Definition of Option Gamma

The Gamma of an option measures the rate of change of the option delta. Its' number is denoted relative to a one point move in the underlying asset. For example, if the gamma for an option shows 0.015 with a delta of 0.45 then a full point move in the stock (i.e. 35 to 36) means the delta will move to 0.465.

Gamma is calculated via an option model such as Black and Scholes or Binomial. The value is the same for both call and put options.

The Gamma of an option is important to know because the delta of an option is not constant; the delta increases and decreases as the underlying moves. Because delta is essentially our position value in the underlying, the gamma therefore tells traders how fast their position will increase or decrease in value vs movements in the underlying asset.

In other words, Gamma shows how volatile an option is relative to movements in the underlying asset. So, watching your gamma will let you know how large your delta (position risk) changes.

When is Option Gamma Highest?

Gamma is not linear. Like Delta, Gamma has curvature and is effected by the inputs that calculate the Gamma, the most notable forces that influence it are factors such as the difference between the strike price and the underlying price, the time to expiration of the option and the implied volatility used in the model. Interest rates and dividends are also factors that effect the value of the Gamma, however, the magnitude of these inputs is minimal when compared to the previously mentioned variables.

The attention on a Gamma's sensitivity is mostly focused on its' position relative to the underlying price. Looking at the above graph you can see that the Gamma reaches its' peak when the option is at-the-money and tapers off either side. When an option position moves towards the ATM level, the changes in the position delta, and hence the position value relative to the stock, change with greater amounts. Options that are either deep ITM or deep OTM experience less variability as the stock price changes and therefore will show low Gamma values.

Time and Volatility

Adding more time to an option contract increases the likelihood of that option expiring in-the-money. Because higher volatility also increases the chances of an option's in-the-moneyness, both volatility and time have the same effect on an option's Gamma value.

The above graphs show how increasing time/volatility value reduces the Gamma of the option and hence it's sensitivity to changes in stock price.

While adding more time to an option increases the VAUE of the option, it generally reduces the option's Gamma. With more time to expiration the option becomes less sensitive to movements in the underlying asset. However, as the option approaches its' maturity date, its' time value will move towards zero and then become more responsive to changes in the underlying price.

These graphs provide a great way to look at how Gamma is effected by the passage of time. Both plot a $25 call option's Gamma across a range of underlying prices, however, on each graph is shown 3 different times to maturity. This is so you can see how the Gamma value becomes the highest when it is both ATM and close to expiration. When this happens, option positions will have the highest fluctuations in position value (Delta).

What is Long Gamma?

Note: The Gamma value is the same for calls as for puts. If you are long a call or a put, the gamma will be a positive number. If you are short a call or a put, the gamma will be a negative number.

When you are "long gamma", your position will become "longer" as the price of the underlying asset increases and "shorter" as the underlying price decreases.

Conversely, if you sell options, and are therefore "short gamma", your position will become shorter as the underlying price increases and longer as the underlying decreases.

This is an important distinction to make between being long or short options - both calls and puts. That is, when you are long an option (long gamma) you want the market to move. As the underlying price increases, you become longer, which reinforces your newly long position.

If being "long gamma" means you want movements in the underlying asset, then being "short gamma" means that you do not want the price of the underlying asset to move.

A short gamma position will become shorter as the price of the underlying asset increases. As the market rallies, you are effectively selling more and more of the underlying asset as the delta becomes more negative.

Gamma in Option Chain

The graphs shown here, display gamma with constant volatility and strike price. In practice, options across different strike prices have different implied volatilities and therefore a different gamma distribution.

The above is an example of what Gamma and Delta values look in practice. This is an option chain of MSFT stock options showing an expiration 10 days out.

Notice how the ATM strike of $76.50 shows the highest Gamma value of 0.233 for the calls and 0.235 for the puts. I'm not sure why they are different here...they really should show exactly the same value for the call and the put - perhaps a rounding issue. Nevertheless, 0.002 difference is fairly immaterial.

If the stock trades up 1 full point to $77.29 then the $76.50 call option Delta will move from 0.464 to 0.697. So while the stock price has only moved 1.3% your effective position in the underlying has increased by 50%.

Black Scholes Gamma

If you're interested in knowing how to calculate option gamma in excel, you can download my option pricing spreadsheet for a working example. Otherwise, here are some code examples:

Excel VBA

Option Gamma Formula

NdOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend) / (UnderlyingPrice * (Volatility * Sqr(Time)))

Where:

NdOne = Exp(-(dOne(UnderlyingPrice, ExercisePrice, Time, Interest, Volatility, Dividend) ^ 2) / 2) / (Sqr(2 * 3.14159265358979)) dOne = (Log(UnderlyingPrice / ExercisePrice) + (Interest - Dividend + 0.5 * Volatility ^ 2) * Time) / (Volatility * (Sqr(Time)))