Option Greeks across a book · RiskAtoms Academy

By the end you will be able to

1Read a book's risk report: delta, gamma, vega and theta by position and in total.
2Add Greeks across positions factor by factor, and spot the cross effects between factors.
3Estimate P&L with the second-order Taylor expansion (delta-gamma, with cross terms and theta) and judge it against full revaluation.
4Convert Black-Scholes Greeks into the units of the book's risk factors with partial derivatives and the chain rule.

Why it matters

Every morning the desk's 1,000 UK index puts cost £10,531 in time decay and would lose £192,680 if the index rose 1%. On day 486 of the history, the worst day in the book's 750, the index fell 6% and implied volatility jumped 7 points: the puts made £2.78m. Which numbers on the morning report would have told the risk manager that was coming, and how close would they have got?

Core explanation

From one option to a book

Illustrative dataRiskAtoms Bank plc's Markets desk

One made-up trading book on one made-up valuation day, in pounds: eight positions in four books, eight risk factors and 750 days of simulated factor moves. The levels (UK index 8,000, GBP/USD 1.30, Brent $80 and so on) are round illustrative numbers, not market data. The same book runs through every lesson in Module 3.2, so the figures agree from lesson to lesson.

The puts' Greeks

L1.7.09: What drives an option’s price named what drives an option's price and L2.5.04: Sensitivities used delta. A risk report needs four Greeks. Black-Scholes gives them per option, in index points; multiplying by 1,000 contracts × £10 a point gives pounds:

Greek	Per put (index points)	Position (£)	Meaning
Delta	−0.2409	−£2,408.5 per point	Index +1%: −£192,680
Gamma	0.000485	£4.847 per point²	Delta rises as the index rises
Vega	12.409	£124,091 per vol point	Vol +1 point
Theta	−1.0511	−£10,511 per day	One calendar day passes

The puts are long gamma: as the index falls, their delta becomes more negative and they gain faster; as it rises, they lose more slowly. That curvature is paid for by theta. For a position with its delta hedged, the gamma gain on a move of $\Delta S$ points is $\tfrac12\gamma(\Delta S)^2$ , which covers one day's theta when the index moves about 66 points, 0.82%. That is close to the daily move implied by 16% volatility (0.84%): an option's price is a bet on whether realised moves beat implied ones.

The risk report

Greeks add across positions, factor by factor, exactly as sensitivities did in L3.2.02: Risk factors and position mapping. The desk's risk report puts them in one table (£ thousands; deltas per 1% move, rates per bp, vega per vol point, UK gamma as the change in UK delta for a 1% move, theta per day):

	UK Δ /1%	US Δ /1%	FX Δ /1%	Brent Δ /1%	2y /bp	10y /bp	IG /bp	Vega /pt	UK Γ /1%²	Θ /day
UK futures	600.0	0	0	0	0	0	0	0	6.0	0
US basket	0	300.0	−300.0	0	0	0	0	0	0	0
UK puts	−192.7	0	0	0	0	0	0	124.1	29.1	−10.5
Gilt	0	0	0	0	0	−79.6	0	0	0	0
2y swap	0	0	0	0	56.6	0	0	0	0	0
FX forward	0	0	200.0	0	0	0	0	0	0	0
Brent	0	0	0	123.1	0	0	0	0	0	0
Corp bonds	0	0	0	0	−21.4	−12.9	−34.3	0	0	0
Book	407.3	300.0	−100.0	123.1	35.1	−92.5	−34.3	124.1	35.1	−10.5

Reading across the bottom row: the desk is long the UK index by £407,320 per 1% even after the puts; it is long £124,091 of vega, so it gains if markets get nervous; it pays £10,531 a day for that protection. UK gamma is £35,096: mostly the puts, plus £6,000 from the futures, because as the index rises 1% the £60m of futures becomes £60.6m and the delta per 1% grows with it.

Where theta comes from in this book

Only the puts carry time in the book's model. In practice the gilt and the corporate bonds also earn carry each day (coupon accrual less funding) and the swap's floating leg resets; real risk reports show those as theta too.

Cross gammas

Some positions depend on two factors at once, so their delta to one factor changes when the other moves. Those are cross gammas, the off-diagonal entries of the book's Hessian (in £ per 1%·1%, per bp², per vol point², or mixed):

	UK index	US index	2y rate	10y rate	GBP/USD	Brent	UK vol	IG spread
UK index	35,096	0	0	0	0	0	−9,567	0
US index	0	3,000	0	0	−3,000	0	0	0
2y rate	0	0	−8.8	4.4	0	0	0	11.8
10y rate	0	0	4.4	79.0	0	0	0	7.1
GBP/USD	0	−3,000	0	0	1,000	−1,231	0	0
Brent	0	0	0	0	−1,231	1,231	0	0
UK vol	−9,567	0	0	0	0	0	3,333	0
IG spread	0	0	11.8	7.1	0	0	0	18.9

Three stand out. UK index × UK vol (−£9,567), the puts' vanna: when volatility rises, the puts' delta becomes more negative, so a fall in the index and a rise in vol (the usual crash pattern) help them more than the two separately. US index × GBP/USD (−£3,000), the basket: a dollar gain is worth fewer pounds if the pound has risen (L3.2.05: FX and commodity risk). Brent × GBP/USD, the same effect for oil. The rates entries (convexity) are tiny per bp² but grow with the square of the move.

