# Options Trading VII, Gamma What does the Number say

This publication is part of a series of posts on the Greeks. These posts are a short summary of a part of the content of my book How to Calculate Options Prices and Their Greeks

As shown in previous publications, gamma trading is based on the change of the delta of an option in relation to a change in the underlying level.

The Delta

The delta of an option can be seen as equivalent to the amount of underlying shares/ futures represented by this option. For instance, for an options portfolio consisting of 100 at the money calls (each call representing 100 shares) with a delta of 50% one can say that if the underlying will move up by 1 $, the value of the options portfolio will increase by 10,000 (uderlying units) times 50 cents (50% of 1 $) resulting in a value increase of $ 5,000. This is the same as assuming that an options portfolio with 100 calls with a 50% delta is equivalent to a position where one is 5,000 shares long. So actually we can trade deltas as if they are shares/ futures.

Dynamic Delta

When having a look at the chart below, one can see that if the market starts moving, the 50 call will have a highly dynamic delta distribution. At 50 it will have a delta of 52% (equivalent to being long 5,200 shares when owning 100 calls), at 54 it will have a delta of 79% (equivalent to being long 7,900 shares), at 48 it will have a delta of 36%. So each time the delta/ amount of shares of the position will change when the market meets new levels. Trading (each time neutralizing the total delta of the portfolio) these delta changes is called gamma trading.

The chart above is showing the delta distribution of a 50 call at 10% and a maturity of 1 year. When being long 100 times the 50 calls, one can trade quite some amount of underlying shares according to the delta changes in relation to the market changes. The initial position of being long 100 50 calls, will be hedged by selling 5,200 shares in order to have a neutral delta position (no risk when there will be an adverse market move). When the market will increase, the options, in combination with the underying hedges, will generate a long delta position which can be hedged. When the market will decrease, the options, in combination with the underlying hedges, will generate a delta short position, which can be hedged.

So the owner of a long options portfolio can generate a decent P&L on the back of these delta (underlying) hedges, as shown below (market moves from 50 to 54 and back, to 48 and then back to 50).

Gamma

So with an option position of 100 50 calls, the delta on the way up from 50 to 54 will increase by 2,700. This means that on average the delta will increase by 675 (shares) when the market moves up 1 $. On the way down, from 50 to 48, the delta position will decrease by 1,600, meaning that on average the delta will decrease by 800 (shares) when the market will drop with 1 $. This average increase on the way up and decrease on the way down, expressed per $, is what is called the gamma.

Quite often the gamma is expressed in a percentage. For instance when the gamma is 8%, this means that the delta will increase by 8% when the market would move up 1$ and that the delta will decrease by 8% when the market would decrease 1$. With a position consisting of 100 calls, the gamma, expressed in share increase/ decrease per $, will thus be: 10,000 times 8% resulting in 800 shares per $.

The chart below shows the gamma distribution of a 50 call at 10% volatility and a maturity of 1 year.

One can see the 8% gamma for the 50 call, however this is not a constant gamma value, it changes at different levels in the underlying. When being at the money it will have its highest value, when being at the boundaries of the probability distribution (i.e. far out of the money, around 35 and 70, see previous article on boundaries) the gamma will be the lowest.

Now let’s assume a trader who has a position with an evenly distributed gamma, meaning that he owns for instance a gamma of 2,000 over a large range. This means that when the market increases, every dollar higher he will get longer 2,000 additional shares, when the market is decreasing he will get 2,000 shares shorter by each drop of 1 $ in the underlying.

So, as for gamma one can expect some sort of exponential P&L growth when not hedging the underlying. The “prudent” trader however, the one hedging every dollar, will end up with a P&L of $ 5,000 after the (large) move from 50 to 55, as shown below.

Gamma Hedging Strategy

How to hedge your gamma is thus key to your performance. The wide hedging strategy as shown in the first table has a far better result than the tight hedging strategy as performed by the “prudent” trader.

In my book; How to Calculate Options Prices and Their Greeks, the tight gamma hedging strategy versus the wide gamma hedging strategy is lengthily discussed.

Any option trader should have a strategy of how to hedge his gamma position. Most often a gamma short hedging strategy is based on the tight strategy and a gamma long hedging strategy on the wide strategy. Usually the hedging strategy is based on personal preferences and experience, however quite often traders are not aware of the necessity of having a gamma hedging strategy in place.

In the next article I will further discuss on gamma hedging strategies.