Lessons from the Pros

Dutiful Delta and the Gamut of Gammas

During one of my option classes, some of my students were really having a hard time grasping the concept of the Greeks, namely the Delta and Gamma, as well as their relationship to each other. In this newsletter, I will focus on explaining those concepts to the readers in a similar way that I did for my option students.

Many words in the English language have multiple meanings and such is the case with the term Delta in options trading. The most common definition of Delta is the change of premium in relationship with a one point change in the underlying. For instance, if the option trader has selected a Delta of 0.87, then for a one point move in the underlying, the option premium would increase by $0.87. However, if the stock moves two points, then the question becomes, does the premium again increase only by the delta of 0.87 or does it increase by more than that? The answer lies in the Greek component known as Gamma. The Gamma is the measure of the acceleration of the change in Delta. In other words, as the Delta changes so does the Gamma, yet not at the same rate. Unlike the Delta, the Gamma’s value is the highest ATM (at the money), from that point on it decreases in value. Hence, for instance, if the stock is trading at $63.88, then the Gamma for a near the money strike price (Jan 65 call) is 0.0983. Conversely, for the same stock, the ATM put (Jan 65 put) has the same Gamma of 0.0983. The figure below is the option chain for JNJ as of the close on December 2nd, 2009.

Figure 1

On the option chain above, I have pulled up the January options for JNJ and reading the columns from left to right, there is the option symbol, volume for individual strike prices, open interest, high of that particular option for that trading session, the GAMMA, low of that particular option for that trading session, the Bid, the Delta, and the Ask. After the strike column, everything repeats for the put side. Notice that I have placed the Deltas in between the Bid and Ask, since the Delta is the heartbeat of option premium, whereas I have placed the Gamma in between the high and low. The red ovals show the Gammas on both the call side and the put side. Observe that they are the greatest ATM or near the money. From there on, the Gamma proportionately goes lower, regardless on which side of the standard deviation curve it is. Also observe on the option chain that the strike price increments are five points wide. The ATM strike is 65, while the one above is the strike of 70, and the one below is the strike of 60. The next figure shows a visual presentation of my point. ITM stands for in the money, while OTM means out of the money.

For many students a visual illustration is helpful when attempting to grasp the concept of the relationship between Gamma and Delta. The chart below shows the increase of the stock by one point and what happens to the Delta and Gamma. Again, these are just approximations, not exact numbers.

Basically, what these numbers mean is the following: as the stock price increases in value, dollar by dollar, the premium of the Jan 60 call increases in value as well. While the stock was at $63.88, the premium for the Jan 60 call at the Ask was $4.25 and the Delta for that strike price was 0.8685, rounded to 0.87. As the underlying goes up in value by a whole point to $64.88, our 0.87 Delta is added to the premium of $4.25 and the new premium is $5.12. For the next one point of increase in the underlying our Delta will increase in value as well. The original Delta was at 0.8685 and a Gamma of 0.0531 needs to be added to get the new Delta for the Jan 60 call when the underlying is at $64.88.

Hence, as the stock is ready to move higher the premium is valued at $5.12 while the new Delta is at 0.9216, rounded to 0.92. When the stock moves to the new level of $65.88 then the new Delta gets added to the premium ($5.12 + 0.92) to produce the sum of $6.04. At that level, $65.88, the new Delta for the 60 call is now at 0.9644, rounded to 0.96, while the Gamma has decreased to 0.0297. At the next point increase in the stock’s value, when the new Gamma is added to the Delta our Delta will be 0.9941, almost 1. As the stock goes up in value our ITM call will virtually track the stock increase penny by penny. The Delta can never be more than one, for in that case it would mean that the premium was increasing more than the increase of the actual underlying. This is completely impossible.

In conclusion, this article has offered a deeper look into Delta and its relationship to Gamma. Some readers might have found this connection somewhat basic; however, the best way to be certain that one does understand the inner workings of options is when one can turn around and teach it. If by any chance you, the reader, are one of those who knew this relationship, I suggest you attempt to explain it to a complete novice without the use of any written material or even a visual aid. It would prove to be an interesting experience. Have fun with it.

- Josip Causic