Option Greeks: Gamma

This article continues our series on the option Greeks, which began with part 1 on delta. In this part, we look at the second Greek letter in options analysis: gamma.

Continuing the example from the previous part: NIFTY is trading at 8000 on the spot market and the call strike price is 8200. This is an OTM option, so delta can be anywhere from 0 to 0.5; here I assume a delta of 0.2. NIFTY’s spot price then rises 300 points, which means the 8200 option moves from OTM to ITM and delta is no longer 0.2 but somewhere between 0.5 and 1. In this case, let us assume delta is 0.8.

Clearly, when the spot price changes and the option premium changes, delta changes too. Delta is a variable that shifts with movements in the underlying price and the premium. To measure how delta changes when the price of the underlying changes, we use gamma.

A gamma example

Example:

NIFTY spot = 8326

Strike price = 8400

Option premium = 26

Delta = 0.3

Gamma = 0.0025

Change in spot price = 70 points

What are the new premium and the new delta?

New premium = 26 + 0.3*70 = 47

Change in delta = Gamma * change in spot price = 0.0025*70 = 0.175

New delta = 0.3 + 0.175 = 0.475

So when NIFTY rises from 8326 to 8396, the premium rises from 26 to 47 and delta moves from 0.3 to 0.475.

Now suppose the spot price rises another 70 points, from 8396 to 8466.

Previous premium = 47

Previous delta = 0.475

From this, the change in premium = 70*0.475 = 33.25.

New premium = 47 + 33.25 = 80.25

New delta = 0.475 + 0.0025*70 = 0.65

Note that, unlike delta, gamma is always positive for both calls and puts.

In the next example, an ATM put has a gamma of 0.004. If the spot price of the underlying moves 10 points, what is the new delta?

Because this is an ATM put, delta is -0.5 (as discussed in the previous article).

Case 1: the spot price rises 10 points

Delta = -0.5

Gamma = 0.004

Change in spot price = 10

New delta = -0.5 + 0.004*10 = -0.46

Case 2: the spot price falls 10 points

New delta = -0.5 + 0.004*(-10) = -0.54

Using gamma to estimate risk

Using gamma to estimate risk

Scenario:

Suppose you trade 10 contracts (10 ATM options contracts, each with a delta of 0.5, which is equivalent to holding 5 futures contracts).

Strike price = 8400

Spot price = 8405

Delta = 0.5

Gamma = 0.005

“Short” position

You sell 10 call options, so the total delta is 10*0.5 = 5. Recalling what we covered on delta in the previous part, a delta of 5 means that when the underlying moves 1 point, the premium on the whole position moves 5 points.

Now, if NIFTY moves 70 points, the position’s new delta is (0.5 + 0.005*70)*10 = 8.5. At this point, a 1-point move in the underlying moves the premium on your position by 8.5 points. The position’s risk has increased because delta has risen from 5 to 8.5.

Because you are short calls, you do not want delta to be high. Take a few minutes to think this through. When you sell a call, however far the underlying falls, your income is limited to the fixed premium you collected. When the underlying rises, your position loses money, and the higher the delta, the more you lose. So the risk of a short call position grows as delta rises. Since gamma measures how delta changes, that is why gamma is used as a tool for estimating risk.

How gamma behaves

Look at this chart of the relationship between gamma and the spot price:

The relationship between gamma and the spot price

The blue, green and red curves show how gamma moves as the spot price changes, for options with strike prices of 80, 100 and 120.

Take the blue line on its own: this is a call with a strike of 80.

• When the spot price equals 80, the short call is ATM.
• When the spot price is below 80, the short call is OTM
• When the spot price is above 80, the short call is ITM
• Gamma is small for OTM options and peaks when the option is ATM
• Gamma is also low for ITM options. So for a 1-point move in the underlying, the delta of an ITM option changes less than that of an ATM option. But an ITM option’s delta is higher than an ATM option’s, so the change in its premium is relatively large.

The green and red lines behave in the same way.

Gamma: key takeaways 

• Gamma measures the rate of change of delta
• Gamma is always positive for both call and put options
• A large gamma can mean large risk
• When you buy calls or puts, you are “long gamma”
• When you sell calls or puts, you are “short gamma”
• Delta changes quickly for ATM options
• Delta changes slowly for ITM and OTM options

Happy trading!