Option gamma: Meaning, Criticisms & Real-World Uses

What Is Option Gamma?

Option gamma is one of the key Options Greeks that measures the rate of change of an option's Delta with respect to a change in the price of the underlying asset. It is a second-order derivative, providing insight into the stability of an option's delta. Within the broader field of options pricing, gamma is crucial for understanding how sensitive an option's delta will be to movements in the underlying security's price. A high gamma indicates that delta will change rapidly as the underlying asset's price moves, while a low gamma suggests delta will be more stable. Understanding option gamma is essential for hedging strategies, particularly for market participants who need to maintain a delta-neutral position.

History and Origin

The concept of option gamma, along with other option sensitivities, gained prominence with the development of formal options pricing models. While rudimentary forms of options have existed for centuries, the modern financial theory underpinning their valuation, including metrics like gamma, largely stems from the groundbreaking work of Fischer Black, Myron Scholes, and Robert Merton. Their seminal 1973 paper, "The Pricing of Options and Corporate Liabilities," introduced what became known as the Black-Scholes model, providing a mathematical framework for valuing European-style options.

The formalization of option pricing coincided closely with the establishment of the first standardized options exchange in the U.S., the Chicago Board Options Exchange (CBOE), which launched in April 1973⁴, ⁵. The CBOE's introduction of standardized options trading meant that market participants could more easily trade and hedge, thereby increasing the practical need for robust metrics like option gamma. The increased sophistication of the market and the advent of theoretical pricing models mutually reinforced the importance of understanding the intricate relationships between an option's price and its various influencing factors.

Key Takeaways

Option gamma measures how much an option's Delta is expected to change for every one-point move in the underlying asset's price.
It is highest for at-the-money options and decreases as an option moves further in or out of the money.
Option gamma helps investors and traders assess the stability of their delta-hedged positions.
Long option positions (buying call option or put option) have positive gamma, while short option positions have negative gamma.
Positive gamma benefits from large price movements in the underlying asset, whereas negative gamma is penalized by such movements.

Formula and Calculation

Option gamma is derived from an options pricing model, most commonly the Black-Scholes model for European options. The formula for gamma is the second derivative of the option's price with respect to the underlying asset price.

For a call option or put option, the gamma ((\Gamma)) can be calculated as:

\Gamma = \frac{N'(d_1)}{S \sigma \sqrt{T-t}}

Where:

(N'(d_1)) = The probability density function of the standard normal distribution evaluated at (d_1).
(S) = Current price of the underlying asset.
(\sigma) = Volatility of the underlying asset.
(T-t) = Time remaining until the expiration date.

The (d_1) term is itself part of the Black-Scholes formula for calculating option prices and delta:

d_1 = \frac{\ln(\frac{S}{K}) + (r + \frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t}}

Where:

(K) = Strike price.
(r) = Risk-free interest rate.
(\ln) = Natural logarithm.

Interpreting the Option Gamma

Interpreting option gamma is crucial for managing option portfolios. A positive gamma indicates that as the underlying asset's price increases, the option's Delta will increase (become more positive for calls or less negative for puts). Conversely, if the underlying price decreases, delta will decrease. This positive relationship means that long option positions generally benefit from larger price swings in the underlying.

Conversely, a negative gamma means that delta will move in the opposite direction of the underlying price change. For example, if an investor is short an option with negative gamma, and the underlying price increases, the absolute value of their negative delta exposure will increase, necessitating more active hedging to maintain a neutral position.

Option gamma is highest for options that are "at-the-money," meaning their strike price is very close to the current price of the underlying asset. As an option moves further in-the-money or out-of-the-money, its gamma tends to decrease, indicating that its delta will become less sensitive to subsequent price changes in the underlying.

Hypothetical Example

Consider an investor who holds a call option on Company XYZ stock with a strike price of $100.

Initially, Company XYZ stock is trading at $100, and the option has a Delta of 0.50 and a Gamma of 0.10. This means that for every $1 increase in the stock price, the option's theoretical value should increase by $0.50.
If Company XYZ's stock price increases by $1 to $101, the option's delta would increase from 0.50 to 0.50 + 0.10 = 0.60.
Now, if the stock price moves up another $1 to $102, the delta will increase again, but from its new value. The delta would become approximately 0.60 + 0.10 = 0.70.

This illustrates how option gamma accelerates the change in delta. For an investor or market maker trying to maintain a hedging portfolio, positive gamma means their position becomes more delta-positive as the underlying rises, and more delta-negative as it falls. This can be beneficial for those anticipating large moves.

Practical Applications

Option gamma plays a vital role in portfolio management and risk control for traders and financial institutions. One of its primary applications is in gamma hedging, where traders aim to create a portfolio whose aggregate option gamma is neutral or within acceptable limits. This is particularly important for market makers who often have short option positions, which inherently carry negative gamma. Negative gamma means that as the underlying asset moves, the Delta of their position moves against them, requiring frequent rebalancing to maintain a delta-neutral hedge.

For instance, if a market maker is short options and the underlying stock rises, their negative delta will become even more negative. To re-establish a delta-neutral position, they would need to buy more of the underlying asset at a higher price. Conversely, if the stock falls, their negative delta would become less negative (or even positive), requiring them to sell the underlying at a lower price. This constant buying high and selling low to maintain a delta-neutral position with negative gamma can lead to significant losses,¹ ² ³