Explain gamma. Why is a long option position positive gamma, and what does that mean for a delta-hedged book?

How to answer

Gamma = ∂²V/∂S² = ∂(delta)/∂S, the curvature in S. Long options (calls or puts) have positive gamma: delta rises as the stock rises, falls as it falls. A delta-hedged long-gamma book automatically buys low and sells high when rehedging, profiting from realized movement. The cost is negative theta — you pay time decay for convexity. Gamma is largest for ATM, near-expiry options.

Key idea: Thinking gamma differs in sign for calls vs puts — both positive when long. Long gamma only pays if realized vol exceeds the implied vol you bought.