Gamma hedging

What Is Gamma Hedging?

Gamma hedging is an options trading & risk management strategy employed by market participants, primarily market makers, to maintain a neutral delta in their portfolio of options and underlying assets. While delta measures the sensitivity of an option's price to changes in the underlying asset's price, gamma measures the rate at which delta itself changes. Therefore, gamma hedging seeks to mitigate the risk that rapid price movements in the underlying asset will cause a portfolio's delta to shift significantly, requiring frequent and potentially costly adjustments. By neutralizing gamma, a trader aims to create a more stable delta, making the overall hedge more robust and reducing the need for continuous rebalancing.

History and Origin

The concept of hedging financial positions has existed for centuries, but the systematic approach to hedging derivatives like options gained significant traction with the formalization of options markets and the development of pricing models. Modern exchange-traded options became widely accessible with the opening of the Cboe Global Markets in Chicago in 1973. This marked a pivotal moment, providing standardized contracts and a central clearinghouse. The subsequent development and widespread adoption of the Black-Scholes model in the 1970s provided a theoretical framework for pricing options and, crucially, for calculating their associated Option Greeks, including delta and gamma. With these quantitative tools, sophisticated hedging strategies, such as gamma hedging, became practical for market participants seeking to manage their exposures.

Key Takeaways

• Gamma hedging is a strategy to stabilize a portfolio's delta exposure to the underlying asset.
• It is crucial for market makers and large options traders to manage risk from price fluctuations.
• A portfolio with neutral gamma means its delta is less sensitive to movements in the underlying asset's price.
• The strategy typically involves buying or selling options to offset the portfolio's overall gamma.
• It is a dynamic process, requiring continuous monitoring and adjustment as market conditions change.

Formula and Calculation

Gamma ($\Gamma$) is the second derivative of an option's price with respect to the underlying asset's price. It quantifies the rate of change of an option's delta for a one-point change in the underlying asset's price.

For a portfolio of options, the total portfolio gamma is the sum of the gammas of individual options, weighted by the number of contracts held:

$\Gamma_{Portfolio} = \sum_{i=1}^{n} N_i \times \Gamma_i$

Where:

• (\Gamma_{Portfolio}) = Total gamma of the portfolio
• (N_i) = Number of contracts for option (i)
• (\Gamma_i) = Gamma of a single contract for option (i)

To achieve a gamma-neutral position, a trader will add or remove options (or the underlying asset itself, though options are preferred for gamma adjustment) to bring the portfolio's total gamma close to zero. This involves calculating the gamma of existing positions and then determining the quantity of a new option position needed to offset it.

For example, if a portfolio has positive gamma, a trader would typically sell options with positive gamma (often options they are short, or other suitable options) or buy options with negative gamma to reduce the net positive gamma. Conversely, if the portfolio has negative gamma, the trader would buy options with positive gamma or sell options with negative gamma.

Interpreting Gamma Hedging

Gamma hedging is interpreted as a measure of how stable a portfolio's delta position will be as the price of the underlying asset moves. A portfolio with high positive gamma indicates that its delta will increase when the underlying asset's price rises and decrease when the price falls. This means that a long gamma position generally profits from large moves in the underlying asset, regardless of direction. Conversely, a portfolio with high negative gamma means its delta will move in the opposite direction of the underlying price change, making it susceptible to losses from large movements.

In the context of risk management, interpreting gamma primarily revolves around minimizing the need for frequent rebalancing and controlling exposure to large price swings. A gamma-neutral portfolio, or one with a small net gamma, suggests that the delta changes less dramatically, requiring fewer adjustments. This is particularly important for market makers who aim to profit from bid-ask spreads rather than directional price movements.

Hypothetical Example

Consider a market maker who has sold a significant number of call option contracts on XYZ stock.

• Initial Position: Sold 100 call options on XYZ stock.
• Current Delta of each option: +0.60
• Current Gamma of each option: +0.05
• Portfolio Delta: (100 \times 0.60 = 60)
• Portfolio Gamma: (100 \times 0.05 = 5)

The market maker is "short gamma" on this position because they sold the options, meaning their portfolio's gamma is effectively (-5) (as they sold options with positive gamma). This negative gamma makes their delta change unfavorably with price moves. To mitigate this, they need to acquire positive gamma.

Suppose the market maker wants to make their portfolio gamma-neutral. They identify another XYZ option (e.g., a longer-dated or different strike price option) with a gamma of +0.02.

