Adjusted average gamma

What Is Adjusted Average Gamma?

Adjusted average gamma refers to a conceptual approach within options trading for evaluating and managing the overall sensitivity of an options portfolio to changes in the underlying asset's price. While "adjusted average gamma" is not a universally standardized metric with a single, agreed-upon formula, it encapsulates the strategic consideration of how a trader or market maker dynamically manages their aggregate gamma exposure over time or across various derivatives positions.

Gamma is one of the "Options Greeks," which are measures used to quantify the sensitivity of an option's price to various factors. Specifically, gamma measures the rate at which an option's delta changes in response to movements in the underlying asset's price²⁵, ²⁶. A higher gamma indicates that the delta will change more dramatically for a given price move in the underlying²⁴. The concept of adjusted average gamma emphasizes the ongoing, active risk management required in dynamic markets.

History and Origin

The concept of "Greeks," including gamma, emerged from the theoretical advancements in options trading in the 20th century. While rudimentary forms of options contracts can be traced back to ancient Greece, with philosopher Thales of Miletus reportedly using them for olive presses²², ²³, modern options pricing theory truly began to take shape with the standardization of options contracts and the development of mathematical models.

A pivotal moment was the establishment of the Chicago Board Options Exchange (CBOE) in 1973, which provided a formalized marketplace for options. In the same year, the groundbreaking Black-Scholes model was conceived by Fisher Black and Myron Scholes. This formula revolutionized options pricing and laid the foundation for understanding the sensitivities of options, from which the Greeks were derived²⁰, ²¹. The Greeks, including delta, gamma, theta, vega, and rho, became essential tools for traders and financial institutions to measure and manage the various risks associated with their options positions¹⁸, ¹⁹. While gamma itself has a clear origin, the notion of "adjusted average gamma" evolved from the practical need for sophisticated traders to manage their aggregate portfolio risk dynamically, particularly in highly volatile markets.

Key Takeaways

• Adjusted average gamma is a conceptual approach to managing the aggregate gamma exposure within an options portfolio.
• Gamma measures the rate of change of an option's delta in response to underlying asset price movements.
• It is crucial for understanding how an option's directional sensitivity (delta) will accelerate or decelerate.
• Positive gamma benefits long option positions, while negative gamma increases risk for short option positions¹⁷.
• Effective management of average gamma is vital for maintaining appropriate hedging strategies and controlling portfolio risk.

Formula and Calculation

"Adjusted Average Gamma" does not refer to a single, standardized mathematical formula in finance. Instead, it describes the process of considering and managing gamma exposure across an entire portfolio or over a specific period, implying active adjustments.

However, the calculation for Gamma for a single option is fundamental to this concept:

Where:

• (\Delta \text{Delta}) represents the change in the option's delta.
• (\Delta \text{Underlying Price}) represents the change in the price of the underlying asset.

In practice, this means if an option's delta is 0.50 and its gamma is 0.10, a $1 increase in the underlying asset's price would cause the delta to increase to 0.60 (0.50 + 0.10)¹⁶. When considering "adjusted average gamma," a trader would conceptually or mathematically average the gamma values of all options positions within their portfolio, possibly weighted by position size or dollar exposure. The "adjustment" then comes from actively adding or removing positions to modify this average gamma to a desired level, often for hedging or speculation.

Interpreting the Adjusted Average Gamma

Interpreting the adjusted average gamma involves understanding its implications for a portfolio's overall risk management and potential profitability. Since gamma itself indicates how quickly a portfolio's delta changes, an adjusted average gamma reflects the aggregate sensitivity of the entire options position to price movements in the underlying assets.

A portfolio with a high positive adjusted average gamma means that as the underlying assets move in the favorable direction (up for long calls/short puts, down for long puts/short calls), the portfolio's delta will increase rapidly, leading to accelerated profits. Conversely, unfavorable movements will see delta decrease, mitigating losses. This characteristic makes high positive gamma desirable for positions anticipating large price swings. However, it also means that the position requires more frequent hedging adjustments to maintain a desired delta neutrality, if that is the strategy¹⁵.

