# Why Gamma shares an inverse relationship with volatility

At this TOS tutorial

On page 3:

Tom Preston: Gamma increases when the option is at the money, the option is close to expiration or volatility is low

I am confused about 'volatility is low' part. When volatility is high, it seems intuitively that the underlying will jump around, hence the delta will also jump around quite a bit, maybe going from +ve to -ve and vice versa very quickly.

So this means , an increased gamma, does it not?

For this, it's important to figure out, what is delta, gamma etc. anyway? Gamma is the derivative of delta, which is the partial derivative of the value as it relates to the spot price, so it's the rate of change of a rate of change, delta relating how changes in the price would affect the value, and gamma relating how delta changes as the price changes.

This can be a little hard to conceptualize without taking some calculus so rather than try to rationalize it with something that makes intuitive sense the easiest thing is to look at the equations...

Delta is `N(d1)`, and d1 includes volatility, but unless the equation for `N()` seems simple for you I think it's worth admitting that you will likely have a hard time understanding what is 'intuitive' about the relationship. Since delta includes volatility as a factor (in d1), regardless of whether volatility is high or low as long as the price change has a proportionate effect on the expected value then delta may not be jumping around as much as you think. The take-home for delta, is it measures how much a marginal change in the price will affect a marginal change in the value. So, if everything else was equal, if the price went up by \$1 and delta was 0.5, the value should increase by ~\$0.5.

For gamma, gamma is how fast delta changes with respect to the underlying price. So if you thought it hard to get an intuitive grasp on delta, then this is one step removed from that! It's important to realize that volatility measures changes of the price over time, so the relationship between it and delta/gamma is not quite straightforward. Rather than trying to come up with an intuitive explanation for why this is, it's easier to look at the equations and then work back from that.

So Gamma is this: σ is volatility. Since it's a part of the denominator, when it increases, gamma decreases, and vice-versa. So, basically as volatility increases, delta isn't going to change as fast, and when volatility is low, delta is going to change faster. This is just the natural result of the Black-Scholes equation itself.

It does make sense if you think about it: if the price tends to change a lot and rapidly, you shouldn't expect the value to change the way it follows the price much: the volatility will already be 'priced in' so for any marginal change in price you shouldn't expect delta to change that much. However, if the price doesn't move much, a single move in price could signal a lot more, and so it makes sense that delta in that case would be more sensitive to price movements.

Some further notes:

1. Vol enters BS always in the product sigma^2 * T. Thus, the "influence" of sigma must be in the same way as the influence of time to expiry (unless overwhelmed by rates, which are affected by time, but not vol).

2. When vol is small (or indeed time to expiry is small), the option is practically worthless when S is below the strike (thus delta = 0), but practically a forward when S is above the strike (thus delta = 1). In other words, delta changes very quickly in the vicinity of the strike. That gives you big gamma.

3. Conversely, if vol goes to infinity, then the call price is basically S. Thus, delta stays at 1, and gamma is zero.