To achieve gamma neutrality, they need to buy enough of these new options to offset their existing negative gamma:

• Desired net gamma: 0
• Current portfolio gamma: -5
• Gamma per new option: +0.02

Number of new options to buy = (\frac{5}{0.02} = 250) options

By buying 250 of these new options, the market maker adds (250 \times 0.02 = 5) to their portfolio gamma, bringing the total portfolio gamma to approximately (0). While achieving gamma neutrality, this action will also alter the portfolio's delta, requiring a subsequent delta adjustment (e.g., by buying or selling shares of XYZ stock) to maintain overall delta neutrality.

Practical Applications

Gamma hedging is a critical component of risk management for entities heavily involved in options trading. Its practical applications include:

• Market Making: Market makers utilize gamma hedging to maintain neutral or near-neutral positions. This allows them to profit from the bid-ask spread without taking significant directional bets on the underlying asset. Rules from regulators like the U.S. Securities and Exchange Commission (SEC) often provide special considerations for market makers engaged in bona fide hedging activities to support market liquidity, although specific exceptions, such as the Options Market Maker Exception to SEC Regulation SHO, have been subject to changes and scrutiny over time.
• Proprietary Trading Firms: These firms often employ complex options strategies and rely on gamma hedging to manage their overall portfolio risk and ensure their exposures remain within defined limits.
• Hedge Funds: Funds that engage in volatility arbitrage or other options-centric strategies use gamma hedging to isolate specific risk factors and control their overall exposure to changes in implied volatility.
• Institutional Hedging: Large institutions with significant options positions, whether from structured products or long-term hedging of equity portfolios, incorporate gamma hedging into their broader risk frameworks. The cost of such hedging can be a significant factor, as highlighted in reports discussing corporate efforts to manage currency volatility using options, as noted by Reuters.

Limitations and Criticisms

Despite its importance in risk management, gamma hedging has several limitations and criticisms:

• Discrete Hedging: In practice, gamma hedging cannot be performed continuously. Rebalancing a portfolio involves transaction costs and is done at discrete intervals. This means that between rebalancing points, the portfolio will still have some gamma exposure, which can lead to hedging errors, especially during periods of high volatility.
• Model Dependence: Gamma calculations rely on options pricing models, such as the Black-Scholes model. If the assumptions of these models are violated (e.g., constant volatility, no jump risk), the calculated gamma may not accurately reflect the true sensitivity. Academic research, such as studies applying rough-path theory to gamma hedging, explores the robustness of these strategies under various market conditions, noting that even with model misspecification, the gamma-hedging strategy can remain effective¹.
• Slippage and Liquidity: Executing trades to adjust gamma can be subject to slippage, particularly for large orders or in illiquid markets. This can increase transaction costs and erode the effectiveness of the hedge.
• Other Greeks: While gamma hedging addresses changes in delta, a portfolio is also exposed to other Option Greeks like Vega (sensitivity to volatility) and Theta (time decay). A pure gamma hedge does not neutralize these other risks, potentially leading to residual exposures.

Gamma Hedging vs. Delta Hedging

The primary distinction between gamma hedging and delta hedging lies in the type of risk each strategy aims to neutralize within an options portfolio.

While delta hedging ensures that a portfolio is directionally neutral at a given moment, it does not account for how that directional exposure will change as the underlying asset's price moves. This is where gamma hedging becomes essential. By neutralizing gamma, a trader ensures that the delta remains relatively stable, reducing the need for constant re-hedging of the delta position. In essence, gamma hedging supports and enhances the effectiveness of delta hedging.

FAQs

Why is gamma important for options traders?

Gamma is important because it tells traders how much their delta will change as the underlying asset's price moves. A high gamma means delta changes rapidly, leading to significant directional exposure if not managed. For market makers, controlling gamma is crucial to maintaining a market-neutral position and avoiding large losses from unexpected price swings.

How does gamma hedging differ from other options Greeks hedging?

Gamma hedging specifically addresses the risk associated with changes in an option's delta. Other Option Greeks hedging strategies focus on different risks: for example, delta hedging neutralizes directional risk, Vega hedging manages volatility risk, and Theta hedging accounts for time decay. Gamma hedging is often used in conjunction with delta hedging to provide a more robust and stable hedge.

Is it possible to be perfectly gamma-neutral?

Achieving perfect gamma neutrality is challenging in real-world trading. Since rebalancing occurs at discrete intervals and involves transaction costs, there will always be some residual gamma exposure between adjustments. Additionally, the calculations depend on theoretical models, and real market behavior can deviate. The goal of gamma hedging is typically to minimize gamma exposure to an acceptable level rather than achieving perfect neutrality.

Who typically uses gamma hedging?

Gamma hedging is primarily used by professional traders, such as market makers and quantitative hedge funds, who manage large and complex options portfolios. Their business model often involves maintaining neutral positions to profit from spreads or small pricing inefficiencies, making precise risk management essential.