Read the report

Delta-gamma against full revaluation

Greeks are partial derivatives at today's prices, so they describe small moves. For the puts:

Index move	Delta only	Delta + gamma	Full revaluation
−20%	+£3,853,601	+£10,058,163	+£10,885,027
−10%	+£1,926,800	+£3,477,941	+£3,817,222
−5%	+£963,400	+£1,351,185	+£1,406,566
−1%	+£192,680	+£208,191	+£208,686
+1%	−£192,680	−£177,169	−£177,676
+5%	−£963,400	−£575,615	−£639,793
+10%	−£1,926,800	−£375,660	−£867,131
+20%	−£3,853,601	+£2,350,961	−£945,728

For ±1% the delta-gamma estimate is within £494. For −10% it misses by £0.34m, and for +20% it predicts a gain. That is why VaR and stress tests on option books use full revaluation (L3.2.10: Historical simulation in practice, L3.2.15: Stress testing the trading book).

The formal version

Second order in many factors (M-B02: Partial derivatives, M-B03: Taylor expansion). With factor moves $\Delta x$ , the delta vector $\delta_i = \partial V/\partial x_i$ and the Hessian $\Gamma_{ij} = \partial^2 V/\partial x_i\partial x_j$ (symmetric, because the order of differentiation does not matter for smooth functions):

\begin{aligned} \Delta V &\approx \sum_i \delta_i\,\Delta x_i + \tfrac12 \sum_i\sum_j \Gamma_{ij}\,\Delta x_i\,\Delta x_j + \theta\,\Delta t \\ &= \delta^\top \Delta x + \tfrac12\,\Delta x^\top \Gamma\,\Delta x + \theta\,\Delta t \end{aligned}

Greeks add across positions because differentiation is linear: $\partial(V_1 + V_2)/\partial x_i = \partial V_1/\partial x_i + \partial V_2/\partial x_i$ , and the same for each $\Gamma_{ij}$ . So the book's delta vector and Hessian are the sums of the positions'.

From Black-Scholes to the book's factors: the chain rule. Black-Scholes differentiates with respect to the index level $S$ . The book's factor is the log return $x$ , with $S = S_0 e^{x}$ , so $dS/dx = S$ :

\begin{aligned} \frac{\partial V}{\partial x} &= \frac{\partial V}{\partial S}\,\frac{dS}{dx} = \Delta\,S \\ \frac{\partial^2 V}{\partial x^2} &= \frac{\partial}{\partial x}\big(\Delta\,S\big) = \frac{\partial \Delta}{\partial S}\,S \cdot S + \Delta\,S = \Gamma S^2 + \Delta S \\ \text{per put:}\quad \Gamma S^2 + \Delta S &= 31{,}022.8 -1{,}926.8 = 29{,}096.0 \text{ points per } 1.00^2 \end{aligned}

Times 10,000 (£ per point for the position) and 0.01² (per 1%), that is £29,096 per 1%², the puts' entry in the report.

The puts in full, as a 2 × 2 problem. In the UK index (in %) and UK vol (in points):

\delta = \begin{pmatrix} -192{,}680 \\ 124{,}091 \end{pmatrix}, \qquad \Gamma = \begin{pmatrix} 29{,}096 & -9{,}567 \\ -9{,}567 & 3{,}333 \end{pmatrix}

The diagonal holds gamma and volga; the off-diagonal entry is vanna. For a move $\Delta x = (x, v)^\top$ :

\begin{aligned} \tfrac12\,\Delta x^\top \Gamma\,\Delta x &= \tfrac12\big(\Gamma_{11}x^2 + 2\Gamma_{12}\,x\,v + \Gamma_{22}v^2\big) \\ &= \underbrace{\tfrac12\Gamma_{11}x^2}_{\text{gamma}} + \underbrace{\Gamma_{12}\,x\,v}_{\text{vanna}} + \underbrace{\tfrac12\Gamma_{22}v^2}_{\text{volga}} \end{aligned}

Try it

Move the index and watch where each approximation leaves the full revaluation curve. Then add a jump in implied volatility, and let time pass.

Try itGreeks against full revaluation: the puts

UK index move−10%

Implied vol change0 pts

Days passed0

Full revaluationDelta + gamma + vega + thetaDelta + gammaDelta only

Full revaluation (index −10%)+£3,817,222

Index move in points−800

Delta only: error−£1,890,421

Delta + gamma: error−£339,281

+ vega + theta: error−£339,281

Errors are approximation minus full revaluation. The approximations use today's Greeks and ignore the cross terms (vanna, volga) that the worked example adds. Past ±10% the parabola from gamma bends the wrong way: the puts' real curvature fades as they go deep in or far out of the money.

Worked example

Day 486, term by term

The risk-off day: UK index −6% (log return), UK implied vol +7 points, along with the moves in every other factor.

