VEGA FIELD GUIDE · 20 · RISK & WORKFLOW
Delta Hedging and P&L Attribution
An option position can make money for several reasons at once. Delta hedging reduces one source of movement and attribution helps explain what remains. The result is a clearer conversation with the trade.
A delta hedge offsets local first-order spot exposure at a moment in time.
Gamma changes delta as spot moves, so a hedge needs rebalancing to remain locally neutral.
Attribution separates option marks, hedge trades, theta, vega, gamma, fees, and unexplained residual.
What a delta hedge does
If one long call has a delta of 0.52 and the contract multiplier is 100, it carries about 52 share equivalents of positive spot exposure. Selling 52 shares creates a locally delta-neutral package at that instant.
The hedge is local because delta changes with spot, time, and implied volatility. A long-gamma position gains delta as spot rises and loses delta as spot falls. Rebalancing sells shares after rises and buys them after declines, which can harvest movement when it is large enough to exceed theta and trading costs.
A practical attribution equation
For a short interval, the option price change can be approximated with delta times the spot change, one-half gamma times the squared spot change, theta for elapsed time, and vega times the implied-volatility change. Rates, dividends, skew movement, higher-order Greeks, quote noise, and discrete timing land in the residual.
Greek units must be explicit. Vega is commonly quoted as option-price change for one volatility point. A move from 30% to 31% is one point. Treating it as a 100-point move creates a spectacular attribution error and a residual that has to hide the evidence.
delta x dS + 0.5 x gamma x dS^2 + theta x dt + vega x dIV + residualWorked example: one hedged call
A long call starts with delta 0.52, gamma 0.04, theta of -$0.08 per day, and vega of $0.12 per volatility point. The stock rises from $100 to $102 over one day and implied volatility falls two points. For one contract, the estimated components are +$104 delta, +$8 gamma, -$8 theta, and -$24 vega. The estimated option gain is $80 before residual.
Suppose the option mark actually gains $76. Residual is -$4. An initial hedge of short 52 shares loses $104 as the stock rises. Option plus hedge P&L is -$28 before transaction costs. The call made money, while the locally hedged volatility package lost because vega and theta outweighed gamma after directional exposure was offset.
| Component | One-contract P&L |
|---|---|
| Delta | +$104 |
| Gamma | +$8 |
| Theta | -$8 |
| Vega | -$24 |
| Residual | -$4 |
| Option mark | +$76 |
| Initial stock hedge | -$104 |
| Combined before costs | -$28 |
Hedge frequency is a tradeoff
Frequent rebalancing keeps delta closer to neutral and produces a cleaner realized-gamma path. It also creates more turnover, spreads, fees, and market impact. Slower rebalancing allows more directional drift and can miss intraday movement that disappears by the close.
A historical market-close replay measures one particular policy. Its results should state the hedge schedule, entry and exit marks, dividend handling, fees, and treatment of missing quotes. Those assumptions are part of the result.
Read the residual before trusting the story
- Reconcile option marks, stock hedge cash flows, dividends, and fees to total P&L.
- Confirm Greek units and contract multipliers.
- Flag stale or crossed option quotes before attribution.
- Measure residual as dollars and as a share of absolute explained P&L.
- Investigate persistent residuals for skew shifts, higher-order effects, timing mismatches, or data quality issues.
Replay the position and its hedge
Inspect daily option marks, hedge trades, Greek drivers, fees, residuals, and reconciliation.