Gamma is a crucial metric for options traders more particularly as it helps them to quantify the rate at which the delta of an option changes in relation to the underlying asset's price. Delta measures the option's sensitivity to changes in the underlying asset's price, while gamma measures the rate at which delta changes in response to changes in the underlying asset's price.

It is equally essential for risk managers to understand the potential risks associated with an option position, particularly when the underlying asset's price is expected to fluctuate significantly. Options with high gamma are particularly risky as their delta will change significantly with even small movements in the underlying asset's price, leading to potentially large and unexpected changes in the position value.

To mitigate this risk, options traders can utilize gamma to make informed decisions and adjust their position size or hedge their positions.

Formally, an option's gamma is a sensitivity measure used to evaluate an option derivative instrument.

Explanation-1: An option's gamma is a rate of change in the option's delta with respect to a change in the price of the underlying asset, other factors being constant.

Explanation-2: An option's gamma is an amount by which the option's delta is expected to change with respect to a $1 change in the price of the underlying asset, other factors being constant.

As per the explanations,

Option Gamma = Δ in Option Delta / Δ in Underlying Price

###### Example:

(this example is in continuation to Understanding Delta: A Deep Dive into Options Greek)

A 1-Day price movement is as follows:

The Strike Price is $1490.00

Underlying Price changes from $1488.05 to $1498.05

Call Option Delta changes from $0.4785 to $0.6435

Put Option Delta changes from $-0.5215 to $-0.3565

Therefore, using the risk sensitivity approach,

gamma of a call option = ( CDt - CDt-1 ) / ( St - St-1 ) = ( $0.6435 - $0.4785 ) / ( $1498.05 - $1488.05 ) = 0.0165 represents that the call option's delta is expected to change by $0.0165 for every $1 change in the price of an underlying asset.

gamma of a put option = ( PDt - PDt-1 ) / ( St - St-1 ) = ( $-0.3565 - $-0.5215 ) / ( $1498.05 - $1488.05 ) = 0.0165 represents that the put option's delta is expected to change by -$0.0165 for every $1 change in the price of an underlying asset.

Option's gamma is a second-order derivative i.e. a derivative (gamma) of a derivative (delta) as it measures the rate of change of an option's delta stating how fast an option changes its directional movement. The important point to note here is that an option's gamma i.e. the rate of change of the option's delta of calls and puts is exactly equal.

"gamma influences the delta of calls and puts in the same direction"

In our example, the price of the underlying asset increases by $10.00 from $1488.05 to $1498.05 and so the call option's delta as well from $0.4785 to $0.6435 reflects a higher probability and higher expected payoff in case the option expires in the money. The change of $0.0165 is getting added to the call option's delta. However, the put option's delta decreases from $-0.5215 to $-0.3565 reflecting a lower probability of the option expiring in the money. Again, the change of $0.0165 is getting added to the put option's delta.

Irrespective of calls or puts, the option's gamma will remain equal as the impact on the option's delta of both calls and puts is equal as highlighted above in the example.

"gamma is always positive for long and negative for short"

In our example, if the price of the underlying asset decreases by $10.00 from $1488.05 to $1478.05, the call option's delta will also decrease from $0.4785 to $0.3162 reflecting a lower probability and lower expected payoff in case the option expires in the money. The change of $0.0162 is getting deducted from the call option's delta. However, the put option's delta increases from $-0.5215 to $-0.6838 reflecting a higher probability of the option expiring in the money. Again, the change of $0.0162 is getting deducted from the put option's delta.

Irrespective of the movement in the price of the underlying asset, the option's gamma is positive for a long as highlighted above in the example.

In other words, gamma is always beneficial to a long option.

### Option Gamma: An Estimation Tool or A Risk Sensitivity?

'gamma' sensitivity is important because it helps, (1). traders to estimate the movement in the option's delta for any change in the price of the underlying asset. (2). risk managers to gauge the directional risk add-on to the first-order risk and is the most crucial risk attribute for analysis as it explains the second-order risk, sometimes referred to as higher-order risk.

### Estimating the Gamma Movements using the Pricing Formula

As discussed here, traders use the option's delta to predict movements in the option's price based on the movements in the price of the underlying asset.