A portfolio with a high negative adjusted average gamma means that favorable movements will see delta decrease, limiting profits, while unfavorable movements will cause delta to increase, accelerating losses. This scenario is riskier, as it requires constant monitoring and potentially costly re-hedging, especially as options approach their expiration dates. Traders aim to adjust their average gamma to align with their market outlook and risk tolerance, often targeting positive gamma when expecting volatility and negative gamma when expecting stability.

Hypothetical Example

Consider a professional portfolio manager, Sarah, who manages a large options portfolio on a tech stock, "Innovate Corp." (INN). Sarah holds a mix of call options and put options with various strike prices and expiration dates.

Initially, her portfolio has an aggregate gamma of +1,500. This means for every $1 increase in INN's stock price, the portfolio's net delta is expected to increase by 1,500. Sarah believes INN is due for a significant price swing following an upcoming earnings announcement, but she is unsure of the direction. She wants to maintain a relatively high positive average gamma to profit from a large move while remaining delta-neutral at the current price.

Currently, INN is trading at $100.

1. Initial Position: Sarah's portfolio has a delta of 0 (delta-neutral) and a gamma of +1,500.
2. Market Move: After the earnings announcement, INN's stock price jumps to $105.
3. Delta Adjustment: Due to the positive gamma, her portfolio's delta quickly becomes positive. The new delta would be approximately (0 + (1,500 \times \text{change in price})), which is (0 + (1,500 \times 5) = 7,500). This means her portfolio now effectively behaves like holding 7,500 shares of INN.
4. Re-hedging: To maintain delta-neutrality and "adjust her average gamma" for future movements, Sarah needs to sell approximately 7,500 shares of INN stock or an equivalent amount of options with negative delta. This action rebalances her delta back to zero, but she retains the positive gamma exposure, allowing her to benefit from subsequent large moves in either direction. This continuous re-hedging process, driven by changes in gamma, is what constitutes the practical application of managing "adjusted average gamma."

Practical Applications

Adjusted average gamma is a vital consideration in various aspects of financial markets, particularly within options trading and sophisticated risk management strategies.

• Market Making and Dealer Hedging: Market makers, who provide liquidity by quoting both bid and ask prices for options, often hold large, dynamic portfolios. They aim to maintain delta-neutral positions to avoid directional risk, but their gamma exposure dictates how frequently they must adjust their hedges. A high positive gamma requires frequent re-hedging (buying stock when the price rises, selling when it falls) to stay delta-neutral, a process known as gamma hedging¹⁴. Managing their "adjusted average gamma" helps them optimize their re-hedging costs and manage their overall exposure to price fluctuations.
• Volatility Trading: Traders engaged in volatility trading often seek to profit from changes in implied volatility rather than directional price movements. Strategies such as long straddles or strangles aim for positive gamma, benefiting from large price swings regardless of direction. Their "adjusted average gamma" helps them gauge their sensitivity to such movements and decide when to adjust their positions.
• Systemic Risk Assessment: The aggregate gamma positioning of large institutional investors, such as hedge funds, can have broader market implications. Periods of concentrated negative gamma among major players can amplify market movements, as these entities are forced to buy into rising markets or sell into falling ones to maintain their delta hedges. This was observed during the GameStop phenomenon in 2021, where a "gamma squeeze" occurred, driven by extensive call option buying, forcing market makers to buy the underlying stock, further driving up its price¹³. Concerns about concentrated vulnerabilities in derivatives markets, including those stemming from gamma exposure, are sometimes highlighted by institutions like the International Monetary Fund (IMF) in their financial stability reports¹².
• Portfolio Management: For institutional investors and sophisticated individual traders, understanding the "adjusted average gamma" of their entire derivatives book is crucial for strategic portfolio construction. It informs decisions about balancing risk and reward, especially when anticipating market phases of high or low volatility.

