P&L attribution

Why split a P&L

A profit on its own doesn't say whether the trade worked for the reason it was put on. A short straddle can make money because the price stayed quiet, as intended, or simply because implied volatility fell, which would have helped any short option. Attribution answers that by explaining the change in value with the position's Greeks at the start.

The formula

For each leg, weighted by its sign and ratio as in position Greeks:

• Spot: delta × dS + ½ × gamma × dS², where dS is the change in the underlying. Second order, because gamma matters for options even over modest moves.
• Volatility: vega × the change in implied volatility, in points.
• Time: theta × the number of days elapsed.
• Residual: the actual change minus the three terms above.

The residual is what the starting Greeks couldn't see: the Greeks themselves change along the way (gamma grows as expiry nears, vega changes with the price), and a large move or a large change in IV makes first-order terms less exact. A small residual means the split is trustworthy; a large one means the position changed character during the period.

A worked example: a short straddle

Seven days ago, with BTC at $100,000, you sold the 30-day $100,000 call and put at 50% IV, $5,714 each: $11,428 for the short straddle. At entry each leg had gamma 0.0000278, vega $114 and theta −$95 a day; the call's delta was 0.529 and the put's −0.471. Short both, the position had delta −0.057, gamma −0.0000555, vega −$228 and theta +$190 a day.

Today BTC is at $103,000, 23 days are left and IV is 46%. The call is worth $6,325 and the put $3,325: $9,650 to buy back. The position is up $1,777.0 at mid (from the unrounded prices, $11,427.5 and $9,650.5).

The spot term splits into −$171 from delta and −$250 from gamma: the $3,000 move hurt, and more than delta alone says. Time paid most of the profit, and the fall in IV paid about two-thirds as much again ($913 against $1,331). Had IV stayed at 50%, the vega term alone says the trade would have made $913 less; repricing both options at 50% gives about $790 less, because the starting vega overstates how much the IV drop was worth after a week of decay and a $3,000 move. The residual of −$45 is small next to the total, so the split is a fair description of what happened.

How BOSS computes it

A strategy page's "If opened N days ago" section prices the page's exact contracts from a recorded snapshot of the whole chain, taken 1, 3, 7, 14 or 30 days back. Snapshots are recorded about once an hour; the one used can be up to three hours older than the target, so a missed recording doesn't blank the comparison.

• The headline P&L is at execution: opening then at the ask for bought legs and the bid for sold legs, closing now on the other side, with trading fees both ways. That is what the round trip would really have made.
• The attribution explains the change at mid, with each leg's Greeks as the venue reported them in the snapshot. dS is the change in each leg's expiry forward, and the days are the exact time elapsed. The residual is shown as "Unexplained" rather than folded into the other terms.
• If there is no snapshot that far back, or the snapshot lacks one of the legs (not listed yet, or outside the recorded strikes), the section says so instead of guessing.

The gap between the execution P&L and the change at mid is the cost of crossing the bid-ask spread twice plus fees.