Gamma (Options Greek): Understanding How Delta Changes with Price Movements

Gamma is one of the key options Greeks that traders use to manage and understand the risk profile of options positions. While many traders are familiar with delta, which measures how much an option’s price is expected to change for a $1 move in the underlying asset, gamma goes a step further. Specifically, gamma measures how much the delta itself will change if the price of the underlying asset moves by $1. In other words, if delta tells you the sensitivity of the option price to the underlying, gamma tells you how that sensitivity changes as the underlying price changes.

Formula: Gamma is mathematically defined as the second derivative of the option price with respect to the underlying price, or the first derivative of delta with respect to the underlying price. In practical terms, it can be expressed as:

Gamma = ∂Delta / ∂Underlying Price

This means that if the underlying asset moves by one dollar, the option’s delta will increase or decrease by the amount of gamma.

Why is Gamma important?

Gamma helps traders understand the stability or instability of their delta hedge. For example, if you have an option position with a high gamma, the delta can change rapidly with small price movements in the underlying, requiring frequent adjustments if you are delta-hedging. Conversely, options with low gamma have deltas that change more slowly, making hedging simpler.

Gamma is typically highest for at-the-money (ATM) options that are near expiration. Deep in-the-money or out-of-the-money options tend to have lower gamma. This is because ATM options are most sensitive to small price changes in the underlying, while deep ITM or OTM options behave more like the underlying or like worthless contracts, respectively.

Real-Life Example:

Imagine you are trading options on a popular stock trading at $100. You hold a call option with a delta of 0.50 and a gamma of 0.10. If the stock price rises from $100 to $101, your delta won’t just stay at 0.50; it will increase by the gamma amount. So, new delta = 0.50 + 0.10 = 0.60. This means your option’s sensitivity to further price changes has increased, reflecting a higher probability that the option will finish in the money.

If you are actively managing a delta-neutral portfolio, this change in delta due to gamma means you need to rebalance your hedge, such as buying or selling shares to maintain delta neutrality.

Common Mistakes and Misconceptions:

One common misunderstanding is that delta remains constant as the underlying price moves. In reality, delta is dynamic and changes continuously — and gamma measures exactly how fast it changes. Traders who fail to account for gamma risk may find their hedges becoming ineffective quickly, especially when the underlying is volatile.

Another misconception is that gamma is always positive. For long options (calls and puts), gamma is positive, but for short options (where you have sold the option), gamma is negative. This means that short option positions can become riskier as the underlying moves, because the delta can change against your position rapidly.

People often ask: “How does gamma affect option pricing?” or “What’s the difference between delta and gamma?” It’s important to remember that delta measures the first-order sensitivity of option price to the underlying, while gamma measures the second-order sensitivity — essentially the curvature of the option price graph in relation to the underlying price.

Additionally, traders may wonder: “How does gamma behave near expiration?” Gamma tends to increase as expiration approaches, especially for ATM options. This can lead to rapid changes in delta and option prices near expiry, making gamma risk particularly important to monitor during the final days of an option’s life.

In summary, gamma is a vital Greek that provides insight into how an option’s delta will change with movements in the underlying price. Understanding gamma is crucial for effective risk management, especially for traders who engage in delta hedging or complex options strategies. Keeping an eye on gamma can help you anticipate when your portfolio’s sensitivity to the underlying might shift sharply and allow you to adjust your positions accordingly.