Gamma Risk: Units, Position Curvature, Hedging, and Full Repricing

Audit Gamma units and signs, position scaling, local Delta-Gamma P/L, cash-gamma naming, discrete hedging, full repricing, and lifecycle risk.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

For an option value V(S,...), Gamma is the local curvature Gamma_i = partial Delta_i / partial S = partial^2 V_i / partial S^2. If Delta is measured in underlying units per option unit and S in dollars, unit Gamma is the change in Delta for a $1 underlying move. A signed position with contract quantity q, multiplier or deliverable M, and stock hedge h has Delta_pos = q x M x Delta_i + h and Gamma_pos = q x M x Gamma_i.

For ordinary vanilla options, unit Gamma is generally positive for a long option and negative after applying a short position sign. That rule must not be generalized to every digital, barrier, adjusted, path-dependent, or exercise-boundary exposure. Apply direction only once: if a data feed already supplies signed, position-scaled Gamma, do not multiply by q or M again.

Build a signed curvature and hedge record

Lock the exact series, valuation timestamp, long or short direction, quantity, multiplier, live deliverable, currency, exercise style, settlement method, and any stock or proxy hedge.
State the price coordinate and units: spot dollars, index points, futures points, percentage move, per-share, per-contract, or already aggregated position sensitivity.
Freeze spot, time, rates, dividends, borrow, volatility surface, and the surface-dynamics rule used by the model; Gamma is not independent of these inputs.
Calculate unit and signed position Delta and Gamma separately, preserve gross and net legs, and identify whether the platform has already applied signs, quantity, multiplier, or cash scaling.
For a small frozen-state move, use Delta_pos_new ~= Delta_pos_old + Gamma_pos x Delta S and Delta V_pos ~= Delta_pos x Delta S + 0.5 x Gamma_pos x (Delta S)^2; label the omitted Theta, Vega, carry, cross-Greek, and cost terms.
Validate with centered finite differences, symmetric up and down shocks, several bump sizes, and full repricing under spot, time, surface, jump, and liquidity scenarios; record hedge orders and actual fills.
Reconcile option, stock, cash, dividends, funding, borrow, transaction costs, margin, exercise, assignment, settlement, fees, tax, and the residual between local attribution and realized P/L.

Cash-Gamma and Dollar-Gamma names are not standardized. One common definition is CashGamma = Gamma_pos x S^2; the Gamma-induced Dollar-Delta change for a return r_move is CashGamma x r_move. Under constant Gamma, the second-order P/L for that move is 0.5 x CashGamma x r_move^2. A vendor may include 0.5, use a one-percentage-point shock, or report a signed position value, so the formula and units must accompany the label.

A Delta-neutral position still has Gamma risk: after a move, local Delta reappears by approximately Gamma_pos x Delta S. Long-Gamma rebalancing can sell after rises and buy after falls, but theoretical curvature revenue must exceed Theta, volatility repricing, spreads, impact, financing, and hedge errors. A closed-market gap cannot be traded continuously.

Four worked examples

Signed position curvature: Ten long calls have M = 100, Delta_i = 0.50, and Gamma_i = 0.08 shares/$. Then Delta_pos = 500 shares and Gamma_pos = 80 shares/$. For Delta S = +$2, estimated Delta is 660 shares and P/L is +$1,160; for Delta S = -$2, estimated Delta is 340 shares and P/L is -$840. Curvature contributes +$160 in each scenario, so the paired curvature sum is +$320; the two total P/L outcomes differ by $2,000.
Cash-Gamma conventions: Five long calls have M = 100, Gamma_i = 0.08 shares/$, and S = $100. Thus Gamma_pos = 40 shares/$ and CashGamma = $400,000. For a 1% move, the Gamma-induced Dollar-Delta component is $4,000, while second-order Gamma P/L is 0.5 x $400,000 x 0.01^2 = $20. For a 2% move, Gamma P/L is $80. A label without the formula cannot reveal which number is intended.
Discrete Gamma scalping: Start Delta-neutral with a constant teaching assumption Gamma_pos = 50 shares/$. A path moves +$2, is rehedged, then moves -$2 and is rehedged again. Local Gamma contribution is $100 on each segment, or $200; Theta is -$120, and the two hedge trades cost $15 each. Net result is $200 - $120 - $30 = $50. This is not a guaranteed trading result because Gamma, fills, surface, and path were simplified.
Local versus full repricing: A European call has S = $100, K = $100, r = 4%, q_div = 1%, sigma = 25%, and tau = 0.5. Black-Scholes gives V0 = $7.7215522303, Delta = 0.5659323171, and Gamma = 0.0221205770. For Delta S = +$5, the second-order estimate is $3.1061687980; full repricing at unchanged IV gives $3.0956949428, an error of $0.0104738552/share or $1.047386 at M = 100. If IV also rises to 30%, total change is $4.4412937626, which cannot be attributed to the original Delta and Gamma alone.

Gamma-control failure modes

Long and short signs are omitted or applied twice.
Quantity and multiplier are omitted, duplicated, or already embedded in vendor output.
An adjusted deliverable or cash point value is treated as a standard share multiplier.
Spot, forward, futures, index-point, currency, or return coordinates are mixed.
A $1 move is confused with a 1% move.
Cash Gamma, Dollar Gamma, Gamma P/L, and Dollar-Delta change are used without formulas.
The Taylor factor 0.5 is omitted or applied twice.
Unit, contract, and position Gamma are compared without normalization.
Spot, time, rates, dividends, borrow, or model inputs are stale.
Sticky-strike, sticky-Delta, and other surface-dynamics assumptions are inconsistent.
IV level, skew, term structure, Vanna, Volga, and cross-Greek changes are ignored.
Theta units and the exact intraday or overnight clock are wrong.
A large move is evaluated with one unchanged local Gamma.
A jump or closed-market gap is treated as continuously hedgeable.
Digital, barrier, American-boundary, or other nonsmooth exposure is treated as vanilla curvature.
Finite-difference bump, grid, interpolation, or solver error contaminates Gamma.
Hedge latency, bid/ask, impact, partial fills, rejects, and fees are omitted.
Short-sale constraints, borrow, dividends, or proxy-basis risk block the hedge.
Margin, buying power, house liquidation, exercise, assignment, or settlement creates extra exposure.
Funding, tax, cash, shares, option removal, and attribution residuals are not reconciled.

Common misconceptions

“Gamma is the option’s price change.” Delta is the first-order price sensitivity; Gamma is the local rate at which Delta changes.
“High Gamma predicts a large market move.” Gamma is sensitivity, not a directional or volatility forecast.
“Long Gamma always profits.” Curvature must overcome Theta, volatility repricing, spreads, impact, financing, and hedge errors.
“Delta-neutral means risk-free.” Gamma recreates Delta as spot moves, while other Greeks and lifecycle risks remain.
“One Gamma number works for a large jump.” It is a local derivative whose model state and value can change sharply across a move.