What Is Gamma Options?
Gamma is one of the "Greeks," a set of measures in derivatives that quantify the sensitivity of an options contracts price to various factors. Specifically, gamma represents the rate of change of an option's delta with respect to a change in the price of the underlying asset. It is a crucial measure in risk management within the broader field of derivatives and portfolio theory. While delta measures the immediate sensitivity of an option's price to the underlying, gamma captures how that sensitivity (delta) itself changes. A high gamma indicates that an option's delta will react sharply to price movements in the underlying asset, making it a key component for traders employing dynamic delta hedging strategies.
History and Origin
The concept of option Greeks, including gamma, arose with the development of sophisticated option pricing models. Before the advent of standardized options exchanges, options trading was largely an over-the-counter market with non-uniform contracts. A significant turning point came in 1973 with the founding of the Chicago Board Options Exchange (CBOE), which standardized options contracts, making them more accessible and facilitating the development of robust pricing theories.3
In the same year, financial economists Fischer Black and Myron Scholes published their seminal work on options pricing, known as the Black-Scholes-Merton model, with contributions from Robert C. Merton. This model provided a mathematical framework for valuing European-style call options and put options, based on factors like the underlying asset's price, strike price, time to expiration, volatility, and the risk-free interest rate. The Black-Scholes model inherently allowed for the calculation of the Greeks, including gamma, which are derived as partial derivatives of the option price with respect to these input variables. The model's insights revolutionized the understanding and application of options in financial markets, making measures like gamma indispensable for traders and risk managers.
Key Takeaways
- Gamma measures the rate of change of an option's delta relative to the price of its underlying asset.
- It is highest for at-the-money options and decreases as options move further in-the-money or out-of-the-money.
- Gamma exposure increases as an option approaches its expiration date, especially for at-the-money options.
- Traders use gamma to assess the stability of their delta hedge and to predict how much their delta will change with price movements.
- Positive gamma generally benefits option holders, while negative gamma is typically associated with option sellers.
Formula and Calculation
Gamma is the second derivative of an option's price with respect to the underlying asset's price. For a call option within the framework of the Black-Scholes model, the formula for gamma (\Gamma) is given by:
Where:
- (N'(d_1)) = The probability density function of the standard normal distribution evaluated at (d_1). This is the derivative of the cumulative standard normal distribution function (N(d_1)) used in the Black-Scholes formula.
- (S) = Current price of the underlying asset
- (\sigma) = Volatility of the underlying asset
- (T-t) = Time remaining until the option's expiration (in years)
The term (d_1) is part of the Black-Scholes formula and is calculated as:
Where (K) is the strike price and (r) is the risk-free interest rate.
Interpreting the Gamma
Gamma provides crucial insight into the responsiveness of an option's delta. A high gamma indicates that delta will change rapidly for small movements in the underlying asset price. This means that positions with high gamma will require more frequent rebalancing of their delta hedging to maintain a neutral position. For instance, an option with a delta of 0.50 and a gamma of 0.10 means that if the underlying asset's price increases by $1, the option's delta would increase to approximately 0.60 (0.50 + 0.10).
Options that are at-the-money (where the strike price is very close to the current underlying price) tend to have the highest gamma. This is because their delta is most sensitive to price changes around this point. As an option moves further in-the-money or out-of-the-money, its gamma typically decreases, approaching zero. This phenomenon is also related to time decay, as options closer to expiration, particularly at-the-money ones, exhibit significantly higher gamma due to their increased sensitivity to price movements in the final hours of trading.
Hypothetical Example
Consider an investor who holds a long call option on ABC stock with a strike price of $100, currently trading at $100. Let's assume the option currently has a delta of 0.50 and a gamma of 0.15.
- Initial State: ABC stock is at $100, option delta is 0.50.
- Stock Price Increase: If ABC stock rises to $101, the option's new delta would be approximately 0.50 + 0.15 = 0.65.
- Stock Price Decrease: If ABC stock falls to $99, the option's new delta would be approximately 0.50 - 0.15 = 0.35.
This example illustrates how gamma causes the delta to accelerate or decelerate based on the direction of the underlying price movement. A positive gamma position (like a long option) means that as the underlying price moves in a favorable direction, the delta increases, further benefiting the position. Conversely, an unfavorable movement leads to a decrease in delta, somewhat mitigating losses compared to a static delta. This dynamic adjustment is key for understanding the true sensitivity of an options contracts.
