Gamma (Γ): Measures the Rate of Change of Delta

Gamma (Γ) is a financial metric that measures the rate of change of delta with respect to the underlying asset's price. It is particularly significant in options trading to evaluate the sensitivity of delta.

Gamma (Γ) is a key risk metric used in the financial derivatives market, specifically within the realm of options trading. It represents the rate of change of an option’s delta in response to changes in the price of the underlying asset. Delta ($\Delta$) itself is a measure of the sensitivity of an option’s price to movements in the underlying asset’s price. Therefore, Gamma provides insights into the convexity of an option’s value.

Formal Definition§

Gamma ($Γ$) is mathematically defined as the second derivative of the option’s price with respect to the underlying asset’s price, or the first derivative of delta with respect to the underlying asset’s price. In notation, it is expressed as:

$$\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2}$$

where:

• $\Delta$ is the delta of the option.
• $S$ is the price of the underlying asset.
• $V$ is the value of the option.

Importance of Gamma in Options Trading§

Gamma is significant for several reasons:

• Risk Management: Higher Gamma indicates that delta is more sensitive to changes in the underlying asset’s price, meaning the position needs to be adjusted more frequently.
• Portfolio Hedging: Traders can use Gamma to estimate how much their delta hedge will need to change as the underlying price moves.
• Predictive Insight: Gamma gives insight into the stability of delta over time, which is crucial for maintaining effective hedging strategies.

Types of Gamma§

Positive vs. Negative Gamma§

• Positive Gamma: This is common in long options positions, where the delta increases as the underlying asset price rises and decreases as the price falls, providing a favorable convexity.
• Negative Gamma: Typically seen in short options positions, resulting in delta decreasing as the asset price rises and increasing as the price falls, which can make the position riskier.

Running Gamma§

Running Gamma is a concept used to describe the dynamic adjustment of Gamma over time and different price levels, providing a more nuanced view in complex trading strategies.

Special Considerations§

At-the-Money vs. In-the-Money Options§

• At-the-money options have the highest Gamma, meaning their delta will change most rapidly with movements in the underlying price.
• In-the-money and out-of-the-money options have lower Gamma.

Time to Expiration§

Gamma tends to increase as the expiration date approaches, particularly if the option is at or near the money.

Examples§

Basic Example§

Consider an at-the-money call option on a stock currently priced at $100 with a delta of 0.5. If the stock price increases to $105 and the delta increases to 0.6, the Gamma (assuming linear change) would be:

$$\Gamma = \frac{\Delta_{\text{new}} - \Delta_{\text{old}}}{S_{\text{new}} - S_{\text{old}}} = \frac{0.6 - 0.5}{105 - 100} = 0.02$$

Practical Example in Trading§

A trader with a long call position on an index might monitor Gamma to anticipate how their delta hedge needs to be adjusted as index levels fluctuate, leveraging Gamma to stay ahead of substantial shifts in market dynamics.

Historical Context§

The concept of Gamma, as part of the “Greeks,” was introduced and evolved through the development of the Black-Scholes model and further refinements in options pricing theories. These developments became integral with the increasing complexity of financial markets.

Applicability in Modern Finance§

Gamma is widely used by:

• Market makers to manage their exposure dynamically.
• Hedge funds for crafting complex trading strategies.
• Retail traders to understand the implications of their options positions better.

Comparison with Related Terms§

• Delta (Δ): Measures the change in option price with respect to changes in the underlying asset’s price.
• Theta (Θ): Measures the sensitivity of the value of the option to the passage of time.
• Vega (ν): Measures sensitivity to volatility of the underlying asset.

FAQs§

Why is Gamma highest for at-the-money options?

At-the-money options are most sensitive to changes in the price of the underlying asset, thus having the highest Gamma.

How does time decay affect Gamma?

As expiration approaches, Gamma tends to increase, particularly for at-the-money options.

How do traders use Gamma?

Traders use Gamma to predict the changes in delta and adjust their hedging strategies accordingly.

References§

• Hull, J. C. (2020). Options, Futures, and Other Derivatives. Pearson.
• Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy.

Summary§

Gamma (Γ) is a crucial metric in the options trading world, providing insights into the sensitivity of an option’s delta to movements in the underlying asset’s price. By understanding Gamma, traders can better manage risks, optimize hedging strategies, and predict market dynamics, ensuring more informed and strategic trading decisions.