Gamma Finance Formula - Financial Discussion

Gamma in options trading measures the rate of change of an option’s delta for a one-point move in the underlying asset’s price. It’s a second-order derivative, representing the curvature of the option’s price with respect to the underlying asset’s price. A higher gamma means the option’s delta is more sensitive to changes in the underlying asset’s price. **Formula and Components** While there isn’t a single, easily digestible “Gamma Finance Formula” in the way one might think of the Black-Scholes formula for option pricing, gamma is a calculated value derived from the option pricing models. It typically leverages the standard normal probability density function. In the Black-Scholes model, gamma is calculated as: Gamma = (N'(d1)) / (S * sigma * sqrt(T)) Where: * **N'(d1):** The standard normal probability density function evaluated at d1. This gives the probability density of a normally distributed variable being equal to d1. d1 is a component of the Black-Scholes model and is calculated as: d1 = [ln(S/K) + (r + (sigma^2)/2) * T] / (sigma * sqrt(T)) * **S:** The current price of the underlying asset. * **K:** The strike price of the option. * **r:** The risk-free interest rate. * **sigma:** The volatility of the underlying asset. * **T:** The time to expiration (expressed in years). * **S:** The current price of the underlying asset (repeated for clarity). * **sigma:** The volatility of the underlying asset (repeated for clarity). * **sqrt(T):** The square root of the time to expiration (expressed in years). **Interpretation and Implications** Gamma is always a positive value for both call and put options. This is because both call and put option deltas move in the same direction as the underlying asset. * **High Gamma:** A high gamma indicates that the option’s delta is very sensitive. This means a small change in the underlying asset’s price can lead to a relatively large change in the option’s delta. Options near the money (at-the-money) typically have the highest gamma. As expiration approaches, the gamma of at-the-money options increases significantly. * **Low Gamma:** A low gamma indicates that the option’s delta is less sensitive. Options that are deep in the money or deep out of the money typically have low gamma. **Practical Use and Hedging** Gamma is crucial for delta-hedging strategies. Delta-hedging aims to create a portfolio that is neutral to small changes in the underlying asset’s price. Because delta changes as the underlying asset’s price moves, gamma helps traders understand how frequently they need to rebalance their delta-hedge. A higher gamma requires more frequent rebalancing to maintain a delta-neutral position. **Gamma Risk:** While gamma is a useful tool, it also represents a risk. This risk, known as “gamma risk,” arises from the uncertainty of how the underlying asset’s price will move. If the underlying asset’s price moves significantly and quickly, a trader’s delta hedge can become ineffective, leading to potential losses. This is often seen during periods of high volatility. **Summary** Gamma is a vital concept for options traders, especially those employing delta-hedging strategies. Understanding gamma helps traders manage their risk and adjust their positions effectively as the underlying asset’s price fluctuates. While the precise calculation involves the standard normal density function, the core idea is understanding the sensitivity of an option’s delta to changes in the underlying asset’s price.