A Taylor expansion turned into a P&L report: each Greek's share of one day on an option book, what is left unexplained, and when the approximation breaks.
Write the day's P&L as a Taylor expansion: delta times the price move, plus half gamma times the move squared, plus vega times the change in implied volatility, plus theta for the time that passed. On 10,000 illustrative one-year at-the-money calls, a rally from 100 to 103 with implied volatility up from 24 to 25 per cent earns 22,681 on full revaluation; the four Greeks explain 22,751, of which delta is 18,384 and vega 3,829. The 70 left over is the unexplained P&L, 0.3 per cent of the day.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The option is Kestrel 100, the illustrative European call used throughout Quantitative Finance: spot and strike 100, a 4 per cent rate, one year to expiry, 24 per cent implied volatility, priced at 11.454068. The Greeks are taken at the start of the day; the day is one calendar day.
| Greek | Meaning | Value |
|---|---|---|
| Delta | Price change per 1 of spot | 0.612816 |
| Gamma | Delta change per 1 of spot | 0.015953 |
| Vega | Price change per volatility point | 0.382882 |
| Theta | Price change per calendar day | -0.018048 |
| Vanna | Delta change per volatility point | -0.000744 |
| Volga | Vega change per volatility point | 0.000213 |
Overnight the stock moves from 100.00 to 103.00 and implied volatility from 24 to 25 per cent. Repricing the call at 103, 25 per cent and 364 days gives 13.722123, a gain of 2.268055 per option, 22,681 on the lot. That is the number the Greeks have to explain.
P&L ≈ Δ × dS + ½ × Γ × dS2 + vega × dσ + Θ × dt
Delta: 0.612816 × 3 = 1.838449, or 18,384 on the lot
Gamma: ½ × 0.015953 × 32 = 0.071790, or 718
Vega: 0.382882 × 1 = 0.382882, or 3,829
Theta: −0.018048 × 1 day, or −180
Explained = 2.275073 per option, 22,751 on the lot
Excel: =Delta*dS + 0.5*Gamma*dS^2 + Vega*dVolPts + Theta*Days, times the number of options
| Term | Per option | On the lot |
|---|---|---|
| Delta | 1.838449 | 18,384 |
| Gamma | 0.071790 | 718 |
| Vega | 0.382882 | 3,829 |
| Theta | -0.018048 | -180 |
| Explained by four Greeks | 2.275073 | 22,751 |
| Actual, full revaluation | 2.268055 | 22,681 |
| Unexplained | -0.007018 | -70 |
The residual comes from terms the four Greeks leave out. The largest here is the cross term between spot and volatility: when both move, delta changes with volatility as well as with spot. Vanna times dS times dσ is −0.000744 × 3 × 1 = −0.002233, or −22. Volga, the curvature in volatility, adds 1. With both, the unexplained figure falls to −49, 0.22 per cent of the day. What remains is higher-order: mostly the change in gamma across the 3-point move and the change in delta over the day, which between them account for about −20 of the −49.
Notice what each line says about the book. Delta alone explains 18,384 of 22,681 and misses 18.9 per cent. The vega line, 3,829 from a single volatility point, is five times the gamma line on a 3 per cent move. For a one-year option, the volatility mark is often the bigger story; what one vol point is worth works that number through.
A Taylor expansion is local. The table holds volatility at 24 per cent and lets spot move over one day.
| Spot move | Actual | Delta + theta | Error | Delta + gamma + theta | Error | Error on the lot |
|---|---|---|---|---|---|---|
| -10 | -5.2962 | -6.1462 | 0.8500 | -5.3485 | 0.0523 | 523 |
| -5 | -2.8749 | -3.0821 | 0.2073 | -2.8827 | 0.0079 | 79 |
| -3 | -1.7826 | -1.8565 | 0.0739 | -1.7847 | 0.0021 | 21 |
| -1 | -0.6227 | -0.6309 | 0.0082 | -0.6229 | 0.0002 | 2 |
| +1 | 0.6025 | 0.5948 | 0.0078 | 0.6027 | -0.0002 | -2 |
| +3 | 1.8902 | 1.8204 | 0.0698 | 1.8922 | -0.0020 | -20 |
| +5 | 3.2373 | 3.0460 | 0.1913 | 3.2455 | -0.0081 | -81 |
| +10 | 6.8438 | 6.1101 | 0.7337 | 6.9078 | -0.0639 | -639 |
The delta-only error grows with the square of the move, which is exactly what the gamma term captures. With gamma in, the error grows with the cube instead: tiny at 1 or 3 points, 523 on the lot for a 10-point fall. On a normal day a Greek-based explain is accurate to a few units; on a crash day it is not, which is why stress losses are computed by full revaluation, not from the Greeks.
Most option books are delta-hedged. Sell 6,128 shares against the lot and the delta line disappears: the hedged position earns 4,296 on the same day, and gamma, vega and theta explain 4,366 of it. The attribution now reads like the trader's actual exposures: 718 from realised movement, 3,829 from the volatility mark, minus 180 of decay. One line is missing from that hedged figure: the cash from the short sale earns interest, about 55 a day on the lot at 4 per cent, which cancels the interest part of theta, leaving the 126 of pure time decay. Whether the gamma earned pays for the theta over time is the question in the gamma-theta break-even move.
All five Greeks, with finite-difference checks and the conversion to a 10,000-contract position, are in the free workbooks for this book.
The difference between the P&L from full revaluation and the sum of the Greek terms. On the illustrative 10,000-call book, a day that earned 22,681 is explained to 22,751 by delta, gamma, vega and theta, so -70 is unexplained. Adding vanna and volga brings it to -49. A small, stable residual is normal; a large or growing one points to a missing risk factor, a wrong input or a booking error.
Delta is the slope at today's spot, and the slope changes as the stock moves. On a 10-point fall, delta and theta alone predict a loss of 6.1462 per option against 5.2962 actually lost, an error of 0.8500. Adding half gamma times the move squared cuts the error to 0.0523. For moves of 1 or 3 points the gamma term reduces the error to 0.0002 and 0.0021.
The delta term is cancelled by the shares, so what remains is gamma, vega and theta. Hedged with 6,128 shares sold, the illustrative 10,000 calls earn 4,296 on a day the stock rises 3 and implied volatility 1 point; gamma, vega and theta explain 4,366 of it.
This article is one calculation from Quantitative Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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