Absolute option gamma

Absolute Option Gamma

Absolute Option Gamma is a measure of the sensitivity of an option's delta to changes in the price of its underlying asset. As a key component of the Option Greeks, which are a set of risk measures in options trading, absolute option gamma quantifies how much the Delta of a Derivative will change for a one-point move in the price of the Underlying Asset. This second-order sensitivity is crucial for traders and portfolio managers in assessing and managing the dynamic risk of their options positions.

History and Origin

The concept of option "Greeks," including gamma, emerged prominently with the development of sophisticated option pricing models. While options and similar financial instruments have existed for centuries, the mathematical framework to systematically price them and understand their sensitivities gained significant traction in the 20th century¹³. A pivotal moment was the publication of the Black-Scholes Model in 1973 by Fischer Black and Myron Scholes¹², with contributions from Robert Merton. This model provided a theoretical valuation for European-style options and laid the groundwork for calculating various sensitivities like delta, gamma, theta, and vega. The Black-Scholes model revolutionized the nascent derivatives market, offering a quantitative approach that enabled a deeper understanding of option price behavior and the risks inherent in options positions¹¹.

Key Takeaways

• Absolute Option Gamma measures the rate of change of an option's delta with respect to the underlying asset's price.
• It is highest for options that are at-the-money and those with less time until their Expiration Date.
• High absolute option gamma indicates that delta will change rapidly as the underlying price moves, leading to significant fluctuations in the overall option position's sensitivity.
• Traders utilize absolute option gamma to understand the convexity of their options portfolio and manage dynamic Hedging strategies.
• Positive absolute option gamma benefits long option positions when the underlying asset moves sharply, while negative absolute option gamma can amplify losses for short option positions.

Formula and Calculation

Absolute option gamma is mathematically represented as the second derivative of the option price with respect to the underlying asset's price. For a European call or put option, within the framework of the Black-Scholes model, the gamma (\Gamma) can be calculated as:

Where:

• (N'(d_1)) = The probability density function of the standard normal distribution evaluated at (d_1).
• (d_1 = \frac{\ln(S/K) + (r - q + \sigma^2/2)T}{\sigma \sqrt{T}})
• (S) = Current price of the Underlying Asset
• (K) = Strike Price of the option
• (T) = Time to Expiration Date (in years)
• (r) = Risk-free interest rate
• (q) = Dividend yield of the underlying asset
• (\sigma) = Volatility of the underlying asset

This formula shows that gamma is dependent on several factors, including the underlying price, strike price, time to expiration, and volatility.

Interpreting the Absolute Option Gamma

Interpreting absolute option gamma involves understanding how an option's sensitivity to price movements, represented by its delta, changes. A high absolute option gamma indicates that the delta of an option will change significantly with small movements in the underlying asset's price. Conversely, a low absolute option gamma suggests that the delta will remain relatively stable.

For investors holding long option positions (buying a Call Option or a Put Option), positive gamma is generally favorable, as it means their delta will increase in magnitude when the underlying asset moves in the desired direction. For example, if a call option has a positive delta and positive absolute option gamma, its delta will increase as the underlying stock price rises, leading to accelerated gains.

Conversely, for those who have written or sold options, a negative gamma implies that their delta will move against them as the underlying price fluctuates, increasing their exposure to price movements¹⁰. Options that are at-the-money (where the strike price is close to the current underlying price) typically have the highest absolute option gamma because their delta is most sensitive to price changes around this point⁹. Similarly, options closer to expiration tend to have higher absolute option gamma, making their deltas more volatile as time runs out. Understanding this dynamic is critical for effective Risk Management in options portfolios.

Hypothetical Example

Consider an investor, Sarah, who owns a Call Option on XYZ stock with a strike price of $100 and a current delta of 0.50. This means for every $1 increase in XYZ stock, the option price is expected to increase by $0.50.

Now, let's consider the absolute option gamma for this call option. Suppose the gamma is 0.10.

• If XYZ stock increases from $100 to $101, the option's new delta will be approximately (0.50 + 0.10 = 0.60).
• If XYZ stock increases further from $101 to $102, the option's delta will increase again to approximately (0.60 + 0.10 = 0.70).

