" when discussing options strategies. Delta¹ is often at the top of the list because it helps estimate how much the value of an option might change for each $1 move in the underlying stock. Theta² can help calculate the impact of time, and vega³ is a measure of how sensitive an option is to changes in volatility. But what does gamma⁴ tell us? Let's find out.

Of the four big greeks, gamma is the only one that is a second-order derivative. Delta is a first order derivative (the change in the price of an option with respect to a change in the stock price). Gamma, which is used to estimate changes in delta, is one more step away from the options price itself. But that doesn't mean gamma should take a backseat to the other greeks. It's the greek that might not seem like an important metric today but could become one tomorrow. When a trade's delta is causing trouble, gamma might be the reason.

Consider it this way: Gamma measures the potential increase or decrease in an option's delta when the stock price changes by $1. Long options—both puts and calls—have positive gamma, and short options have negative gamma.

Say XYZ stock is trading at $100. A 102-strike call has .40 delta and .03 gamma. A 97-strike put expiring the same month has –.30 delta and .02 gamma. If XYZ goes up $1 to $101, all things being equal, the delta of the 102-strike call is estimated to increase to .43, while the delta of the 97-strike put is estimated to decrease to –.28.

On the other hand, if XYZ drops $1 to $99, all things being equal, the delta of the 102-strike call is estimated to decrease to .37 and the delta of the 97-strike put to increase to –.32.

As you can see, gamma "manufactures" call deltas when a stock rallies and put deltas when a stock drops. It "un-manufactures" call deltas when the stock drops and put deltas when the stock rallies. In practice, if a trader is short a call and the stock rallies, the position's delta risk increases as the stock price goes up. Gamma measures the expected increase in the call's delta. That's why gamma is important. If you were to look at delta in terms of risk, XYZ's 102-strike call with .40 delta has a theoretical risk similar to 40 shares of XYZ. When XYZ's price went up, the 102-strike call had a risk similar to 43 shares of XYZ.

The risk of an options position in delta terms, then, isn't static. It generally changes when the stock price moves. But the gamma of the option estimates how much the risk will change. The higher the gamma, the more delta is expected to change when a stock price moves. The lower the gamma, the less the delta is expected to change when a stock price moves.

Gamma is highest for at-the-money⁵ (ATM) calls and puts. It gets successively lower as the calls and puts move further out of the money⁶ (OTM). All things being equal, the delta of an ATM option will theoretically change more than the delta of an OTM option when the stock price changes.

Gamma is also highest for ATM options closer to expiration. It gets successively lower the more time to expiration an option has. All things being equal, gamma is lower when there's more time to expiration and higher with less time to expiration.

Let's look at an example. With the S&P 500^® index (SPX) at 3020, the 3020-strike call with seven days to expiration (DTE) has a gamma of .0088, while the 3020-strike call with 60 DTE has a gamma of .0031—almost three times smaller (see figure below). The deltas of those 3020-strike call options were relatively the same at .52 and .54 for the seven-day and 60-day expiration, respectively. Although both calls had roughly the same delta exposure, their deltas had significantly different gamma exposure. As an option approaches expiration, its gamma can increase a lot more than its delta. Therefore, simply looking at delta doesn't tell the whole story about risk. The gamma indicates how stable the delta would be if the stock or index should move.

Gamma can move around even without a stock price changing. That's why traders want to keep an eye on their position's gamma.

Regarding DTE and strike price, gamma can also be like theta. The closer an options strike price is to the stock price, the higher its gamma and theta. And close to expiration, an option with a strike price close to the stock price can have very high gamma and theta. That can lead to a trading conundrum. If a trader is short an option in pursuit of positive theta, an option with less time to expiration will have a theoretically high positive theta. But that option will also have theoretically high negative gamma. As expiration approaches, the risk on a short option can go from mild to hair-raising quickly if the stock price moves around a lot. What's a trader to do?

For option traders, there are three general ways to manage gamma risk: close, roll, or hedge. Let's review each.