"delta captures the linear movement and delta changes momentarily"

Mathematically speaking, if the price of an underlying asset increases (or decreases), the option's delta of both call and put options increases (or decreases) but at a decreasing rate. If it would have been a constant increase or decrease, the question of second-order derivative wouldn't have been in the picture. And this is because of the non-linear relationship between the option's price and the price of the underlying asset as shown below.

And because of this relationship, the delta curves of both call and put options are not tangent/straight lines as these curves represent that relationship highlighted above.

Instead, these curves are seeming to be more like s-shaped and behave differently at different points of moneyness.

for out-of-the-money options: The delta values of both call and put options are almost near 0, a small change in the price of the underlying asset won't be impacting much on the probability, and therefore, the delta value remains reasonably stable resulting in gamma to be almost flat at 0.

for at-the-money options: The delta values of both call and put options are almost 0.5, a small change in the price of the underlying asset will be impacting the delta values to a greater extent as the sensitivity of the option's delta to a change in the price of the underlying asset i.e. option's gamma is highest at this point.

for in-the-money options: The delta values of both call and put options are almost near 1 (absolute-wise), a small change in the price of the underlying asset won't be impacting much on the probability, and therefore, the delta value remains reasonably stable resulting in gamma to be almost flat at 0.

However, it may also change due to the change in the volatility and time-decay, but that is the next topic of our financial book.

The formula that has been stated above with an example is a sensitivity-based formula that strictly goes with explanations 1 and 2. However, professional traders determine the price of an option using some sophisticated models that often resemble the Black-Scholes-Merton (BSM) model. This model provides a mathematical function to estimate the theoretical price of an option by considering market risk factors.

Explanation-3: An option's gamma determines the variability in the expected payoff in case the option expires in the money.

As discussed here, N(d1) is the probability that determines the expected payoff in case the option expires in the money. And,

As per explanation 3, [ St*σ*√T ] determines the variability in the expected payoff in case the option expires in the money.

Therefore,

Gamma = [ (LN(St/K) + (r-q+σ^2/2)*T) / σ*√T ] from standard normal distribution / (St*σ*√T)

###### Example:

[option's delta-gamma series across the spot ladder at a particular strike price]

Using the Black-Scholes equation,

[ (LN(1488.05/1490.00) + (0.06+0.2145^2/2)*0.0055) / 0.2145*√0.0055 ] from standard normal distribution / (1488.05*0.2145*√0.0055)

Gamma = 0.0169

The below table represents options delta-gamma values and prices across the movement in the price of the underlying asset. This can be calculated with the help of the above equation.

Observation-1: Considering an ATM option, the probability of the option landing ITM is 50%, and therefore, the delta value of the call option is near 0.5, a small change in the price of the underlying asset will be impacting the delta values to a greater extent as the sensitivity of the option's delta to a change in the price of the underlying asset i.e. option's gamma is highest at this point.

As the price of the underlying asset increases (from ATM to ITM to deep ITM i.e. from $1488.05 to $1498.05 to $1508.05), the delta of the call option also increases (from 0.4785 to 0.6435 to 0.7843) but at a decreasing rate. As the option gets further ITM, the rate of increase in the option's delta further decreases, and therefore, the sensitivity of the option's delta i.e. option's gamma approaches to 0.

As the price of the underlying asset decreases (from ATM to OTM to deep OTM i.e. from $1488.05 to $1478.05 to $1468.05), the delta of the call option also decreases (from 0.4785 to 0.3162 to 0.1825) but at a decreasing rate. As the option gets further OTM, the rate of increase in the option's delta further decreases, and therefore, the sensitivity of the option's delta i.e. option's gamma approaches to 0.

Observation-2: Considering the other options trading deep ITM or OTM has a greater probability of them getting exercised or lapsed, respectively, as a small change in the price of the underlying asset won't be impacting much on the probability, and therefore, the delta value remains reasonably stable resulting in gamma to be almost flat at 0.

### Understanding Gamma is key to becoming an Expert in Trading Options

Professional traders are more interested in monitoring the entire options chain having both options prices and their corresponding delta-gamma values across strike prices as it helps them to, (1). estimate any potential change in the underlying price impacting all the positions that they have in the options market. (2). adjust their positions by buying or selling those options.