Gamma risk

What Is Gamma Risk?

Gamma risk refers to the risk associated with changes in an option's delta, which measures an option's sensitivity to the price movements of its underlying asset. It is a key concept within options Greeks, a group of measures used in the field of derivatives risk management to quantify the various factors affecting an option's price. Specifically, gamma quantifies how much an option's delta is expected to change for every one-point move in the underlying asset's price. When an option position has high gamma exposure, its delta can change rapidly, leading to significant and often sudden shifts in the overall risk profile of a portfolio. This dynamic sensitivity is what constitutes gamma risk, making accurate hedging more challenging, especially during periods of high volatility.

History and Origin

The foundational understanding of option pricing, and subsequently the risks associated with them, largely stems from the development of the Black-Scholes model. Introduced by Fischer Black and Myron Scholes in their seminal 1973 paper, "The Pricing of Options and Corporate Liabilities," this model provided a theoretical framework for valuing European-style options⁴. While the paper initially focused on the valuation formula, the development of the model inherently led to the derivation of the "Greeks," including delta, gamma, vega, and theta. These measures allowed market participants to better understand and quantify the various sensitivities of option prices, moving beyond simple price-based analysis to a more nuanced view of derivative exposure. Gamma, as the second derivative of the option price with respect to the underlying asset's price, emerged as a critical measure of an option's convexity and the stability of its delta, thus giving rise to the concept of gamma risk.

Key Takeaways

• Gamma measures the rate of change of an option's delta relative to the underlying asset's price.
• High gamma indicates that an option's delta will change rapidly, leading to increased sensitivity of the option's value to small price movements.
• Positive gamma is beneficial for option buyers, as it means their delta exposure increases when the underlying moves favorably and decreases when it moves unfavorably.
• Negative gamma is a concern for option sellers (writers), as their delta exposure accelerates losses when the underlying moves against their position.
• Gamma risk is highest for at-the-money options and decreases as options move further in-the-money or out-of-the-money, or as the expiration date approaches.

Formula and Calculation

Gamma ((\Gamma)) is the second derivative of the option's price with respect to the underlying asset's price. For a European call option in the Black-Scholes model, the formula for gamma is:

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) = Implied volatility of the underlying asset's returns.
• (T-t) = Time to expiration.

The term (d_1) is part of the Black-Scholes formula and is given by:

Where:

• (K) = Strike price.
• (r) = Risk-free interest rate.

This formula illustrates that gamma is influenced by the underlying asset's price, volatility, and time to expiration.

Interpreting the Gamma Risk

Interpreting gamma risk is crucial for active traders and portfolio management. A high positive gamma indicates that an option's delta will change significantly with small movements in the underlying asset's price. For example, if an option has a delta of 0.50 and a gamma of 0.10, a $1 increase in the underlying asset's price would increase the delta to approximately 0.60. Conversely, a $1 decrease would reduce the delta to 0.40. This means that positions with positive gamma become more sensitive to favorable price movements and less sensitive to unfavorable ones, which is desirable for option buyers.

Conversely, positions with negative gamma (typically held by option sellers) imply that delta will move against the position. If a short option has a delta of -0.50 and a gamma of -0.10, a $1 increase in the underlying would decrease the delta to -0.60 (making the short position more negative delta and thus more exposed to further upside moves). A $1 decrease would increase the delta to -0.40. This means losses can accelerate quickly as the market moves unfavorably, making negative gamma positions particularly susceptible to gamma risk. Gamma tends to be highest for options that are near the strike price and with less time to expiration, as the sensitivity of the option's value to the underlying's movements is maximized at these points.

Hypothetical Example

Consider an investor holding a long call option on XYZ stock, which is currently trading at $100. The call option has a strike price of $100, a delta of 0.50, and a gamma of 0.05.

1. Initial State:

   • XYZ Stock Price: $100
   • Option Delta: 0.50 (meaning the option's price will increase by $0.50 for every $1 increase in XYZ stock, all else being equal).

2. Scenario 1: XYZ stock increases by $1.

   • New XYZ Stock Price: $101
   • Change in Delta due to Gamma: (0.05 \times $1 = 0.05)
   • New Delta: (0.50 + 0.05 = 0.55)
   • This means the option's price will now increase by $0.55 for every subsequent $1 increase in XYZ stock, demonstrating increased sensitivity.

3. Scenario 2: XYZ stock decreases by $1.

   • New XYZ Stock Price: $99
   • Change in Delta due to Gamma: (0.05 \times -$1 = -0.05)
   • New Delta: (0.50 - 0.05 = 0.45)
   • Here, the option's price will now increase by $0.45 for every subsequent $1 increase, or decrease by $0.45 for every subsequent $1 decrease, showing reduced sensitivity to further negative movements.

