Gamma: Understanding Changes in Delta

Why Do We Need Gamma?

In the previous chapters, we learned that Delta is not fixed.

As the underlying price changes, the option can move from OTM to ATM or from ATM to ITM. As this happens, its Delta also changes.

So a natural question arises:

How quickly does Delta change when the underlying price moves?

This is where Gamma becomes important.

Understanding the Idea Through Speed

A simple way to understand Gamma is to first think about a moving vehicle.

Suppose a car travels:

• 4 km in 5 minutes during one part of its journey.
• 8 km in the next 5 minutes.

The car is now travelling faster.

So, we are not only interested in its speed. We also want to know how quickly its speed is changing.

That change in speed is called acceleration.

In mathematics:

• Change in distance over time = Velocity
• Change in velocity over time = Acceleration

Velocity is considered a first-order derivative, while acceleration is a second-order derivative.

Connecting This to Options

Now apply the same idea to options.

We already know:

Delta = Change in Option Premium for a Change in Underlying Price

But Delta itself changes when the underlying moves.

Therefore:

Gamma = Change in Delta for a Change in Underlying Price

In simple words:

Gamma tells us how quickly an option's Delta changes when the underlying price moves.

A Simple Option Example

Suppose an index is trading at 8,000.

A Call Option with a Strike Price of 8,200 is OTM.

Assume its Delta is 0.20.

Now imagine the index rises sharply.

The Call may move closer to ATM or even become ITM.

Its Delta will no longer remain at 0.20. It may increase significantly.

Gamma helps us estimate this change in Delta.

So:

Underlying moves → Delta changes → Gamma measures that Delta change

Delta vs Gamma

It is important not to confuse the two.

The two work together.

Delta tells us how sensitive the option is today.

Gamma tells us how that sensitivity may change as the market moves.

Gamma as the Second-Order Derivative

For readers interested in the mathematical connection:

• Delta is the first-order derivative of the option premium with respect to the underlying price.
• Gamma is the second-order derivative of the option premium with respect to the underlying price.

This does not mean traders need advanced calculus to use Gamma.

The practical idea is much simpler:

Delta tells you the current sensitivity. Gamma tells you how that sensitivity changes.

Why Gamma Matters to Traders

Imagine you have calculated your position's Delta and believe your market exposure is manageable.

If the underlying moves significantly, Gamma can cause the Delta to change.

Your position may therefore become:

• More sensitive to price movements.
• Less sensitive to price movements.
• More exposed than you originally expected.

This is particularly important for traders holding larger option positions.

Gamma and Moneyness

Gamma is closely related to the option's moneyness.

As we will explore further, Gamma tends to be particularly important around ATM options, where Delta can change more rapidly.

For now, the important point is:

Delta changes as the option moves through different moneyness levels, and Gamma measures the rate of this change.

The detailed behaviour of Gamma across OTM, ATM and ITM options will be covered in the next chapter.

Practical Insight

Don't look at Delta as a permanent number.

If the underlying moves significantly, the option's Delta can change.

Gamma helps traders understand this changing exposure and therefore provides a deeper view of option risk.

Common Beginner Mistake

A common mistake is calculating the position's Delta once and assuming that the exposure will remain the same throughout the trade.

In reality, Delta changes as the underlying price changes.

Ignoring Gamma can therefore lead to an incomplete understanding of directional risk.

Key Insight

Delta measures sensitivity. Gamma measures how that sensitivity changes.

Understanding both gives a more complete picture of how an option position may behave when the underlying moves.

Key Takeaways

• Delta is not a fixed number; it changes as the underlying price changes.
• Gamma measures the rate of change of Delta.
• Delta is the first-order derivative of the option premium.
• Gamma is the second-order derivative of the option premium.
• Velocity and acceleration provide a useful way to understand the relationship between Delta and Gamma.
• Delta tells us the option's current sensitivity to the underlying.
• Gamma tells us how that sensitivity may change.
• Gamma becomes especially important when the underlying makes a significant move.
• Understanding Gamma helps traders assess changing directional exposure.