Stock Options Greeks: Gamma Explained

For option traders, the greeks are your pals because they help you assess risk and potential opportunity. Delta lets you bet on the direction of the stock price, vega lets you bet on the direction of volatility, and theta lets you bet on time passing. Heck, even rho lets you bet on interest rates. But what does gamma let you bet on?

Good Ol’ Gamma

Of the four big greeks, gamma is the only 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 is one more step away from the options price itself. That makes gamma somewhat tougher to use as a speculative tool. But that doesn’t mean you should ignore it. Gamma is the greek that might not seem like a big deal today but could become a big deal tomorrow. When a trade’s delta is getting you into trouble, gamma is likely to blame.

Consider it this way: Gamma increases or decreases an options position’s delta when the stock price changes. Long options—both puts and calls—have positive gamma, and short options have negative gamma.

Say XYZ stock is trading at $100. The 102 call has 0.40 delta and 0.03 gamma. The 97 put has -0.30 delta and 0.02 gamma. If XYZ goes up $1 to $101, all things being equal, the delta of the 102 call goes to 0.43, while the delta of the 97 put goes to -0.28.

On the other hand, if XYZ drops $1 to $99, all things being equal, the delta of the 102 call goes to 0.37 and the delta of the 97 put goes to -0.32.

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 you’re short a call and the stock rallies, the position’s delta risk increases as the stock price goes up. In fact, it’s gamma that’s increasing the call’s delta. That’s why gamma is important. If you were to look at delta in terms of risk, XYZ’s 102 call with 0.40 delta has a theoretical risk similar to 40 shares of XYZ. When XYZ’s price went up, the 102 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 determines how much the risk changes. The higher the gamma, the more delta can change when a stock price moves. The lower the gamma, the less the delta can change when a stock price moves.

Gamma is highest for at-the-money (ATM) calls and puts. It gets successively lower as the calls and puts move further out of the money (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 SPX at 3020, the 3020 call with seven days to expiration (DTE) has a gamma of 0.0088, while the 3020 call with 60 DTE has a gamma of 0.0031—almost 3x smaller (see figure 1). The deltas of those options were relatively the same at 0.52 and 0.54 for the five-day and 60-day 3020 calls, 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 the delta doesn’t tell you all you need to know about risk. The gamma indicates how stable the delta would be if the stock or index should move.

The Gamma Riddle

As you can see, gamma can move around even without a stock price changing. That’s why you need to keep an eye on your position’s gamma.

Regarding DTE and strike price, gamma can also be like theta. The closer an option 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 leads to a trading conundrum. If you’re 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.

Close

During expiration week when the gamma of an option is growing and making your options position’s delta less stable, you may choose to close a position if it has a profit. 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 and potentially create a losing trade.

For example, if the Dow Jones Industrial Average index options (DJX) are at 260, the 259 put with four DTE has a delta of -0.39 and a gamma of 0.08. Assume you sold that put for a $4 credit a couple months before. Now it’s worth about $1.25, giving you a profit (without taking transaction costs into account). If you hold that put and think you’re comfortable theoretically making $39 if DJX goes up $1 or losing $39 if DJX goes down $1, you need to look at the gamma. If DJX drops $1, the short 259 put’s delta would theoretically go from -0.39 to -0.47, more than a 20% increase in risk. More than you’re comfortable with? Sometimes discretion is the better part of valor. Closing the trade, exiting the risk, taking the profit, and moving on to the next one can be something to consider.

On the other hand, if the options trade is losing money, particularly if it’s a long position, it might be worth holding it 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.

Roll

When you’ve got two or three weeks to expiration and you want to maintain a position but reduce its gamma, rolling is a potential strategy. Say DJX is at 260 and you have a 259 DJX put that has, say, 18 days until expiration. It has -0.43 delta and 0.04 gamma. All things being equal, its short gamma will start to grow. But if you still want to maintain a bullish, positive delta position in the DJX using a short put, you could consider rolling the short put to a further expiration and to a strike that has a similar delta as your current 259 put. Or, you could consider rolling the 259 put with 18 DTE to the 259 put with 53 DTE that has -0.41 delta but only 0.02 gamma. Rolling maintains roughly the same delta exposure but cuts the delta risk in half. Yes, you’re giving up some positive theta because the 259 put with 53 days has less theta than the 259 put with 18 days. But that’s the tradeoff you have to make if you want to reduce your short gamma. Note that rolling strategies will incur additional transaction costs.

Hedge

When an options trade has three weeks or more to expiration, you may choose to reduce its 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 you have a short OTM put with negative gamma you deem too high, you 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 and will offset the negative gamma of the short put. Buying a further OTM put can significantly decrease the gamma risk.

Consider a 259 DJX put with 50 DTE with the DJX at 260. It has -0.43 delta and 0.033 gamma. You could reduce your gamma risk by buying a 254 put in the same expiration that has -0.28 delta and 0.025 gamma. That would create a short 254/259 put vertical with -0.15 delta and 0.008 gamma. While you significantly reduce your gamma risk with this kind of hedge, you also reduce the delta, giving the short put vertical less risk overall than the naked short put.

Naturally, these are only suggestions, and you can choose to close, roll, or hedge an options strategy at any time. However, it’s important to understand how gamma is affected by the number of DTE. Consider closing options with short times to expiration, rolling medium-term options, and hedging longer-term options. It’s one way to incorporate that gamma information. Just make sure you control your gamma, lest it ends up controlling you.