This demonstrates how positive absolute option gamma causes the delta to accelerate in the direction of the underlying price movement, amplifying the option's sensitivity. Conversely, if XYZ stock were to fall, the delta would decrease, lessening the negative impact on the call option price. This dynamic change in delta is a key characteristic that absolute option gamma helps quantify.

Practical Applications

Absolute option gamma is a critical measure for various market participants, particularly in the realm of Hedging and risk assessment. Market Makers, who facilitate options trading by providing liquidity, constantly manage their exposure to price movements of the underlying assets⁸. They often employ delta-hedging strategies, which involve adjusting their positions in the underlying asset to offset the delta of their options portfolio. However, because delta changes as the underlying price moves, market makers with significant options inventories must consider gamma to dynamically rebalance their hedges⁷.

A notable real-world application of absolute option gamma's influence is seen during a "gamma squeeze." This phenomenon occurs when a rapid increase in demand for Call Option contracts forces market makers to buy large amounts of the underlying stock to hedge their increasingly positive delta exposure. This buying pressure, amplified by the options' rising delta (due to high gamma), can create a self-reinforcing cycle, leading to a sharp and rapid increase in the underlying stock's price⁶. Examples of such events include the GameStop surge in January 2021, where significant retail investor activity in options contributed to a dramatic price movement driven by market maker hedging⁴, ⁵. Understanding absolute option gamma is therefore crucial for both options traders and regulators assessing market dynamics. The U.S. Securities and Exchange Commission (SEC) provides resources to help investors understand the basics and risks associated with options trading³.

Limitations and Criticisms

While absolute option gamma is a valuable tool for understanding options sensitivity, it has inherent limitations. Like all option Greeks, gamma is a point-in-time measure based on specific inputs and model assumptions. Its value can change rapidly, particularly for options nearing their Expiration Date or those precisely at the Strike Price. This means a gamma-neutral position can quickly become gamma-exposed as market conditions change, requiring frequent rebalancing and incurring higher transaction costs for portfolio managers².

Furthermore, the calculation of absolute option gamma relies on assumptions about future Volatility, which is often estimated using Implied Volatility. If the actual realized volatility deviates significantly from the implied volatility, the gamma calculation may not accurately reflect the true sensitivity. Academic research highlights how gamma positioning can impact market quality and volatility, with negative gamma positions potentially increasing market instability¹. While useful for professional Risk Management, relying solely on gamma or other Greeks without considering broader market context and potential deviations from theoretical models can lead to unexpected outcomes.

Absolute Option Gamma vs. Delta

Absolute option gamma and Delta are both fundamental option Greeks, but they describe different aspects of an option's price sensitivity.

While delta indicates the immediate directional exposure of an option position, absolute option gamma describes how that directional exposure will itself change as the underlying asset moves. In essence, delta tells a trader where their position is going, while absolute option gamma tells them how fast their directional sensitivity is changing. Traders often consider both delta and gamma together for comprehensive Risk Management and more sophisticated hedging strategies.

FAQs

What does a high absolute option gamma mean?

A high absolute option gamma means that the delta of an option will change significantly with even small movements in the price of the Underlying Asset. This implies that your option position's sensitivity to the underlying asset's price will fluctuate rapidly. For long option positions, high positive gamma can be beneficial, leading to accelerated profits as the underlying moves favorably.

How does absolute option gamma affect options trading strategies?

Absolute option gamma is crucial for strategies involving dynamic [Hedging], especially for [Market Maker]s and large institutions. It helps them anticipate how frequently they will need to adjust their hedges (e.g., buying or selling the underlying stock) to maintain a desired level of delta neutrality. Strategies that are "gamma positive" benefit from large price swings in the underlying, while "gamma negative" strategies are hurt by them.

Is absolute option gamma more important for short-term or long-term options?

Absolute option gamma tends to be highest for options with less time remaining until their Expiration Date. Therefore, it is generally more important for short-term options, as their deltas will change more dramatically with underlying price movements compared to long-term options. This makes short-term options inherently more volatile in terms of their sensitivity profile.