Unit 6 of 15 · Intermediate

Delta: direction, hedge, probability

In one read

Delta is the first greek professionals read because it answers three questions at once. As a rate, delta is how much an option's price moves per $1 move in the underlying — a 0.50 delta call gains about $0.50 when the stock rises $1. As a hedge ratio, delta is the number of shares that option behaves like, so a 0.50 delta call moves like 50 shares and tells you how many shares would offset it. As a rough probability, delta approximates the chance the option finishes in-the-money — a 0.30 delta OTM call is loosely a 30% shot. Calls carry positive delta, puts negative. This unit reads one at-the-money call's delta through all three faces on a generic contract, so a single number stops being jargon and becomes the fastest read on a position you have.

Delta is one number with three faces

Delta is the first greek a professional reads, because a single value answers three questions:

  • Direction (a rate). Delta is how much the option's price changes per $1 move in the underlying. A 0.50 delta call gains ~$0.50 when the stock rises $1, and loses ~$0.50 when it falls $1.
  • Hedge (a share-equivalent). A 0.50 delta call behaves like 50 shares of the underlying (delta × 100). That is how many shares would offset the position — hence hedge ratio.
  • Probability (a rough odds read). Delta loosely approximates the chance the option finishes in-the-money. A 0.30 delta OTM call is, very roughly, a 30% shot.

Signs matter: calls have positive delta (0 to +1), puts have negative delta (−1 to 0). Deep-ITM options approach ±1 (they track the stock nearly one-for-one); far-OTM options approach 0.

A worked example

Hold one GENCO $100 call, a month to expiry, with GENCO trading at $100. At-the-money, its delta is about 0.50 — the position we will carry through the next three greeks.

  • As a rate: GENCO ticks to $101, and the call gains roughly $0.50 (×100 shares = ~$50 on the contract).
  • As a hedge: the call moves like 50 shares of GENCO. Short 50 shares against it and, for a small move, the two roughly cancel.
  • As probability: ~0.50 says this ATM call is close to a coin-flip to finish ITM.

One number, read three ways — which is why it is the number you read first.

See it in kestrel

Delta describes response to movement, so it is clearest against a session that moves directionally. Run one and watch the underlying travel:

npx kestrel.markets sim fomc-rate-decision-whipsaw

That runs a deterministic simulation over a generic index session — managed licensed data, no wall time, no signup, no card — and prints a certified proof URL. Kestrel expresses the hedge directly (buy 1 atm C against shares is a delta the tool can hold and grade), so the share-equivalent is something you run, not just picture. Recompute the whole record on your own machine, byte for byte:

npx kestrel.markets certify https://kestrel.markets/proof/art_d29415f0cf502f4a218a9cba

Keep the read 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_d29415f0cf502f4a218a9cba
Keep readingGamma: delta's rate of change

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