Unit 7 of 15 · Intermediate

Gamma: delta's rate of change

In one read

Gamma is the rate at which delta itself changes as the underlying moves. If delta is speed, gamma is acceleration — it tells you how fast your directional exposure grows or shrinks with each $1 move. Gamma is largest for at-the-money options and largest close to expiry, and those two facts combine into the behaviour every options trader learns to respect: a near-expiry at-the-money option whose delta can swing from near 0 to near 1 over a small move in the underlying, so the position's direction changes under your feet. That convexity is why long options can feel calm and then violent, and why the last day before expiry is the twitchiest. This unit reads gamma on the same generic at-the-money call from the delta unit, so acceleration becomes something you watch happen rather than a definition you memorise.

Gamma is the acceleration behind delta

Delta is not constant — it moves as the underlying moves, and gamma is the rate of that movement. If delta is speed, gamma is acceleration: it tells you how much your delta changes for each $1 move in the stock.

Two facts set where gamma lives:

  • Gamma is largest at-the-money — right where delta is most sensitive to price.
  • Gamma grows as expiry approaches — a contract with days left is far twitchier than the same strike months out.

Put those together and you get the effect professionals watch for: a near-expiry ATM option whose delta can swing from near 0 to near 1 across a small move. The direction of the position changes under your feet. That curvature — delta bending as price moves — is convexity, and it is why a long option can sit quiet and then move violently.

A worked example

Carry the same GENCO $100 call from the delta unit — ATM, delta ~0.50. Say its gamma is 0.08 per $1.

  • GENCO rises $1 to $101: delta climbs from 0.50 to about 0.58. The position now moves like 58 shares, not 50 — it got more directional as it went your way.
  • GENCO falls $1 to $99: delta drops to about 0.42 — less directional as it moves against you.

Now shrink the clock. With a month left that shift is gentle; on the last day before expiry, the same ATM call's gamma is far higher, and a $1 move can whip delta most of the way from 0 to 1. Same strike, same stock — the clock turned a calm position into a jumpy one. That is gamma.

See it in kestrel

Gamma shows itself in fast, convex moves, so it is clearest against a violent session. Run one and watch how sharply the underlying travels:

npx kestrel.markets sim s-p-500-etf-pandemic-volatility-crash

That runs a deterministic simulation over a generic broad-ETF session with sharp expansion — managed licensed data, no wall time, no signup, no card — and prints a certified proof URL. Because the tape is recorded tick by tick, the way an option's delta would accelerate across it is something you can recompute, not just assert. Run it again on your own machine, byte for byte:

npx kestrel.markets certify https://kestrel.markets/proof/art_66d7dda7f0466f69c123463c

Keep the curvature one command away: drop the kestrel.markets MCP server into your client and the next session opens where this one left off — no account in between.

Recompute it

Every claim in this unit recomputes from a certified proof — no account, no card.

/proof/art_66d7dda7f0466f69c123463c
Keep readingOne position, all four greeks

Agent twin: /education/options/opt-gamma/content.md