Practical Applications
Gamma is a critical component for professional options traders and market makers who engage in hedging strategies. Its primary application is in dynamic delta hedging. A portfolio that is "delta-neutral" has a delta of zero, meaning it is theoretically insulated from small price movements in the underlying asset. However, without considering gamma, this delta neutrality is fragile.
Market makers, who often sell options and thus have negative gamma, must constantly adjust their hedges as the underlying price moves to maintain delta neutrality. This rebalancing act involves buying or selling the underlying asset. For example, if a market maker has negative gamma, and the underlying stock price increases, their negative delta position would become even more negative, requiring them to sell more of the underlying asset to re-establish delta neutrality. This constant adjustment, driven by gamma, can contribute to market liquidity and price discovery. Regulations from bodies like the U.S. Securities and Exchange Commission (SEC) often touch upon the proper management and disclosure of complex derivatives positions, implicitly covering the risks that gamma helps quantify.2
Limitations and Criticisms
While gamma is an essential measure for understanding and managing options risk, it operates within the assumptions of option pricing models, such as the Black-Scholes model. These models simplify market realities, assuming constant volatility, continuous trading, and no transaction costs. In reality, volatility is not constant, and large price movements or sudden shifts in market sentiment can render static gamma calculations less accurate.
Furthermore, hedging solely based on delta and gamma requires continuous rebalancing, which incurs transaction costs. For very short-dated, at-the-money options, gamma can be extremely high, leading to significant changes in delta with even minuscule price movements. This "gamma risk" means that maintaining a perfect delta-neutral hedge can be practically challenging and costly, especially in volatile markets. Academic research has explored the challenges of insuring against or managing this specific gamma risk, highlighting its potential for heavy-tailed loss distributions for hedged portfolios.1 This inherent sensitivity can lead to unexpected hedge slippage if not managed rigorously.
Gamma Options vs. Delta Options
Gamma and delta are both crucial option Greeks, but they describe different aspects of an option's sensitivity.
Feature | Gamma Options | Delta Options |
---|---|---|
Definition | Measures the rate of change of an option's delta with respect to a change in the underlying asset price. | Measures the sensitivity of an option's price to a $1 change in the underlying asset price. |
Analogy | Analogous to acceleration (how fast the speed changes). | Analogous to speed (how fast the position changes). |
Role in Hedging | Indicates how often a delta-hedged portfolio needs to be rebalanced; higher gamma means more frequent adjustments. | Determines the initial hedge ratio for a portfolio to be delta-neutral. |
Maximum Value | Highest for at-the-money options, especially near expiration. | Approaches 1.00 for deep in-the-money call options and -1.00 for deep in-the-money put options. |
Perspective | Represents the second-order sensitivity. | Represents the first-order sensitivity. |
While delta options provides a static measure of sensitivity at a given moment, gamma provides insight into how that sensitivity will evolve as the underlying price moves. Investors primarily concerned with static exposure might focus on delta, whereas active traders and market makers engaging in dynamic hedging strategies will closely monitor gamma to anticipate and manage their rebalancing needs. Confusion often arises because both terms relate to the underlying asset's price movement, but gamma specifically addresses the change in delta, making it a measure of convexity for the option's price curve.
FAQs
What does positive gamma mean?
Positive gamma means that as the underlying asset price increases, the option's delta will increase, and as the underlying price decreases, the option's delta will decrease. Holding long options (whether call options or put options) provides positive gamma, which is generally beneficial to the option holder, as it causes their position to become more responsive in a favorable direction.
How does time affect gamma?
As an option approaches its expiration date, its gamma typically increases, especially if the option is at-the-money. This is because the option's sensitivity to price movements becomes extremely pronounced in the final days or hours before expiration, leading to rapid changes in its delta for even small shifts in the underlying asset's price. This heightened sensitivity is often referred to as "gamma risk" or "gamma blow-up."
Is high gamma good or bad?
Whether high gamma is "good" or "bad" depends on the investor's position. For option holders (long options positions), positive, high gamma is generally desirable because it means their delta will move in their favor if the underlying moves significantly. For option sellers (short options positions), negative, high gamma can be risky. It means their delta-hedge needs frequent and potentially costly adjustments as the underlying asset price moves, making them vulnerable to large losses if the market moves against their position quickly.