What is Gamma? Modeling the Acceleration of Delta

Gamma represents the convexity of an option. Learn why high Gamma near expiration leads to 'pinning' and significant hedging risks for market makers.

I analyzed the 'Gamma Squeeze' of 2021 for a week to understand how massive retail call buying forces dealers to buy the underlying at an accelerating rate. What is Gamma ($\Gamma$)? It is the second derivative of the option price with respect to the underlying price, or the first derivative of Delta.

The formula is: $\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}$. Gamma measures the 'convexity' of a position. A high Gamma means that Delta is highly sensitive to price changes, making a Delta-neutral hedge unstable.

Market makers who are short Gamma must 'hedge into the move', buying as the price rises and selling as it falls, which accelerates market volatility. Ignoring Gamma risk in a high-volatility environment is the fastest way to suffer 'slippage' that exceeds your entire premium income.

Calculate the Gamma for an at-the-money option with 1 day to expiry and compare it to 30 days; the difference explains why '0DTE' options are so volatile.