Let's face it. Almost everything about options pricing sounds complicated: Black-Scholes model, Greeks, and up and down delta. If I had concentrated on delta's mathematical definition--the partial derivative of the Black-Scholes model with respect to the underlying's price--in my introduction of the Greeks two weeks ago, all but the most mathematically inclined might have stopped reading.
Concepts about option pricing don't have to be complicated. They certainly aren't where it concerns gamma. Delta tells us how much an option's price might change for each one-point move in the underlying; gamma tells us how much delta itself will change when that one-point move is made. Almost everything about gamma seems intuitive when we think about how options' prices change.
We already know that the delta of an option doesn't remain constant as the underlying moves. It can't if delta is near 1.0 for deep in-the-money calls, near 0.50 for at-the-money calls and near 0.00 for far out-of-the-money calls. It can't if the delta of puts changes from 0 to -1.0. Delta must be changing as the price of the underlying moves.
Gamma tells us how much it's changing.
Let's look at an example. At the close of trading on October 26, GOOG closed at 674.60. A NOV 680 call was quoted as follows:
Imagine that GOOG had risen in the last few minutes of trading that day. How would the value of that NOV 680 call have changed?
If we ignore the up-delta effect that was explained in last week's article and all other inputs such as volatility were to stay the same, we could expect value of that NOV 680 to escalate by 0.48 if GOOG were to rise by a single point. But what if GOOG had risen two points? Gamma tells us that for every point GOOG rises, delta will increase by 0.01, so after GOOG had risen that first point, delta would no longer be 0.48. Instead, it would be 0.49. So, still ignoring that up-delta effect, we could expect that NOV 680 call to escalate by $0.97 ($0.48 + 0.49) if GOOG had risen two points, rather than $0.96 ($0.48 x 2).
Gamma is always positive but ranges down to 0.0. That's intuitive, too. We know already that if a call is deep in the money, for example, its value is going to change nearly point for point with any price change in the underlying. It's going to keep changing nearly point for point unless there's a big move. Since delta won't change much unless there's a big change in price, gamma must be zero or nearly zero when options are deep in the money, we reason, and we'd be right.
Gamma is also zero or nearly zero when an option is far out of the money. In this case, delta is also approaching zero. The far OTM option's price isn't going to change much when the price of the underlying changes, and it isn't going to change much until the underlying makes a big move. If gamma measures how much delta is going to change for each one-point change in the underlying, then it's reasonable to expect that it's zero or near zero for an option that's far out of the money. That reasonable assumption turns out to be right, too.
GOOG's option chain on October 26 shows us that our intuitive guesses were right. With GOOG at 674.60, the follow data was found for one ITM and OTM put:
(ITM) GOOG NOV 740 Put:
(OTM) GOOG NOV 600 Put:
With the far OTM 600 put, a value of 0.00 for gamma shows us that GOOG would have to make a big move before that OTM's delta started changing, and, therefore, before the put's price started changing much for each point move in GOOG. With the deep in the money 740 put, the put's value will escalate about $0.87 for each point drop in GOOG until GOOG makes a big enough move to change that value.
If gamma moves closer toward zero the further an option is in the money or out of the money, logic tells us that it would be highest for at-the-money options. We know from the previous study of delta that delta changes the most rapidly for ATM options, once the underlying's price moves. Again logic would be right. In our GOOG example, with GOOG at 674.60, the NOV 670 put and call revealed the following gamma values:
NOV 670 Put
NOV 670 Call
That's not a big value, is it? It might not be, but it's a common gamma value for ATM options that day, still 19 days from option expiration. For example, with the OEX at 717.18, gamma for NOV 715 calls and puts was 0.01.
We learned previously that delta began to change more rapidly for ATM options once option expiration approached. It's possible to compare gamma for OEX NOV 715 calls and puts to the weekly 715's, which were only a week away from expiration on October 26. The weeklies had a gamma of 0.02. That's still not a big gamma, although it's double the gamma of the 715's that had 19 days until expiration.
If you've ever bought an ATM OEX call on expiration Friday and the OEX has risen two points, you know that the call is by then moving almost point-for-point for each gain the OEX makes. Delta has quickly risen to 1.00 or nearly 1.00. While gamma might be only 0.01-0.02 for an ATM option weeks before expiration, it might have risen to 0.20-0.25 the last day an option trades before expiration.
It's obvious that gamma for ATM options increases as expiration approaches. However, the opposite is true of OTM options. Gamma more quickly approaches zero as expiration approaches. Think about delta again as the prediction that the option will be in the money at expiration. As the time to expiration decreases, the probability that the OTM option will be in the money at expiration grows smaller, and its not going to change much unless the underlying makes a big move. Similarly, gamma more quickly approaches zero for in-the-money options as the time for expiration approaches.
Gamma is also impacted by volatility. The lower the volatility of an underlying, the higher the gamma will be for the ATM options. Does this make sense logically? Sure it does. Again, go back to the concept of delta as a prediction of whether the option will be in the money at expiration. If a stock or index isn't volatile and is sitting at the money as expiration approaches, any small movement is likely to move it permanently into or out of the money, so delta will change at a greater rate for any small movement in the price of the underlying. A higher gamma for these low-volatility ATM options predicts that delta will change more rapidly for ATM strikes in a less volatile underlying.
However, if the underlying is volatile, a move of a point or two might quickly be reversed, so doesn't greatly change the prediction of whether the option will be in the money by expiration. Delta isn't going to change much for small moves of the underlying, so gamma should be smaller for the more volatile ATM stock. Two examples show this effect.
DLX @ 40.09 on October 26
OMI @ 40.12 on October 26
The less volatile stock's ATM call had a higher gamma.
The opposite effect is seen with OTM options, however. There, the more volatile stock has the highest gamma.
McMillan notes that when an underlying is extremely volatile, gamma values should remain stable across all strikes when there are at least three months remaining until expiration. He explains that this indicates that the delta of all options could change quite a bit for even a small move in the underlying. I admit finding this concept not as intuitive. In fact, when I took a look at JUN GOOG options, my charting service showed gamma values of 0.00 for nearly all options. I guess that is stable, right?
Perhaps my brain has just had enough logical thinking for one setting, and
perhaps yours has, too. Next week, it will be time to discover how people use
gamma in setting up their options positions, so give your brain a rest.