Who this page is for — Anyone managing options who needs to understand why a delta hedge stops being neutral as the underlying moves.
Gamma measures the local change in delta with respect to the underlying price. It is the second derivative of option value:
Gamma = ∂Delta / ∂S = ∂²V / ∂S²If an option has delta 0.40 and gamma 0.03 per one-unit underlying move, an
increase of roughly one unit takes delta toward 0.43 in the local
approximation. The calculation does not establish delta after a large shock:
gamma itself changes with the market, time and volatility.
Unit and scale
Gamma is not a pure number in every display. If delta is expressed in
equivalent underlying units and S in dollars or euros per unit, gamma is the
change in delta for a one-currency-unit move in the underlying. A system may
instead report it:
- per point or per cent of price movement;
- per contract or per individual option;
- for a 1% relative underlying move;
- as cash gamma, dollar gamma or gamma P&L for a selected shock;
- after multiplying by quantity, multiplier and underlying price.
Two columns labelled “gamma” can therefore differ by large factors. Reports must state denominator, multiplier, currency, quantity and normalising shock. At position level:
position gamma = quantity × multiplier × unit gammaIf instruments have different underlyings or currencies, adding raw gamma requires coherent factor mapping; otherwise the result has no auditable economic unit.
Sign: long option and short position
A long vanilla call or put normally has positive gamma in standard models: its delta rises when the underlying rises. Selling the same option makes position gamma negative because the position value has the opposite sign. “Options have positive gamma” is incomplete in portfolio analysis: it refers to a long contract, not every position.
Exotics, barriers, digitals and compound payoffs can have more complex profiles and discontinuities near contractual levels. Even among vanilla instruments, aggregate gamma of a spread depends on quantities, strikes, expiries and leg signs. A strategy name does not determine it.
Gamma in local P&L
A Taylor approximation holding other factors fixed is:
ΔV ≈ Delta × ΔS + ½ × Gamma × (ΔS)²The quadratic contribution is positive for positive gamma and negative for negative gamma under the stated assumptions. It is not guaranteed profit or loss: implied volatility, time, rates, dividends, smile, gaps and execution costs can change during the same interval. Large movements also make higher- order terms and full revaluation more important.
Scale must match the formula. If the shock is in currency units, gamma must be
consistent with currency squared; if the shock is relative, the normalised
version is needed. Omitting ½ or mixing point gamma with percentage shocks is
a material calculation error.
Gamma and delta-hedge management
A portfolio that is delta-neutral at t₀ does not stay neutral automatically.
Non-zero gamma moves its delta with the underlying. Rebalancing can return it
to target but requires decisions on:
- frequency and intervention thresholds;
- spread, commissions, slippage and market impact;
- gaps that cross a rebalancing range;
- basis between spot, futures and the hedge instrument;
- model error and data latency;
- simultaneous changes in vega and theta.
“Gamma scalping” describes a family of rebalancing procedures, not an automatic source of profit. The outcome depends on the realised underlying path, implied volatility paid or received, costs and hedge rules.
Where gamma can concentrate
For standard vanilla options, absolute gamma is often larger near the strike and can become concentrated near expiry for near-at-the-money options. This is conditional on the model, not a universal threshold. Volatility surface, rates, dividends, exercise style and barriers change the profile. Near expiry, a small underlying move can produce a sharp delta change and make discrete hedging fragile.
Open interest by strike does not reveal the market's net gamma. It does not show who is long or short, which positions are hedged, how contracts are split between clients and dealers or which model is used. A gamma-exposure estimate therefore needs additional assumptions that must be identified as assumptions.
Aggregation and scenario limits
Portfolio gamma should be retained by underlying, currency and relevant expiry or strike bucket. A single net number can mask large opposing gross positions and discontinuities. A zero-gamma snapshot also does not remove delta, vega, theta, jump or liquidity risk.
Full scenario revaluation is the primary control for movements large enough to invalidate the local approximation. Comparing scenario P&L with delta-gamma P&L helps identify residual model terms rather than proving the approximation will continue to hold.
Common mistake — Turning open interest at a strike into signed “dealer gamma”. Open interest counts contracts, not ownership or the hedging activity of both sides.
Checklist
- Is gamma unit, per contract, cash or relative-shock normalised?
- Which quantity, multiplier, currency and sign are included?
- Is the underlying shock in points or percentages?
- Which model and volatility surface generate the measure?
- How does gamma change in relevant price and time scenarios?
- Which costs and gaps limit delta rebalancing?
- Does the report separate observed data from positioning assumptions?
Sources
- CME Group, Options Gamma — The Greeks.
- Cboe Options Institute, Learning the Greeks: An Expert's Perspective.
- Fischer Black and Myron Scholes, The Pricing of Options and Corporate Liabilities, 1973.
- Basel Committee on Banking Supervision, MAR20 — Standardised approach: general provisions and structure — distinction among delta, vega and curvature risk.
- The Options Clearing Corporation, Characteristics and Risks of Standardized Options.