Gamma ($\Gamma$) is a crucial concept in options trading and derivatives markets. It represents the rate of change of Delta ($\Delta$) with respect to changes in the underlying asset’s price. Specifically, Gamma is the second derivative of the option price concerning the asset price.
Where:
- \( C \) is the option price.
- \( S \) is the underlying asset price.
Gamma informs traders of how much the Delta of an option is expected to change as the underlying asset’s price changes. For instance, if Delta (which measures the sensitivity of the option’s price to changes in the asset price) is the slope of the option’s price curve, then Gamma is the curvature.
Significance of Gamma
Rate of Change of Delta
Gamma measures how quickly and significantly Delta can change as the price of the underlying asset changes. High Gamma values indicate that Delta can change rapidly in response to changes in the asset price.
Position Management
Gamma is vital for traders managing portfolios of options. Understanding Gamma helps in predicting the behavior of the portfolio in various market conditions, ensuring better risk management.
Stability and Volatility
Options with high Gamma are more sensitive to changes in the price of the underlying asset, indicating higher volatility and risk. Conversely, options with low Gamma signify more stability.
Types of Options and Gamma
Near-the-Money Options
Options that are near-the-money (the current price of the underlying asset is close to the strike price) typically exhibit the highest Gamma. This is because small changes in the asset’s price are more likely to affect whether the option finishes in-the-money or out-of-the-money.
Deep In/Out-of-the-Money Options
Deep in-the-money and deep out-of-the-money options have lower Gamma. In these cases, large price movements are required to significantly impact Delta.
Practical Examples
Example 1: High Gamma
Consider an at-the-money call option on a stock trading at $100. If the option has a Gamma of 0.05, and the stock price increases by $1, the Delta will increase by $0.05.
Example 2: Low Gamma
A deep out-of-the-money put option on the same stock might have a Gamma of 0.01. Therefore, if the stock price decreases by $1, the Delta would increase by just $0.01.
Historical Context
The concept of Gamma, along with other Greek letters, was developed in the context of the Black-Scholes model, formulated by Fischer Black and Myron Scholes in 1973. This model revolutionized options pricing and significantly advanced the field of financial derivatives.
Applicability and Comparisons
Relation to Other Greeks
- Delta ($\Delta$): First derivative of the option price concerning the underlying asset price.
- Theta ($\Theta$): Measures the sensitivity of the option price to time decay.
- Vega ($\nu$): Measures the sensitivity to volatility.
- Rho ($\rho$): Measures sensitivity to the interest rate.
Each Greek represents a different dimension of risk in the pricing and hedging of options.
Comparison with Delta
While Delta provides the rate of change of the option’s price, Gamma provides a higher-order insight by considering how Delta itself changes. This is essential for fine-tuning hedging strategies and understanding the option’s dynamic nature.
FAQs
Why Is Gamma Important in Options Trading?
How Does Gamma Affect Hedging?
Can Gamma Be Negative?
References
- Black, F., & Scholes, M. (1973). “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy.
- Hull, J. C. (2017). “Options, Futures, and Other Derivatives.” Pearson.
- Wilmott, P. (2006). “Paul Wilmott Introduces Quantitative Finance.” Wiley.
Summary
Gamma is a vital measure in the world of financial derivatives, representing the rate of change of Delta relative to changes in the asset price. It serves as an indicator of an option’s sensitivity and stability and plays an essential role in risk management and dynamic hedging strategies. Understanding Gamma alongside other Greeks provides a comprehensive view of the risks and behaviors associated with trading options.
