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The second-order Greeks are a bit more complicated. Rather than looking at the impact on the option itself, they measure how a change in one of the same underlying parameters leads to a change in the value of a first-order Greek. An important second-order metric is gamma. In fact, it is the only second-order Greek that option traders use with any regularity. Gamma measures the rate of change of the delta with respect to the underlying asset. As delta is a first derivative of the price of an option, gamma is a second derivative. To understand what all this means, we first need to take a step back and define what is the delta of an option. Understanding Delta Options Gamma Math It’s not necessary to understand the math behind gamma (please feel free to go to the next section if you want), but for those interested gamma is defined more formally as the partial derivative of delta with respect to underlying stock price. The formula is below (some knowledge of the normal distribution is required to understand it). Source: http://iotafinance.com Delta refers to the change of a price of an option in regard to the price of the underlying security. For calls, delta ranges from 0 to 1. For puts, it has a value of -1 and 0. Delta expresses how much the price of an option has increased or decreased when the underlying asset moves by 1 point. Usually, when options are at the money, you can expect to see a delta of between 0.5 and -0.5. When options are far out of the money, they have a delta value close to 0, and when they are deep in the money, the delta is close to 1. This means that, typically, call owners make a profit when the underlying stock increases in price, as this leads to a positive delta. In contrast, as puts have a negative delta value, put owners see gains when underlying stock falls. It’s important to note that this is not always the case: when another factor is large enough, it can offset the data. Calculating the Impact of Delta To use the above in an example, imagine a call has a delta of 0.5. If the underlying stock increases by $1, the price of the call should rise by around $0.50. If the underlying asset decreases by $1, the price will drop by about $0.50. This assumes, of course, that no other pricing variables change. Now imagine that a put has a delta of -0.5. If the underlying stock increases by $1, the price of the put will drop by $0.50. If it decreases by $1, though, the price will rise by $0.50. Option holders will notice that the delta of an option increases rapidly at a certain price range — this is called the exploding delta. For the buyer, this is great news, as it can lead to big profits. Of course, the opposite is true for sellers on the other end of an exploding delta. In fact, an exploding delta is a major reason why selling unhedged options incurs such a high risk. Bear in mind, though, that whereas delta hedging can reduce directional risk from movements in price of the underlying asset, such a strategy will reduce the alpha along with the gamma. We’ll now see why that matters. What Is Gamma? Gamma specifies how much the delta will change when the underlying investment moves by $1 (a unit of gamma is 1/$). In other words, whereas the delta tells you at what speed the price of the option will change, the gamma will tell you at what acceleration the change will happen. This means that you can use gamma to predict how the delta will move if the underlying asset changes — and, therefore, how the value of the option will change. Gamma is important because delta is only useful at a particular moment in time. With gamma, you can figure out how much the delta of an option should change in the case of an increase or decrease in the underlying asset. Why Do We Need Gamma? To emphasize why gamma matters and how it adds another level of understanding to options that goes beyond delta, let’s take an example. Imagine two options have the same delta but different gamma values. There’s no need to even use numbers in this example: it’s enough to say that one has a low gamma and the other a high gamma. The option with the high gamma will be riskier. This is because if there is an unfavorable move in the underlying asset, the impact will be more pronounced. In other words, if an option has a high gamma value, there is an increased likelihood of volatile swings. As most traders prefer options to be predictable, the option with the low gamma is preferable. Another way to explain this is to say that gamma measures how stable the probability of an option is. How Gamma Changes with the Passage of Time As the delta of an option is dynamic, the gamma must also be constantly changing. Even minuscule movements in the underlying stock can lead to changes in the gamma. Typically, the gamma reaches its peak value when the stock is near the strike price. As we already saw, the maximum delta value is 1. As the delta decreases as the option moves further into or out of the money, the gamma value will move closer to 0. Using Gamma to Measure Change in Delta Calculating a change in the delta using gamma is quite straightforward. As an example, imagine ABC stock is trading at $47. Let’s say the delta is 0.3 and the gamma is 0.2. In the case that the underlying stock increases in price by $1 to $48, the delta will move up to 0.5. If, instead, the