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Market gamma

What Is Market Gamma?

Market gamma is a crucial concept in options pricing and risk management, quantifying the rate at which an option's delta changes in response to movements in the underlying asset's price. As a second-order derivative, market gamma provides insights into the sensitivity of an option's price acceleration. It is a key metric for traders and market makers, helping them understand how quickly their delta exposure will shift as the price of the underlying stock, exchange-traded fund (ETF), or index fluctuates. Understanding market gamma is essential for managing risk in portfolios containing derivatives, particularly for those involved in hedging strategies.

History and Origin

The framework for understanding how options prices behave, including concepts like market gamma, largely emerged from the development of sophisticated options pricing models in the 1970s. Prior to this period, options trading was primarily conducted in over-the-counter (OTC) markets with less standardization. The launch of the Chicago Board Options Exchange (CBOE) in April 1973 marked a significant turning point, introducing standardized options contracts and fostering a more transparent and liquid market.6, 7 This standardization, alongside academic breakthroughs like the Black-Scholes model, provided the theoretical foundation for quantifying various sensitivities, known as "Greeks," which include delta, theta, vega, rho, and market gamma. The ability to measure these sensitivities precisely enabled more advanced risk management and arbitrage strategies for financial instruments like options contracts.

Key Takeaways

  • Market gamma measures how an option's delta changes for a given movement in the underlying asset's price.
  • It is a critical component for market makers to manage their hedging strategies, particularly when aiming for a delta-neutral position.
  • Positive market gamma benefits option holders, as their delta increases when the option moves in-the-money and decreases when it moves out-of-the-money, amplifying gains and mitigating losses.
  • Negative market gamma, typically held by option writers, increases risk by forcing greater hedging activity as the underlying asset moves.
  • High market gamma can contribute to increased volatility in the underlying asset, especially during periods of significant price movements.

Formula and Calculation

Market gamma is mathematically defined as the second derivative of an option's price with respect to the underlying asset's price, or equivalently, the first derivative of delta with respect to the underlying asset's price. For a European call option, the formula for market gamma ((\Gamma)) can be expressed as:

Γ=N(d1)SσTt\Gamma = \frac{N'(d_1)}{S \sigma \sqrt{T-t}}

Where:

  • (N'(d_1)) is the probability density function of (d_1), which is a component of the Black-Scholes formula related to the likelihood of the option expiring in-the-money.
  • (S) is the current price of the underlying asset.
  • (\sigma) is the implied volatility of the underlying asset.
  • (T-t) is the time remaining until the option's expiration date.

This formula shows that market gamma is influenced by factors such as the underlying asset's price, its volatility, and the time left until expiration. It is a measure of the curvature of the option premium with respect to the underlying price.

Interpreting Market Gamma

Interpreting market gamma involves understanding how an option's directional exposure, represented by its delta, will change. For an option with positive market gamma, its delta will increase as the underlying asset's price rises (for a call option) or falls (for a put option). Conversely, its delta will decrease if the underlying asset moves against the option's favor. This provides a favorable characteristic for option buyers: their position gains more sensitivity as it moves into profit and loses sensitivity as it moves into loss.

For example, if a call option has a delta of 0.50 and a market gamma of 0.05, a $1 increase in the underlying stock price would cause the delta to increase to 0.55. This means the option would then behave more like 55 shares of stock for subsequent price movements, rather than 50. Conversely, a $1 decrease in the stock price would reduce the delta to 0.45. This "delta acceleration" or "deceleration" is a key aspect of options trading. Market gamma is highest for at-the-money options and decreases as an option moves further in-the-money or out-of-the-money.

Hypothetical Example

Consider an investor, Sarah, who buys a call option on Stock XYZ with a strike price of $100 and 30 days until expiration.

  • Current Stock XYZ price: $100
  • Option Delta: 0.50
  • Market Gamma: 0.03

If Stock XYZ rises by $1 to $101:
Sarah's option delta would increase from 0.50 to (0.50 + 0.03 = 0.53). This means that for further small movements, the option would behave as if Sarah owned 53 shares of Stock XYZ, providing greater leverage on upward moves.

If Stock XYZ falls by $1 to $99:
Sarah's option delta would decrease from 0.50 to (0.50 - 0.03 = 0.47). Now, the option behaves more like 47 shares, reducing her exposure to further downward movements.

This hypothetical example illustrates how positive market gamma inherently benefits the option buyer by increasing their delta when the trade is favorable and decreasing it when unfavorable, amplifying potential gains and mitigating losses on subsequent price changes.

