Gamma-Theta Tradeoff: Units, Carry, Paths, and Full Repricing

Audit Gamma and Theta with signed position units, explicit time conventions, PDE carry, executable hedge ledgers, full repricing, and lifecycle controls.

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

Direct answer

The Gamma-Theta tradeoff is a local model relationship, not a fixed price for convexity. For signed quantity q and multiplier M, define Delta_pos = q x M x Delta, Gamma_pos = q x M x Gamma, and Theta_pos = q x M x Theta. Unit Gamma must state the underlying-price unit, while Theta must state whether it is per year, per calendar day, or per trading day.

For ordinary long vanilla calls and puts, Gamma is usually positive and Theta usually negative under the stated model. That sign intuition can fail for portfolios, exotic payoffs, exercise boundaries, or different market regions. Actual performance comes from full option marks, hedge inventory, cash, carry, and costs; a Greek attribution explains that performance but must not be added to it again.

Build one consistent Gamma-Theta record

Lock every option series, signed quantity, multiplier, deliverable, currency, exercise style, settlement method, existing underlying inventory, and Delta target.
Declare the Greek source and coordinate: per-share or per-contract, already position-scaled or not, spot-price unit for Gamma, calendar-time convention for Theta, and whether Vega is per 1 vol point or per 1.00 volatility.
Capture synchronized executable option and underlying bid, ask, size, spot, forward, volatility surface, rates, dividends, borrow, FX, time to expiration, and exact timestamps.
Aggregate signed gross and net Delta, Gamma, Theta, and Vega exactly once. For a frozen local state, use Delta V_pos ~= Delta_pos x Delta S + 0.5 x Gamma_pos x (Delta S)^2 + Theta_pos x Delta t + Vega_pos x Delta sigma, with compatible units and signs.
Treat the local threshold as a diagnostic only. Under Black-Scholes assumptions with continuous dividend yield q_div, the identity is Theta + 0.5 x sigma^2 x S^2 x Gamma + (r - q_div) x S x Delta - r x V = 0, so financing and dividend carry are part of the relation.
Full-reprice spot, time, volatility level, skew, term structure, rates, dividends, jumps, trends, oscillations, gaps, and alternative hedge schedules. If hedging dynamically, update stock and cash only from actual fills and keep the self-financing ledger separate from Greek attribution.
Before and after expiration, reconcile closing trades, exercise, assignment, physical or cash settlement, adjusted deliverables, residual stock, hedge unwind, funding, borrow, dividends, fees, tax, and every model-versus-actual residual.

Four worked examples

Frozen one-day threshold with costs. One long contract has unit Gamma 0.08 Delta/$/share, displayed Theta -$0.18/share/calendar day, multiplier M = 100, and round-trip costs of $4/contract. The cost-free local threshold is sqrt(2 x 0.18 / 0.08) = $2.1213203436; allocating the $0.04/share cost gives sqrt(2 x (0.18 + 0.04) / 0.08) = $2.3452078799. A $2 move gives $16 - $18 - $4 = -$6; a $2.50 move gives $25 - $18 - $4 = +$3. These are frozen-Greek one-period diagnostics, not executable or expiration break-evens.
PDE carry is not pure Gamma versus Theta. For a European call with S = K = $100, T = 0.5, r = 4%, q_div = 1%, and sigma = 25%, suppose V = $7.721552230288, Delta = 0.565932317133, Gamma = 0.022120576997, and Theta = -$8.301615173889/share/year. The Gamma term is $6.912680311701/share/year; the combined financing and dividend term is $1.388934862188/share/year; together they equal $8.301615173889, or -Theta. Dividing by 365 gives about -$0.022744151161/share/calendar day, but a vendor’s displayed daily convention may differ.
Actual-fill self-financing path. A Delta-hedged long-Gamma position begins and ends with spot at $100. After a rise, it sells 100 shares at $101.95; after the reversal, it buys 100 shares at $100.05. Hedge cash is +$190; two fills cost $10; and the full option mark changes by -$150. Combined P/L is $190 - $10 - $150 = +$30. Do not add an idealized $200 hedge gain or a local Gamma term again.
Local Greeks versus full repricing. With the same model inputs, after one calendar day and a move to S_1 = $105, the local per-share terms are Delta $2.829661585667, Gamma $0.276507212468, and Theta -$0.022744151161, totaling $3.083424646974. The unchanged-volatility full value is $10.794375525849, so the actual change is $3.072823295561 and the Taylor error is $0.010601351413. The initial Delta hedge leaves $0.243161709894/share. If volatility rises to 30%, the full value becomes $12.136197003065, the option change is $4.414644772777, and the hedged result is $1.584983187110/share; that difference is not pure Gamma, Theta, or realized variance.

Measurement, execution, and lifecycle risks

Applying the wrong position sign, quantity, multiplier, deliverable, point value, or FX conversion.
Multiplying a feed that is already contract- or position-scaled a second time.
Leaving Gamma’s spot-price unit undefined or mixing dollar and percentage shocks.
Mixing annual, calendar-day, trading-day, weekend, and time-to-expiration Theta conventions.
Multiplying a Vega quoted per vol point by a decimal volatility change without conversion.
Treating an initial Delta hedge as if it remains neutral through spot and time changes.
Extrapolating one local Gamma through a large move while Speed and higher orders change.
Assuming Gamma and Theta remain constant as spot, time, and volatility move.
Omitting rates, dividends, borrow, convenience yield, and option financing from the PDE carry relation.
Using the Taylor approximation outside its local model and coordinate domain.
Adding Greek attribution to a full option mark and counting the same change twice.
Treating one endpoint shock as a dynamically hedged path or Gamma-scalping cash flow.
Mixing realized-variance sampling, day counts, overnight returns, and jumps.
Treating one entry IV as the variance paid or as a sufficient profit condition.
Ignoring volatility level, skew, term structure, Vanna, Volga, Charm, and cross terms.
Substituting midpoint or theoretical values for executable quotes, size, and actual fills.
Ignoring partial fills, rejects, latency, spread, impact, fees, and failed hedge triggers.
Assuming a gap, halt, closed session, or stop order preserves the modeled hedge path.
Omitting margin, funding, dividends, borrow availability, recall, and forced liquidation.
Missing early exercise, assignment, adjusted contracts, physical or cash settlement, residual inventory, tax, or final reconciliation.

Common misconceptions

“High Gamma alone makes an option attractive.” Price, Theta, surface dynamics, path, and executable costs still control the result.
“Positive Theta is daily interest or free income.” It is a conditional model derivative accompanied by other risks and carry terms.
“Gamma and Theta trade at a fixed rate all day.” Both change with the model state, position, and time convention.
“The local threshold is the trade or expiration break-even.” It is neither an implied move nor a guaranteed executable trigger.
“A Delta hedge or limited premium removes the risk.” Gaps, volatility repricing, liquidity, margin, assignment, and repeated losses remain.