What Is Amortized Option Gamma?
Amortized option gamma refers to the dynamic nature and eventual decay or adjustment of an option's gamma over its life, particularly as the option approaches its expiration date or as its sensitivity is managed through hedging. In the realm of financial derivatives, gamma is one of the Options Greeks, a set of measures that quantify an option's sensitivity to various factors influencing its price. Specifically, gamma (Γ) represents the rate at which an option's delta changes in response to a one-point movement in the underlying asset's price.48 It is a crucial second-order derivative that provides insight into the convexity of an option's price.
While "amortized option gamma" is not a formally defined term in the same way as delta or theta, it conceptually captures the idea that gamma's influence is not static. Gamma is highest for at-the-money options and tends to increase significantly as an option nears its expiration date.47 Conversely, its impact "amortizes" or diminishes as options move further in-the-money or out-of-the-money, or as its effects are offset through active risk management strategies.
History and Origin
The concept of gamma, as a measure of the sensitivity of an option's delta, emerged as part of the broader development of options pricing models, most notably the Black-Scholes model in 1973. This model provided a mathematical framework for valuing options and, by extension, for calculating their "Greeks" – the partial derivatives that explain how an option's price changes with respect to various inputs. Gamma specifically addresses the non-linear relationship between an option's price and the underlying asset's price, recognizing that delta itself is not constant.
46Over time, as options markets became more sophisticated and traders engaged in complex hedging strategies, the importance of understanding gamma's dynamic behavior became paramount. The need for constant rebalancing in delta-hedged portfolios, driven by changes in gamma, led to the practical application of managing gamma risk. Academic research has further explored the implications of gamma on market dynamics and hedging effectiveness, with studies, such as "Is the gamma risk of options insurable?" published on ResearchGate, delving into the specific risks associated with gamma exposure.
45## Key Takeaways
- Amortized option gamma refers to the changing nature of an option's gamma over its lifetime, particularly its decay or management.
- Gamma measures the rate of change of an option's delta in response to underlying asset price movements.
- Gamma is highest for at-the-money options and increases as an option approaches expiration.
- Effective risk management often involves strategies like dynamic hedging to manage gamma exposure.
- Understanding amortized option gamma is crucial for traders to anticipate how their portfolio's sensitivity to price changes will evolve.
Formula and Calculation
Gamma is formally defined as the second partial derivative of the option's price with respect to the underlying asset's price. While the precise calculation involves complex partial derivatives from option pricing models (like the Black-Scholes model), a simpler way to conceptualize it is as the change in delta divided by the change in the underlying price:
44$$
\text{Gamma} = \frac{\Delta \text{Option Delta}}{\Delta \text{Underlying Price}}