stock was to decrease in price by $1 to $46, the delta would drop to 0.1. Long and Short Options with Gamma For holders of long options, gamma means an acceleration in profits every time the underlying asset moves $1 in their favor. They are long gamma. This is because the gamma causes the delta of an option to increase as the option moves closer to the money or as it becomes further in the money. Therefore, every dollar of increase in the underlying asset means a more efficient return on capital. This same concept means that when an underlying asset moves $1 against the holder’s favor, losses decelerate. On the flip side, the gamma poses a risk for sellers of options — since, if there’s a winner in the equation, there also has to be a loser. Just as gamma accelerates profits for holders of long options, it accelerates losses for sellers. Similarly, as it causes losses to decelerate for the holder, it leads directional gains to decelerate for the seller. The Importance of Correct Forecasts No matter if you’re buying or selling, having an accurate forecast is essential. As a buyer, a high gamma that you forecast incorrectly could mean the option moves into the money and the delta moves toward 1 faster than you expect. This will mean the delta will then become lower more quickly than you predicted. If you’re a seller, an incorrect forecast is just as problematic. As the option you sold moves into the money, a high gamma may mean your position works against you at an accelerated rate. In the case your forecast is accurate, however, a high gamma could mean the sold option loses money faster, yielding positive results for you. How Volatility Impacts Gamma The gamma of options at the money is high when volatility is low. This is because low volatility occurs when the time value of an option is low. Then, you’ll see a dramatic rise when the underlying stock nears the strike price. When volatility is high, however, the gamma is usually stable across strike prices. The reason for this is that when options are deeply in the money or out of the time, the time value tends to be substantial. As options approach the money, there is a less dramatic time value. In turn, this leads the gamma to be both low and stable. Expiration Risk One more aspect to take into consideration is the expiration risk. The closer an option is to expiration, the more narrow the probability curve. The lack of time for the underlying assets to move to far out-of-the-money strikes reduces the probability of them being in the money. The result is a more narrow delta distribution and a more aggressive gamma. The safest way to use understanding of gamma to your advantage is to roll and close your positions at least seven (or perhaps as many as 10) days before expiration. If you wait longer than seven days out, there’s a greater chance you’ll see drastic swings — where losing trades convert into winners and vice versa. Buyers may be able to benefit from this trend, but it is particularly risky for sellers. List of gamma positive strategies Long Call Long Put Long Straddle Long Strangle Vertical Debit Spread List of Gamma negative strategies Short Call Short Put Short Straddle Short Strangle Vertical Credit Spread Covered Call Write Covered Put Write Iron Condor Butterfly Long Calendar Spread Summary Gamma measures the rate of change for delta with respect to the underlying asset's price. All long options have positive gamma and all short options have negative gamma. The gamma of a position tells us how much a $1.00 move in the underlying will change an option’s delta. We never hold our trades till expiration to avoid increased gamma risk. About the Author: Chris Young has a mathematics degree and 18 years finance experience. Chris is British by background but has worked in the US and lately in Australia. His interest in options was first aroused by the ‘Trading Options’ section of the Financial Times (of London). He decided to bring this knowledge to a wider audience and founded Epsilon Options in 2012. Related articles Options Greeks: Theta, Gamma, Delta, Vega And Rho Options Delta Explained: Sensitivity To Price Options Theta Explained: Price Sensitivity To Time Options Vega Explained: Price Sensitivity To Volatility Options Rho: Sensitivity To Interest Rates Gamma Risk Explained Why You Should Not Ignore Negative Gamma What Is Gamma Hedging Market Neutral Strategies: Long Or Short Gamma? Estimating Gamma For Calls Or Puts What Is Gamma Hedging And Why Is Everyone Talking About It? Short Gamma Vs. Long Gamma

But just as Delta is an estimate, so is Gamma. It should be applied only to better comprehend the relationship found between Delta and the underlying security. In other words, Gamma denotes the momentum or acceleration in Delta’s movement. Options traders consider Gamma a representation of risk because it measures Delta in this manner. High Gamma is seen as higher risk and low Gamma is seen as lower risk. But these conclusions should be drawn with caution. As with all estimates, it is not realistic to act on the assumed probability of outcomes, but to use Gamma and other “Greeks” as portions of a broader analysis of risks. Price changes in the underlying and premium changes in the option are clear indications of varying levels of risk, and many options traders try to match these price movements to individual risk