Practical Applications

Market gamma is particularly important for market makers and professional traders who aim to maintain a delta-neutral position. Since delta changes with every movement in the underlying asset, market makers must constantly adjust their hedges by buying or selling the underlying stock to remain delta-neutral. Market gamma indicates how often and by how much these adjustments will need to occur. High market gamma means that delta changes rapidly, requiring more frequent and potentially larger hedging adjustments.

One notable real-world application of market gamma's impact occurred during the GameStop (GME) "gamma squeeze" in early 2021. As retail investors aggressively bought call options, particularly out-of-the-money calls, market makers who sold these options accumulated large negative market gamma positions. To hedge this exposure and remain delta-neutral, these market makers were compelled to buy increasing amounts of GME stock as its price rose. This forced buying created a positive feedback loop, further driving up the stock price and forcing more hedging, contributing to a rapid increase in volatility and price for GME shares.4, 5

Furthermore, regulatory bodies like the Financial Industry Regulatory Authority (FINRA) establish rules, such as FINRA Rule 2360, that cover options trading, including aspects like position limits and reporting requirements to ensure market integrity and investor protection, indirectly acknowledging the systemic impact of options-related sensitivities like market gamma.2, 3

Limitations and Criticisms

While market gamma is a powerful tool for understanding options sensitivity, it has limitations. Like other "Greeks," market gamma is a theoretical measure derived from options pricing models, and its accuracy depends on the validity of the model's assumptions. Real-world market conditions, such as sudden shifts in supply and demand, liquidity constraints, or unexpected news events, can cause options prices and their sensitivities to behave differently than the models predict.

Another criticism is that market gamma only accounts for the rate of change of delta with respect to the underlying price, assuming all other variables remain constant. In reality, implied volatility can change significantly, especially during periods of high market volatility, which can profoundly impact an option's price and its other Greeks. A rapid increase in volatility can counteract or amplify the effects of market gamma. As noted by the Federal Reserve Bank of San Francisco, stock market volatility itself is influenced by numerous factors and can display persistence, highlighting the dynamic environment in which market gamma operates.1 Over-reliance on market gamma without considering other risk dimensions or real-world market frictions can lead to incomplete risk assessment.

Market Gamma vs. Market Delta

Market gamma and market delta are both fundamental "Greeks" used in options analysis, but they measure different aspects of an option's sensitivity to the underlying asset's price.

Market Delta measures the direct sensitivity of an option's price to a $1 change in the underlying asset's price. For example, if a call option has a delta of 0.60, its price is expected to increase by $0.60 for every $1 increase in the underlying stock price. Delta indicates the directional exposure of an option position and ranges from 0 to 1 for call options and -1 to 0 for put options. It tells traders how much their option position will move with the underlying asset.

Market Gamma, on the other hand, measures the rate of change of market delta. It indicates how much the delta itself will change for every $1 movement in the underlying asset's price. In essence, delta tells you where you are, while market gamma tells you how fast your position's sensitivity is changing, providing a measure of the convexity of the option's price curve. Traders often focus on delta for their immediate directional exposure, but consider market gamma for anticipating how that exposure will evolve with price movements, especially for managing short-term hedges.

FAQs

What does positive market gamma mean?

Positive market gamma means that an option's delta will increase when the underlying asset moves in the option's favor (price rises for a call, falls for a put) and decrease when it moves against the option's favor. This is generally beneficial for option buyers, as it means their profits accelerate, and their losses decelerate as the underlying price changes.

How does market gamma affect market makers?

Market makers typically try to maintain delta-neutral portfolios to avoid taking on directional risk. Market gamma presents a challenge because it causes their delta exposure to change as the underlying asset moves. Market makers holding negative market gamma must frequently buy or sell the underlying asset to re-hedge their positions, which can exacerbate price movements, especially during periods of high options activity.

Is high market gamma good or bad?

Whether high market gamma is "good" or "bad" depends on your position. For an option buyer, high positive market gamma is generally desirable because it amplifies gains and dampens losses by favorably adjusting delta. For an option seller or writer, high negative market gamma implies greater risk, as they will be forced to make larger and more frequent hedging adjustments, potentially at unfavorable prices, which can lead to significant losses, particularly in fast-moving markets.

How is market gamma related to volatility?

Market gamma tends to be highest for options that are at-the-money and decreases as options move further out-of-the-money or in-the-money. This means that options closest to the current underlying price are most sensitive to changes in delta. During periods of high volatility, underlying prices can move significantly, causing options to rapidly change their delta and thus their value. While market gamma is a measure of delta's rate of change, it also indirectly reflects the potential for rapid price shifts due to the interplay between options and the underlying market, as seen during events like a short squeeze.