During expiration week, when the gamma of an option is growing and suggesting an options position's delta is becoming less stable, some traders may choose to close the position. Here, the position's delta can change so dramatically when gamma is high that a relatively small change in the stock price can wipe out a profit, turn a winning trade into a losing trade, or add to losses on an existing trade that's already under water.

For example, if the Dow Jones Industrial Average^® Index (DJX) is trading at 260, the 259-strike put with four DTE might have a delta of –.39 and a gamma of .08. Assume a trader sold that put a couple of months earlier for a $4 credit. Now it's worth about $1.25, giving an unrealized $2.75 profit (without taking transaction fees). Assuming the trader sold one contract, the .39 delta means theoretically making $39 if DJX rises $1 or losing $39 if DJX drops $1. And gamma offers an estimate of how much delta will change after the $1 move. If DJX drops $1, the short 259-strike put's delta would theoretically go from –.39 to –.47, more than a 20% increase in risk. Rather than assume that risk, some traders might prefer to close the trade, exiting the risk, take the profit, and move on to something else.

On the other hand, if the options trade is losing money, particularly if it's a long position, some traders might prefer to leave the position open to see if the higher gamma can manufacture deltas the "right" way to either reduce the size of the loss or even turn it into a profit. However, given how fast delta and gamma will change going into the expiration, the risks of leaving the position open and being on the "wrong" side are substantial, especially because selling puts on an index is among the most advanced options strategies with unlimited risk as the index moves higher.

Some traders might want to maintain a position but reduce gamma by rolling the options prior to expiration. Let's look at an example. DJX is at 260 and the 259-strike put has, say, 18 days until expiration. It has –.43 delta and .04 gamma. All things being equal, its short gamma will start to grow. To maintain a bullish outlook with a positive delta but less gamma, one potential adjustment is to roll the short put⁷ to a further expiration and to a strike that has a similar delta as the current 259-strike put. For example, an adjustment might involve rolling the 259-strike put with 18 DTE to the 259-strike put with 53 DTE and a –.41 delta but only .02 gamma. Rolling maintains roughly the same delta exposure but cuts gamma in half. Yes, some positive theta is sacrificed because the 259-strike put with 53 days has more theta, but it will decay at a slower rate than the 259-strike put with 18 days. But that's the trade-off to reduce short gamma. Note: Rolling strategies will incur additional transaction fees.

Some traders might look to reduce gamma exposure with a hedge. For example, short options have negative gamma, and long options have positive gamma. Combined, a long and short options position will have a lower gamma than each option alone. So, if a trader has a short OTM put with negative gamma they deem is too high, they could reduce the position's gamma by buying a further OTM put. That long put will turn the short put into a short put vertical spread and the adjustment helps offset the negative gamma of the short put. Buying a further OTM put can significantly decrease gamma risk.

Consider a DJX 259-strike put with 50 DTE with the index at 260. The put has –.43 delta and .033 gamma. Buying a 254-strike put in the same expiration that has –.28 delta and .025 gamma could potentially reduce some gamma risk. The adjustment would create a short 254-259 put vertical spread with –.15 delta and .008 gamma. While reducing gamma risk with this kind of hedge, delta is also lower, giving the short put vertical less risk overall than the short put. The trade-off is that the cost of the long put detracts from any returns the short put could generate.

Closing, rolling, or hedging are alternatives from which to choose based on each trader's forecast for the underlying stock or index. The important point is to understand how gamma is affected by the number of DTE. Some traders prefer closing options with short times to expiration, rolling medium-term options, and hedging longer-term options. 

Here's a quick tip. You can beta weight⁸ the deltas of your portfolio in the Position Statement section of the Monitor tab on the thinkorswim platform (see figure below). And when you do that, you also beta weight a portfolio's gamma. Choose gamma from the list of available items to display in the columns. You'll then see the overall gamma of your portfolio and the gamma of each position in terms of the beta-weighting symbol. It's a handy tool to identify positions that have relatively greater gamma risk.