Gamma Scalping: Self-Financing Hedge Ledgers, Paths, and Costs

Audit Gamma scalping with signed Delta targets, executable hedge fills, self-financing cash, full option marks, variance attribution, costs, and settlement.

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

Direct answer

Gamma scalping combines a positive-Gamma option position with repeated underlying trades that bring signed net Delta toward a target. If Delta_opt = sum(q_j x M_j x Delta_j), a zero-Delta target requires h_target = -Delta_opt underlying units. For a hedge change Delta h = h_new - h_old filled at S_fill, the cash account changes by B_new = B_old - Delta h x S_fill - fees, before interest, dividends, borrow, and tax.

Performance is the total self-financing ledger, not isolated stock trades. At a mark, equity is E = O + h x S + B, where O is the signed option value, h x S is the underlying inventory value, and B is cash. Full option marks and actual hedge fills determine P/L; Greek attribution is a separate explanation and must not be added again.

Maintain one executable, self-financing record

Lock every option leg, signed quantity, multiplier, deliverable, currency, exercise style, settlement method, initial premium, existing underlying inventory, and Delta target.
Define the rebalance policy before trading: clock interval, spot move, Delta band, risk limit, order type, executable size, and what happens after partial fills, rejects, halts, or closed sessions.
Capture synchronized option and underlying bid, ask, size, spot, forward, rates, dividends, borrow, FX, volatility surface, and exact timestamps.
Recalculate signed Delta and Gamma after material moves and time changes; submit the target hedge, then update inventory and cash only from actual fills rather than intended orders or model prices.
Maintain O + h x S + B with premium, stock cash, interest, dividends, borrow, spreads, impact, fees, collateral, and realized and unrealized components. Keep this ledger separate from Greek or model attribution.
Full-reprice trends, oscillations, gaps, volatility crush or rise, skew rotation, time decay, liquidity loss, margin, and alternative hedge schedules; use local Gamma-Theta relations only as diagnostics.
Before and after expiration, reconcile option removal, exercise, assignment, physical or cash settlement, residual stock, hedge unwind, financing, fees, tax, and every difference between model and actual P/L.

Under smooth, continuously hedged model assumptions, a local variance attribution may resemble dPi ~= 0.5 x Gamma_pos x S^2 x (sigma_realized^2 - sigma_model^2) x dt. Real trading is discrete: Gamma changes through time and spot, implied volatility and skew move, jumps may occur between fills, and the paid option price is not summarized by one entry IV. Realized volatility above entry IV is therefore not a sufficient profit rule.

More frequent hedging can reduce local Delta tracking error while increasing spread, impact, latency, borrow, and operational costs. Wider bands reduce turnover but leave more directional and gap exposure. The optimal rule depends on the position, path, liquidity, funding, and loss constraints; it is not simply “hedge as often as possible.”

Four worked examples

Executable round-trip ledger: Initially O0 = $5,000, S0 = $100, Delta_opt = +500 shares, h0 = -500 shares, and B0 = $45,000, so E0 = $0. At S = $102, option Delta becomes +660 shares; sell 160 shares at $101.98 and pay $8. When spot returns to $100, option Delta is again +500 shares; buy 160 shares at $100.02 and pay $8. Cash ends at B2 = $45,297.60; with O2 = $4,750 and h2 = -500 shares, E2 = $47.60. Equivalently, hedge gross is $313.60, fees are $16, option change is -$250, and total P/L is $47.60.
Variance-versus-carry diagnostic: Freeze S = $100, Gamma_pos = 40 shares/$, sigma_model = 25%, and dt = 1/252. Model Gamma carry is 0.5 x 40 x $100^2 x 0.25^2 / 252 = $49.6031746032. A one-day +2% move contributes $80, for ideal net $30.3968253968; a +1% move contributes $20, for -$29.6031746032. The cost-free break-even absolute move is $100 x 0.25 / sqrt(252) = $1.5748519709. This is a frozen-Gamma teaching attribution, not a cash ledger or promise.
Full repricing and attribution: Ten European calls with M = 100 have V0 = $7.7215522303, Delta = 0.5659323171, and Gamma = 0.0221205770. For Delta S = +$5, the position Delta term is $2,829.6615855, the Gamma term is $276.5072125, and Taylor change is $3,106.1687980. Unchanged-IV full change is $3,095.6949428, so Taylor error is $10.4738552. With the initial Delta hedge, option plus stock is $266.0333573. If IV rises to 30%, full option change is $4,441.2937626 and the hedged result is $1,611.6321771; the difference is not purely realized-variance profit.
Expiration lifecycle: Five physically settled calls have K = $100, M = 100, and premium $6/share = $3,000. A prior short hedge of 500 shares was sold at an average $104, receiving $52,000. At S_T = $108, exercise pays -$50,000, receives +500 shares, and covers the short hedge; ending shares are 0, and total cash is -$3,000 + $52,000 - $50,000 = -$1,000. A cash-settled version pays +$4,000, while buying 500 shares at $108 to close the hedge costs -$54,000; -$3,000 + $52,000 + $4,000 - $54,000 = -$1,000. The ledgers differ even when these controlled economics match.

Hedge-ledger failure modes

Option quantity, long or short sign, multiplier, deliverable, point value, or FX scaling is wrong.
The Delta target or proxy hedge does not match the option’s underlying exposure.
Initial premium, stock inventory, or cash is omitted from the opening ledger.
Full option marks and Greek attribution are added together and double-count P/L.
Realized and unrealized option, stock, and cash P/L are mixed.
Midpoints or model prices replace executable bid, ask, size, and actual fills.
Partial fills, rejects, latency, cancel races, or stale Delta leave unintended exposure.
Spreads, market impact, routing, fees, and option or stock liquidity are understated.
A higher hedge frequency is assumed to improve results without a cost trade-off.
Gamma is held constant through spot, time, and surface changes.
Theta uses the wrong sign, day unit, intraday clock, or carry convention.
IV, skew, term structure, Vega, Vanna, and Volga repricing is omitted.
Realized volatility uses a different sampling window, day count, overnight rule, or jump treatment.
Entry IV is treated as the exact variance paid by the position.
A gap, halt, limit state, or closed market is treated as an executable intermediate path.
Funding, interest, dividends, borrow, locate, recall, collateral, or FX cash flows are omitted.
Margin, buying power, concentration, or house liquidation interrupts the strategy.
Corporate actions or adjusted deliverables break the hedge ratio.
Exercise, assignment, cash or physical settlement, cutoffs, and residual inventory are mishandled.
Tax, accounting, model version, trade confirms, cash, shares, and attribution residuals are not reconciled.

Common misconceptions

“Profitable stock scalp trades are the strategy profit.” They must be combined with the option, cash, carry, and all costs.
“Returning to the starting spot guarantees profit.” The path, hedge fills, option surface, Theta, financing, and costs determine the result.
“Realized volatility above entry IV guarantees profit.” The relevant variance exposure is Gamma-weighted and affected by model, surface, sampling, jumps, and execution.
“Hedging more often always earns more.” It can reduce tracking error while increasing turnover, impact, spread, and operational loss.
“Delta-neutral and long-Gamma means safe or unlimited profit.” Delta reappears, premium decays, liquidity and margin can fail, and gaps may be unhedgeable.