In our last article, "Option Deltas - A Crude Model," we hinted at the next Greek we would cover, "gamma." Before learning about gamma, it is essential to understand the concept of delta. So far, we have published five articles on delta, starting with "Introduction to Delta." If you have not yet read those articles, we strongly urge you to start there.

If you look up gamma in an options trading textbook, you are likely to find a definition that reads like this:

"Gamma is the sensitivity of an option's delta to changes in the underlying asset's price. To find an option's gamma, one needs to solve for the second derivative of one's option pricing formula with respect to price."

If you recall the textbook definition that we gave for "delta" in "Option Deltas - The Basics," this definition of gamma looks quite similar. Indeed only a few words have been swapped. Like that definition, this one has a few problems. First, gamma is not exclusive to financial options. Second, referencing calculus is as needlessly complicated in defining gamma as it was in delta. On the other hand, the nice thing about the textbook definition is it clearly shows the relationship between delta and gamma.

Gamma is how delta changes when price changes. If a trader is long gamma, his position's delta will increase in a rising market and decrease as prices fall. If a trader is short gamma, the inverse will happen. As prices rally, the short gamma trader's delta will decrease, and as the market drops, his delta will increase. If gamma were free, smart traders would want to own as much as possible.

Traders typically measure their gamma using the same units used to measure delta with an added per-price-change component. In the grain and soybean markets, this usually means equivalent futures per penny. A trader who estimates his gamma to be positive-1.00 expects his delta to increase or decrease by the equivalent of a futures contract for each penny the market rises or falls, respectively. A trader with negative-1.00 gamma will expect to be shorter or longer an equivalent futures contract for each penny prices rise or fall, respectively.

One of the benefits of purchasing options is that doing so also buys gamma. However, there is no free lunch here; this is one reason options have a time premium. As time passes, the premium in those options should also decrease, all else being equal. Traders use the Greek "theta" to measure this time-decay process. We will cover theta in a future article. Suffice it to say that gamma and theta are usually opposing forces in an options portfolio.* This opposition starts to hint at the idea of option replication. Replication is a process where traders who own options lean on gamma to offset theta, or traders who are short options lean on theta to offset gamma. We will cover option replication in a separate article.

Gamma does not always come from options. For example, grain and soybean producers are naturally short gamma. Because price and yield tend to move in opposite directions, producers often have less to sell in rising markets and more to sell when prices fall. This tendency looks a lot like a short gamma position. Mix that with the fact that producers are naturally long delta, and it makes sense that owning puts feels like such a natural hedge for most producers. It is also why put options play a prominent role in our Quartzite Precision Marketing program for grain and soybean producers.

Of course, there is no silver bullet. Even the best portfolios require regular attention and management. There is risk in everything. Keeping tabs on that risk and managing it is almost certainly better than ignoring markets and hoping they go the right way over the long run. Understanding how delta, gamma, and the other Greeks can help you manage your risk more precisely is just another tool in the toolbox. As always, thanks for taking the time to read, and we look forward to your questions and feedback.

*Usually, but not always

There are some complex positions where traders may appear to have both their gamma and theta be positive or negative. This appearance is often only temporary and indicates a portfolio dependent on the path prices take, not just volatility. It is for this reason that experience is necessary when managing complicated portfolios containing multiple options. Blindly trusting a model can lead to disastrous results. Models are only as good as the traders using them.