Option Book PnL Attribution

A company's income statement doesn't just report "we made $4 million this quarter" — it breaks that number into revenue, cost of goods sold, interest expense, and taxes, so management knows why the number came out the way it did. An options desk needs the same thing every single day. The book's total profit or loss is one number on a screen, but that number is the sum of several separate bets running at once — a bet on direction, a bet on how much the stock moves, a bet on time passing, a bet on volatility itself changing. PnL attribution is the exercise of splitting the one number back into those separate bets, using the Greeks the desk was already tracking.

The decomposition, one term per Greek

Over one trading day, a book's change in value can be approximated by adding up each Greek's contribution:

$$\Delta V \;\approx\; \Delta \cdot dS \;+\; \tfrac{1}{2}\,\Gamma \,(dS)^{2} \;+\; \theta \, dt \;+\; \nu \, d\sigma \;+\; \varepsilon .$$

In plain English: $\Delta$ ("delta") times $dS$ (how far the stock moved) is the P&L you'd expect from direction alone, as if the option were just a fixed number of shares. $\tfrac12 \Gamma (dS)^2$ is the extra gain from the curve being bent — gamma P&L, the same term from delta-hedging (see delta hedging frequency and costs), always non-negative for a bought option since it depends on $dS$ squared. $\theta$ ("theta") times $dt$ (one day) is the cost of time passing, almost always negative for an option holder. $\nu$ ("vega") times $d\sigma$ is the P&L from implied volatility itself rising or falling, independent of where the stock went. And $\varepsilon$ (epsilon) is the residual — everything the four named Greeks didn't explain: higher-order effects, the passage of time changing gamma and vega themselves, and any hedging or measurement slippage.

Worked example 1 — a clean day

A book holds a position with delta 500 (behaves like 500 shares), gamma 40 (delta grows by 40 shares for every $1 the stock moves), theta -$300 per day, and vega $150 per volatility point. Today: the stock rises $2, and implied volatility falls by 0.5 points.

• Delta P&L: $500 \times 2 =$ $1{,}000.
• Gamma P&L: $\tfrac12 \times 40 \times 2^{2} = \tfrac12 \times 40 \times 4 =$ $80.
• Theta P&L: -$300 (one day passed, collected regardless of what the stock did).
• Vega P&L: $150 \times (-0.5) =$ -$75.

Sum of explained pieces: $1{,}000 + 80 - 300 - 75 =$ $705. Suppose the book's actual, measured P&L for the day — from the trading system, marking every position to the close — was $720. The residual is $720 - 705 =$ $15: small, and consistent with rounding and minor second-order effects. A desk seeing a $15 residual on a $705 predicted move calls the day fully explained.

Worked example 2 — a day the Greeks don't fully explain

Same book, same starting Greeks. Today the stock jumps $6 (a much bigger move) and implied volatility jumps up 3 points at the same time.

• Delta P&L: $500 \times 6 =$ $3{,}000.
• Gamma P&L: $\tfrac12 \times 40 \times 6^{2} = \tfrac12 \times 40 \times 36 =$ $720.
• Theta P&L: -$300, unchanged.
• Vega P&L: $150 \times 3 =$ $450.

Sum of explained pieces: $3{,}000 + 720 - 300 + 450 =$ $3{,}870. Suppose actual measured P&L was $4,200. The residual is $4{,}200 - 3{,}870 =$ $330 — over 8% of the day's move, far larger than worked example 1's rounding error. Where did it come from? The stock and volatility moved together, and there's a cross-effect — vanna, how delta itself changes when volatility changes — that the four-term formula above doesn't include at all. On a quiet day that cross-term is negligible; on a day where the stock jumps and volatility jumps at the same time, it can be the single largest unexplained piece. This is exactly why attribution is a diagnostic, not just an accounting exercise: a growing residual tells the desk which Greek they're missing before it costs real money.

If you tracked the daily residual, $\varepsilon$, over many trading days and plotted its distribution, it should look roughly like the curve above: centered near zero, with occasional larger outliers on days like worked example 2. A residual distribution that is not centered near zero — consistently positive or negative — means a Greek is being measured wrong, not just that the day was unusual.

What this means in practice

Every options desk runs a version of this attribution nightly, comparing "what the Greeks predicted" to "what actually happened" for every book. It's the single fastest way to catch a hedging error, a stale volatility mark, or a bug in the risk system — long before it shows up as a large, unexplained loss. A trader who can't explain today's P&L in terms of delta, gamma, theta, and vega either had an unusual day (big cross-terms, like example 2) or has a Greek that's wrong, and the residual is what tells them which.

Practice

1. A book has delta 200, gamma 10, theta -$120/day, vega $80/point. The stock falls $3 and implied vol rises 1 point. Compute each of the four P&L pieces and their sum.
2. If actual measured P&L in question 1 was $95, what is the residual, and is it large enough (relative to the total move) to be worth investigating?