# Gamma

If a call option is out of the money (OOTM) its delta is typically less than .5. If the stock price increases, however, the option’s delta also increases. We already know from previous option commentaries and articles, for instance, that if the underlying price increases to the extent that the option is deep in the money, the delta becomes very close to 1.0. In fact, it changes gradually, increasing slowly from almost 0, when the option is deep out of the money, to 1.0, as the underlying price rises and causes the call to become deep in the money (ITM).

A put’s delta also increases as the underlying price increases. When the underlying price is very low and the put is deep in the money, we know that its delta is close to -1.0. As the underlying price increases and the put becomes at the money, the delta rises from -1.0 to

-0.5–i.e., rises to a value not as low as it was. And as the underlying price increases further and the put becomes out of the money, its delta rises to almost 0.

Just as the option value increases as the price of the underlying increases and the rate of increase is measured by the option’s delta at any given stock price, so delta itself also increases as the price of the underlying increases. The quantity that measures the sensitivity of delta to changes in the price of the underlying is called “gamma.” And just as delta changes when the price of the underlying changes, gamma changes also.

Gamma is always positive. This means that an increase in the underlying price always leads to an increase in delta, for both puts and calls as we noted above.

What does a gamma of .05 mean? It means that if the price of the underlying changes by $1, the value of delta changes by (approximately) $1*0.05, or by $0.05; and if the price of the underlying changes by $2, the value of delta changes by (approximately) $2*0.05, or by $0.10. So if the option is slightly in the money and delta equals 0.6, a $1 increase in the underlying price will lead to a delta of .60 + .05 = .65; a $1 decrease in the underlying price will lead to a delta of .60 – .05 = .55; a $2 increase in the underlying price will lead to a delta of .60 + .10 =.70, and a $2 decrease in the underlying price will lead to a delta of .60 – 0.10 = .50.

If delta equals -.60 (meaning that we have a slightly in the money put), a $1 increase in the underlying price will cause the put to be closer to at the money and result in a delta of -.60 + .05 = -.55; a $1 decrease in the underlying price will cause the put to become more in the money and lead to a delta of -.60 – 0.05 = -.65; similarly, a $2 increase in the underlying price will lead to a delta of -.60 + .10 = -.50, and a $2 decrease in the underlying price will lead to a delta of -.60 -.10 = -.70

Gamma is an especially important measurement if you are attempting to maintain delta neutrality in your position. If gamma is large, delta will change rapidly as the underlying price moves and you will have to make large changes in your position as the price of the underlying changes. This may create problems such as high commissions, frequent adjustments, constant monitoring, etc. If gamma is small, then the delta neutrality of your position is relatively insensitive to movement in the underlying and your position will require little adjustment to maintain delta neutrality.

What is a large gamma? To decide this, incorporate the volatility of the underlying. If it is, say, 2% per day and gamma is .05 and the price of the underlying is $100, you can expect the underlying to move about $2 per day. And a $2 change in the underlying leads to a $2*.05 = $0.10 daily change in the delta of the position. At a volatility of 4% per day, you will get a daily change in the delta in the position of $0.20 per day, and you will have to adjust your position twice as frequently to maintain the same level of risk as when the volatility is only 2% per day.

You will find that the largest gammas occur in the at the money (ATM) options. In fact, you can sketch gamma against the price of the underlying just by using your intuition, as follows: Low underlying prices have deltas close to 0 in the case of calls and close to -1.0 in the case of puts. Small changes in the underlying leave the deep out-of-the-money options still deep out of the money, and deep in-the-money options still deep in the money, and thus cause the delta to change little in either case.

Similarly, for high values of the underlying, small changes leave the deep OOTMs still deep OOTM and the deep ITMs still deep ITM, so gamma is small for high values of the underlying. The region where gamma changes most is the ATM region, when the underlying is around the strike price. So the sketch looks a little like a bell curve.

The gamma of a spread is just the combination of the gammas of the various options in the spread, using the same ratios. For instance, if you are long 2 calls with gamma =.05 and short 5 calls with gamma =.03, the net gamma of your position is 2*0.05 – 5*0.05 = -0.05. This position has a negative gamma, and thus a delta that decreases as the underlying increases.