If you read our article, "Introduction to Delta," you probably have an idea that this article will be about converting the directional exposure of options into standard units. If you have not yet read it, starting there is a good idea. That article explains why we would want to use standard units. It is worth noting that option deltas are a deep rabbit hole and that this article will only scratch the surface. Future articles will dive more deeply into the specifics.

In most option trading textbooks, you are likely to find a definition of option deltas that reads something like this:

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

This definition is somehow both too complicated and too simple at the same time. Too complicated because, as we mentioned in "Introduction to Delta," delta is simply a measure of directional exposure. Confusing that idea with references to calculus is needlessly complicated. Remember, the "d" in "delta" is the same as the "d" in "direction." Keep that thought in mind as we go along. We are about to dive into why the definition above is too simple. Having that foundational thought in the back of your mind will help. 

Standard listed option deltas exist in a range from negative-one to positive-one. Deep-in-the-money put deltas will approach negative-one, and deep-in-the-money call deltas will close on positive-one. In this case, "one" refers to one underlying contract. Standard listed puts will always have a negative (or zero) delta, and calls will always have a positive (or zero) delta. Interestingly, adding the absolute value of the put and call deltas of the same strike and expiration will total nearly one. However, they rarely add to exactly one because of the different cash flows of options and their underlying assets. This relationship is related to a phenomenon known as "put-call parity." Put-call parity and the relationships that result from it will be the subject of a future article.

One of the most important aspects of option deltas is that they are continually changing. Standard equations for calculating option deltas have several inputs. These inputs include the option type (call, put, American, European, and so on), the option strike, the price of the underlying, interest rates, time to expiration, volatility expectations, and more. Changing any of these variables will change an option's delta, which is why the textbook definition is too simple. The textbook assumes all of these variables remain constant while the price alone changes. That is never the case in the real world - another topic we will cover more deeply in future articles.

Understanding the way option deltas change is essential in the context of a well-designed hedging strategy - like Quartzite Precision Marketing. Without understanding these changes, hedgers may find themselves with too big or too small of a hedge. If the hedger is trying to manage risk accurately, that is a real problem. In "Introduction to Delta," we added a sidebar stating that traders occasionally use a crude probability model to understand (or teach) how option deltas respond to different input changes. Our next article, “Option Deltas - A Crude Model,” will explore how we can use this model to understand option deltas and how they change.