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Summary

Delta measures an option's price sensitivity per $1 move in the underlying and doubles as share equivalence and a rough ITM probability. Gamma measures how delta itself changes — the curvature that makes long options gain faster than they lose. Together they define the direction-and-acceleration profile of any option position, and their tension with theta is the core trade-off of all volatility trading12.

Delta: Three Framings, One Number

Framing Meaning Example
Hedge rate $ change of option per $1 stock move 0.50-delta call gains $0.50 per $1
Share equivalence delta × contracts × 100 = share-equivalent exposure 5 × 0.43-delta calls ≈ 215 shares
ITM probability (rule of thumb) ~probability of expiring in-the-money 0.75 delta ≈ "75% chance"

Mathematically the probability reading is imprecise, but it is universally used as a first screen for whether a payout is fair1. By put–call parity, |call Δ| + |put Δ| ≈ 1.00 for the same month/strike.

Behavior: deltas pull toward 0.50 with more time or higher IV (ITM shrink, OTM grow); ATM call deltas run slightly above 0.50 from the interest tilt; near ex-dividend, ITM calls approach parity and synthetic deltas can exceed ±1.001.

Gamma: Convexity and Its Behavior

  • Long options have positive gamma (gains accelerate, losses decelerate); short options the mirror. Stock changes only delta — gamma comes entirely from options2.
  • Moneyness/time: gamma peaks at-the-money and explodes for ATM options as expiration nears, while deep ITM/OTM gamma collapses to zero2.
  • Volatility interaction: higher IV raises gamma in well-OTM options (below ~15 delta) — significant for condor/strangle traders, whose positions get more delta sensitivity than the model predicts; ATM gamma falls as IV rises3.
  • Position gamma = Σ(gamma × contracts × 100); long-gamma P&L vs. price is a "smiley," short-gamma a "frown"2.
  • Average delta over a $1 move ≈ Δ ± Γ/2 — the quick way to estimate hedge drift2.

The Gamma–Theta Tension

Gamma and theta travel together: long gamma costs theta daily; short gamma earns theta while adverse moves accelerate. Every volatility trade is a contest between them2. TOMIC's gamma-scalping framework makes the price explicit3:

  • Pay-the-decay method: required daily move = √(2.8 × |Θ| / Γ) (the 7/5 factor accounts for weekend decay; use |Θ|, the positive decay magnitude). Example: SPY straddle with 8.25 gamma and theta of −6.09 — the worked example's theta is signed, so |Θ| = 6.09 enters the square root — needs a $1.44 daily move; scalp at 50% of that.
  • Delta/gamma ratio method: flatten deltas when |Δ| = |Γ| (1:1; 2:1 for momentum names), using front-month options for the purest hedge deltas.

Gamma is thus the vehicle for trading realized volatility, with IV setting the theta price of that gamma — compare realized vol against IV to judge which side is favored2.

Links

Footnotes

  1. Passarelli, Trading Option Greeks — delta topic: four framings, sign/range, behavior across time/vol/rates, position delta. ↩↩↩

  2. Passarelli, Trading Option Greeks — gamma topic: convexity, moneyness/time/IV behavior, gamma–theta trade-off, realized-vol trading. ↩↩↩↩↩↩↩

  3. Chen & Sebastian, The Option Trader's Hedge Fund — Greeks topic: gamma behavior in OTM options, gamma-scalping methods. ↩↩