A moderate move, day 486 for the puts, then the whole book

Step 1 of 6

1
Warm-up: a moderate move (index −240 points, vol +2)
In index points, with vanna −£119.6 per point per vol point and volga £3,333 per vol point² (the same example as M-B02: Partial derivatives):
$\begin{aligned} \text{1st order: } (-2{,}408.5)(-240) + 124{,}091 \times 2 &= 826{,}223 \\ \text{2nd order: } 826{,}223 + \tfrac12(4.8473)(240^2) + (-119.6)(-240)(2) + \tfrac12(3{,}333)(2^2) &= 1{,}029{,}896 \end{aligned}$
Full revaluation: +£1,023,546. For a 3% move the second-order estimate is within £6,349.

Regulatory lens · looking ahead

The Basel standardised approach for market risk is built from these numbers: a delta charge and a vega charge from sensitivities, and a separate curvature charge that captures what delta misses by revaluing options under large up and down shocks (M4.3: Market risk capital (FRTB) and CVA). Volatility surfaces, smile dynamics and the models behind them are in M6.3: Option pricing and volatility.

Spark Atoms

Spark AtomPortfolio insurance, October 1987

In the mid-1980s many US pension funds bought “portfolio insurance”: a strategy that copied a put option's delta by selling index futures as prices fell. On 19 October 1987 the Dow Jones Industrial Average fell 22.6% in a day. The Brady Commission's report found that portfolio insurers' futures selling, done mechanically as delta changed, added to the pressure. Copying an option's delta is not the same as owning its gamma when markets gap.

Spark AtomVega in a day, 5 February 2018

The VIX index of S&P 500 implied volatility more than doubled in one session on 5 February 2018. Exchange-traded products that were short volatility lost most of their value overnight, and one of the largest was closed down within weeks. Short vega positions had earned steady theta for years.

Key takeaways

Delta, gamma, vega and theta are partial derivatives; they add across positions factor by factor, giving the book's risk report.
Long options are long gamma and pay theta; the break-even daily move is close to the move implied volatility prices in.
Cross gammas (vanna, quanto-style FX effects) sit off the Hessian's diagonal and matter when factors move together, as in a crash.
The chain rule turns Black-Scholes Greeks into the book's factor units: ∂²V/∂x² = ΓS² + ΔS for a log-return factor.
Delta-gamma (plus vega, vanna, volga and theta) is good for moderate moves; large moves need full revaluation.

Reaction

Check what stuck

Question 1 · objective 3

An option book has delta −£2,000 per index point and gamma £5 per point². The index falls 100 points. What is the delta-gamma estimate of its P&L, in £?

£

Question 2 · objective 1

The same book has vega £100,000 per vol point and theta −£8,000 a day. Implied volatility rises 3 points over two days with the index unchanged. Estimate the P&L, in £.

£

Question 3 · objective 4

An index option has Black-Scholes gamma 0.0004 and delta −0.3, with the index at 5,000. What is its second derivative with respect to the index's log return, S²Γ + SΔ (index points per 1.00²)?

Question 4 · objective 1

The book's report shows positive UK gamma and negative theta. What does that describe?

AA short option position that earns time decayBA long option position: it gains from large index moves either way and pays for it day by dayCA position with no option riskDA position that loses if volatility rises

Question 5 · objective 3

For a 20% rise in the index the delta-gamma estimate for the puts is badly wrong. Why?

ABecause gamma is negative for putsBBecause the puts' gamma is not constant: as they go far out of the money their curvature fades, but the estimate keeps adding ½γΔS²CBecause the index cannot rise 20%DBecause theta has been left out

Glossary

Delta: The first partial derivative of value with respect to a factor: the P&L for a small move, all else fixed.
Gamma: The second derivative with respect to the underlying: how fast delta changes as the underlying moves.
Vega: The P&L for a 1 point rise in implied volatility.
Theta: The change in value as one day passes with nothing else moving: the cost (or income) of time.
Long gamma: Holding options (net): gains from large moves in either direction, paid for by negative theta.
Cross gamma: A second partial derivative with respect to two different factors: how one factor's delta changes when the other moves.
Vanna: The cross gamma between the underlying and implied volatility: how delta changes when volatility moves.
Volga: The second derivative with respect to implied volatility: how vega changes when volatility moves.
Hessian: The matrix of all second partial derivatives of value with respect to the risk factors.
Risk report: The daily table of a book's sensitivities by position and factor, summed to the book, that traders and risk managers read each morning.

Deep Dives & Origins

Deep Dives

J. C. Hull, Options, Futures, and Other Derivatives, 11th ed., Pearson, 2021, the chapter on the Greek letters. Intermediate.
N. N. Taleb, Dynamic Hedging: Managing Vanilla and Exotic Options, Wiley, 1997. Advanced.

Origins

F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy 81(3), 1973.
Presidential Task Force on Market Mechanisms (the Brady Commission), Report of the Presidential Task Force on Market Mechanisms, US Government Printing Office, 1988.
Basel Committee on Banking Supervision, Basel Framework, MAR21: Standardised approach: sensitivities-based method (delta, vega and curvature), Bank for International Settlements.

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