tolerance. To accomplish this, a trader needs to appreciate how Gamma performs and what it reveals. Gamma develops as a “volatility risk premium” factor and may be more pronounced whenever volatility increases over the option’s lifetime. It is also true that ignoring an option’s gamma can lead to incorrect inference on the magnitude of the volatility risk premium … the S&P options are used as a test case to demonstrate the impact of ignoring gamma on the estimation of the market priced of volatility risk. The findings show that the more prices fluctuate, the greater the variability in the estimation of volatility risk premium when gamma is ignored. [Doran, James S. (2007). The influence of tracking error on volatility risk premium estimation. The Journal of Risk 9 (3), 1-36] This means that growing momentum of price movement (volatility) makes conclusions less reliable when Gamma is not considered, and this is the point that should not be overlooked. If a trader follows Delta but ignores Gamma, the momentum factor will not appear as readily, and a false sense of certainty can result. A related factor is moneyness of the option. When the option is at or very close to the money, Gamma is likely to have maximum value. As the option moves deeper in or out of the money, Gamma will become increasingly lower. Complicating this even more, proximity to expiration is also a factor. When the option’s expiration date approaches, Gamma tends to move higher for ATM options and lower for ITM and OTM contracts. For these reasons, relying on Delta alone is dangerous. Reviewing both Delta and Gamma (often called the Delta of the Delta), you get a more accurate picture of how volatility affects premium and changes overall risks. As an overview of Gamma behavior, you may simplify these relationships by observing that low volatility causes higher ATM Gamma and lower ITM and OTM Gamma, with the lowest point being zero Gamma. In this simplified version, Gamma directly summarizes Delta behavior (and Delta directly summarizes option volatility and behavior). Gamma, to the extent that it reveals behavior in implied volatility of the option, still should be recognized as being influenced by the underlying’s historical volatility. Just as Delta directly reveals this historical volatility and its influence of option pricing, Gamma indirectly does the same thing by showing how Delta movement occurs. The Gamma trend is most readily seen in the responsiveness of option premium to changes in time value. The longer the time to expiration, the less reaction will be seen in the option premium. A phenomenon witnessed by all options traders; and the closer the time to expiration, the more responsive premium becomes. The combined behavior of Delta and Gamma can be used to spot volatility trends with the expiration timeline in mind. The formula for Gamma is developed to summarize the second derivation of the option’s value, [Chriss, Neil A. (1997). Black-Scholes and Beyond. New York: McGraw-Hill, pp. 311-312] or Γ = ∂2V ÷ ∂2S Γ = Gamma ∂2V = second derivation of the option ∂2S = second derivation of the underlying A simplified calculation is to calculate the difference in Delta from one period to another, divided by changes in the underlying. However, this may be inaccurate because underlying price movement is not the same for all issues. A one-point price move for a $100 stock is much more substantial than a one-point move for a stock currently valued at $50 or another at $100. Using this method for calculating Gamma is only entirely reliable when used to compare Delta and Gamma for two underlying stocks with identical prices. This is not practical, however. It presents the same problem as that of how chart scaling is done. A “move” in price for one stock cannot be compared to that of another and, likewise, simplified Gamma calculations for stocks at different prices can be substantially distorted if prices are not at least close to one another. Options traders seeking reliable Delta and Gamma outcomes also must be aware of the influence of unusually strong price movement. For example, if an earnings surprise moves the underlying 10 points when typical daily movement is one to two points, what does it mean for Gamma? Because this is not typical in terms of degree of movement, this is where Delta, Gamma and all other Greeks can easily become distorted. It makes sense to think of Gamma and other Greeks as circumstantial evidence of risk changes. You still need for concrete evidence and may view Delta and Gamma as confirming indicators of ever-changing volatility in the option premium. Remember, though, that even non-conclusive results can be revealing and useful. Or , putting this another way, “Some circumstantial evidence is very strong, as when you find a trout in the milk.” [Thoreau, Henry David, Journal, November 1850] Michael C. Thomsett is a widely published author with over 80 business and investing books, including the best-selling Getting Started in Options, coming out in its 10th edition later this year. He also wrote the recently released The Mathematics of Options. Thomsett is a frequent speaker at trade shows and blogs on his website at Thomsett Guide as well as on Seeking Alpha, LinkedIn, Twitter and Facebook. Related articles The Options Greeks: Is It Greek To You? 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