Limitations and Criticisms

While gamma, and by extension, the concept of adjusted average gamma, is a powerful tool in options trading and risk management, it comes with several limitations and criticisms:

• Theoretical Nature: Options Greeks, including gamma, are theoretical measures derived from pricing models like the Black-Scholes model. Their accuracy depends on the assumptions of these models, which may not always hold true in real-world markets. Factors like sudden market shocks, liquidity crises, or extreme market behavior can cause actual price movements to deviate significantly from model predictions.
• Dynamic and Non-Linear: Gamma itself is not static; it changes as the underlying asset's price moves, particularly accelerating as options move closer to being "at-the-money" and as time decay approaches expiration¹⁰, ¹¹. This dynamic nature means that an "adjusted average gamma" is constantly evolving, requiring continuous monitoring and potentially frequent re-hedging, which can lead to increased transaction costs and operational complexity⁹.
• Gamma Slippage: Constant re-hedging to manage gamma exposure can lead to "gamma slippage." This occurs when the high-frequency buying and selling required to maintain a delta-neutral position results in losses due to bid-ask spreads and market impact, especially in illiquid or volatile markets⁸.
• Focus on Price Sensitivity Only: Gamma focuses solely on the rate of change of delta with respect to the underlying price. It does not account for other critical factors that impact option prices, such as changes in implied volatility (measured by Vega) or the passage of time (measured by Theta). A holistic derivatives strategy must consider all Greeks.
• Misinterpretation and Over-Reliance: For less experienced traders, an over-reliance on gamma without a full understanding of its interplay with other market factors and the overall portfolio can lead to unexpected losses. The GameStop "gamma squeeze" demonstrated how concentrated retail buying of call options could trigger rapid price increases as market makers delta-hedged, creating a feedback loop⁷. While potentially profitable for some, such events highlight the volatile and unpredictable nature of gamma-driven movements and the risks to those caught on the wrong side.

Adjusted Average Gamma vs. Gamma

The distinction between "Adjusted Average Gamma" and "Gamma" lies primarily in scope and application.

In essence, gamma is a specific metric for a single option, while "adjusted average gamma" refers to the broader, ongoing process of assessing and strategically altering one's aggregate gamma exposure across multiple positions to achieve a desired portfolio risk profile.

FAQs

Q: Why is gamma called a "second-order" Greek?

A: Gamma is considered a "second-order" Greek because it measures the rate of change of delta, which is itself a "first-order" Greek measuring the rate of change of the option's price with respect to the underlying asset's price. Gamma essentially tells you how quickly the option's directional sensitivity (delta) will change.³, ⁴

Q: Does "Adjusted Average Gamma" have a specific formula like Delta or Gamma?

A: No, "Adjusted Average Gamma" is not a standardized term with a singular, universally accepted formula like the other Options Greeks. It's more of a conceptual approach to dynamically manage and consider the aggregate gamma exposure across a portfolio of options positions, involving ongoing adjustments and strategic rebalancing.

Q: Why would a trader want to "adjust" their average gamma?

A: Traders would want to adjust their average gamma to align their portfolio's sensitivity with their market expectations and risk management objectives. For example, if a trader anticipates high volatility, they might adjust their positions to increase their positive average gamma, benefiting from larger price swings. Conversely, if they expect a stable market, they might reduce their gamma to minimize re-hedging costs. This is often part of a broader hedging strategy.

Q: Is positive or negative gamma better?

A: Neither positive nor negative gamma is inherently "better"; their desirability depends on a trader's strategy and market outlook. Positive gamma is generally preferred by those looking to profit from large price movements, as it means their delta will increase as the market moves in their favor, accelerating profits. However, it also means more frequent hedging is required. Negative gamma can be riskier in volatile markets as it amplifies losses, but might be part of strategies that benefit from stable or declining volatility. Long options (both call options and put options) always have positive gamma, while short options have negative gamma.¹, ²