Why does Gamma fall when Implied Volatility Increases?

As many of you know, at Option Pit we are kind of the volatility experts (which when you think about it, kind of makes us the trading experts).  One of the questions even my brightest option mentoring students get confused by is what happens to gamma when one raises implied volatility?  The answer is it depends on where the option is along the curve.  ATM options, the ones most traders are concerned with will actually see a decrease in absolute value terms.  I thought we could take a step back from dealing with market volatility and talk theory.  So, here is why ATM gamma falls when IV increases:

The one thing we know is that as volatility increase (to a point) all options become ATM options.  Thus if IV increases on OTM options they actually would gain absolute delta.  A 20 delta option becomes a 25 delta option, and 30 delta options becomes a 35 delta option et cetera.  If every option become closer to 50 then there is actually less delta movement between the strikes.  Here is an example

We have 3 strikes with the following deltas:

Notice that there are 15 delta’s between each strike.  Thus, if the underlying were to rise from 50 to 55, the 55 calls would pick up 15 deltas from the movement in the underlying as that option became the ATM call (and thus had a delta of 50).  If the underlying were to drop to 45 the 45 calls would lose 15 deltas as the 45 became the ATM option (and thus had a 50 delta)

Now, let’s increase IV.  As we know, the ITM option will become less ITM and the OTM option will become more ATM.  The new deltas are as such:

With the increase in implied volatility how many delta’s does the 55 call need to pick up to become a 50 delta option if the underlying rallies 5 points.  If the underlying falls 5 points how many deltas do the 45 calls need to lose to become a 50 delta option?  In both cases it is only 10.

Thus, the price movement of 5 dollars now has less impact on delta than it did before IV increased.  Before vol increased a 5 point move would cause a 15 delta swing on an option’s delta, after the IV increase a 5 point move only created 10 deltas. 

As you can see, while counterintuitive initially, when one thinks about gamma and volatility, it is actually quite simple when dealing with ATM options.  Just remember this if you are still confused, when IV goes up, it makes price movements matter less.  If gamma is a measure of how price movements affect an option (which it is in an oversimplified way) then gamma must decrease for ATM options.

Tomorrow we will get into OTM options.  Don't forget to follow us on twitter @optionpit and check out all of our free and pay content!