This example illustrates how positive gamma benefits the option buyer by making their position more sensitive to favorable moves and less sensitive to unfavorable moves. The inverse effect applies to option sellers, for whom gamma risk exacerbates losses.

Practical Applications

Gamma risk is a critical consideration for participants in the derivatives market, particularly market makers and active traders who employ options strategies. These professionals frequently engage in hedging to manage their overall portfolio exposure. When a portfolio has significant gamma exposure, especially negative gamma, it means that its delta can change rapidly, requiring more frequent and often larger adjustments to maintain a delta-neutral position. This process, known as dynamic delta hedging, incurs transaction costs and can be difficult to execute effectively in fast-moving markets.

Regulators emphasize the inherent complexities and risks of options trading. As outlined by the Options Clearing Corporation (OCC), the "Characteristics and Risks of Standardized Options" document informs investors that options, like other securities, carry no guarantees, and investors can lose all of their initial investment or even more³. The U.S. Securities and Exchange Commission (SEC) also provides investor bulletins, highlighting that options writers (sellers) may face unlimited potential losses due to the nature of certain contracts². Gamma risk directly contributes to this potential for rapidly accelerating losses for option sellers, as adverse price movements can quickly increase their delta exposure, making their positions more sensitive to further losses. For this reason, understanding and managing gamma risk is integral to effective risk management in options trading.

Limitations and Criticisms

While gamma is a vital tool for understanding options sensitivity, its practical application has limitations. One significant criticism is that gamma, like other Greeks, is a point-in-time measure. It provides an instantaneous snapshot of how delta is expected to change based on a small movement in the underlying asset. However, markets are dynamic, and large, sudden price swings can render the gamma calculation less accurate for predicting behavior over a wider range of movement. This means that while a position might be delta-neutral and gamma-neutral at one moment, a significant market event could quickly shift these sensitivities, exposing the trader to unforeseen risks.

Furthermore, managing gamma risk through continuous re-hedging can be costly and impractical, particularly for individual investors. Rapidly changing deltas require frequent adjustments to the underlying position, which generates transaction fees and can lead to slippage, eroding potential profits. Jessica Inskip, Director of Education and Product at OptionsPlay, has noted that for those who are short options, "you actually have what's called gamma risk and you have a huge issue of being unprofitable very quickly" when holding options until expiration¹. This highlights how gamma risk can accelerate losses, especially as options approach their expiration date, where gamma tends to peak for at-the-money options. The complexity of dynamic hedging and the costs involved mean that active gamma management is typically the domain of institutional traders and sophisticated market participants rather than retail investors.

Gamma vs. Delta

Gamma and delta are both essential options Greeks that measure an option's price sensitivity, but they capture different aspects of that sensitivity. The primary distinction is that delta measures the initial directional exposure, while gamma measures how that directional exposure changes.

While delta tells a trader how much their option position will move with the underlying, gamma tells them how much their sensitivity will change as the underlying moves. A portfolio that is "delta-neutral" has no immediate directional exposure, but if it has high gamma, it could quickly become delta-positive or delta-negative as the underlying price changes, requiring continuous adjustments.

FAQs

What does positive gamma mean?

Positive gamma indicates that your option position's delta will increase when the underlying asset's price moves in a favorable direction and decrease when it moves unfavorably. This is generally beneficial for option buyers, as it means their profits can accelerate as the underlying moves in their favor, while losses are mitigated as the underlying moves against them.

Why is gamma risk important for option sellers?

For option sellers (or writers), gamma risk is particularly important because they typically hold negative gamma positions. This means that if the underlying asset's price moves against their position, their delta will increase rapidly, accelerating their losses. Managing this rapid change in directional exposure requires constant re-hedging, which can be costly and challenging, especially in volatile markets.

How does time to expiration affect gamma?

Gamma is typically highest for options that are at-the-money and have a short time until their expiration date. As an option approaches expiration, its gamma can increase sharply, especially if it remains near the strike price. This phenomenon, often called "gamma squeeze," can lead to extremely rapid changes in delta and significant price swings for the option.

Can gamma be hedged?

Yes, gamma can be hedged, typically by adjusting the number of options or underlying shares in a portfolio to achieve a "gamma-neutral" position. This involves taking offsetting positions with opposite gamma exposures. For example, a trader with negative gamma from selling options might buy other options to introduce positive gamma into their portfolio. Gamma hedging is a complex aspect of risk management and is primarily undertaken by professional traders and institutions.