4.11.2 Delta, Gamma, Vega

Basic Prerequisites: 01 · From Zero and Pillar 3 · The Greeks & Dynamic Hedging.

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1. Intuition & Practical Objective

Pillar 3 derives the Greeks so that it can replicate an option and price it. Here the same five numbers are read the other way round: they are the risk factors themselves. A desk’s option book is summarised, minute to minute, by five columns - delta, gamma, vega, theta, rho - and the limits, hedges and capital are written against those columns, not against the option prices.

The practical objective of this page is to make the aggregation step concrete. Sensitivities are linear in position size: if you hold $q$ options, your exposure is $q$ times the per-option Greek. Therefore

$${\Delta }_{\text{book}}=\sum _i{q}_i{\Delta }_i,{\Gamma }_{\text{book}}=\sum _i{q}_i{\Gamma }_i,{\nu }_{\text{book}}=\sum _i{q}_i{\nu }_i,\dots$$

The book has one delta even though it has forty trades. That collapse - from a book of instruments to a vector of risk factors - is the whole trick, and it is also where the danger lives: the sum hides the distribution of curvature across strikes and maturities.

The two relations to internalise:

• Gamma–theta (the carry identity). A delta-hedged option earns $\frac{1}{2}\Gamma (\Delta S{)}^2$ when the market moves and pays $\Theta \Delta t$ for the passage of time; risk-neutrally they cancel exactly:

$$\frac{1}{2}\Gamma S^2{\sigma }^2=-{\Theta }_{\text{driftless}}(\text{Haug §2.15}).$$

Being long gamma is being long realised variance and short implied variance. This single sentence is the P&L of every option desk.

• Delta-neutral P&L (Hull eq. 19.3). $\Delta \Pi \approx \Theta \Delta t+\frac{1}{2}\Gamma (\Delta S{)}^2$ - the local expansion that every risk report is built on.

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2. Mathematical Ground Truth & Derivations

The sensitivity table (generalized BSM; Haug §2, Hull Ch 19, both verified). Notation: $c,p$ European call/put, $b$ cost of carry, $N$ the standard normal CDF, $n(x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}$, $e=e^{(b-r)T}$.

Sensitivity	Definition	Call	Put	Sign
Delta	$\partial V/\partial S$	$eN(d_1)$	$e(N(d_1)-1)$	$+/-$
Gamma	${\partial }^{2}V/\partial S^2$	$\frac{en(d_1)}{S\sigma \sqrt{T}}$	(identical)	$+$
Vega	$\partial V/\partial \sigma$	$Sen(d_1)\sqrt{T}$	(identical)	$+$
Theta	$-\partial V/\partial T$	$-\frac{Sen(d_1)\sigma }{2\sqrt{T}}-(b-r)SeN(d_1)-rX{e}^{-rT}N(d_2)$	$-\frac{Sen(d_1)\sigma }{2\sqrt{T}}+(b-r)SeN(-d_1)+rX{e}^{-rT}N(-d_2)$	usually $-$
Rho	$\partial V/\partial r$	$TX{e}^{-rT}N(d_2)$	$-TX{e}^{-rT}N(-d_2)$	$+/-$

Conventions that matter in a risk system. Vega and rho are quoted per 1 vol/rate point $=$ raw$/100$; theta per day $=$ raw$/365$. Gamma is a pure second derivative and has no point-scaling convention - a frequent source of cross-desk confusion. Note the structural facts a risk manager exploits: gamma and vega are identical for calls and puts, gamma peaks at-the-money and decays as $1/\sqrt{T}$; ATM vega grows as $\sqrt{T}$ (it is gamma’s scale factor $S^{2}T$); theta is the most negative at-the-money; rho is negligible for short-dated options.

The key identities (Haug §2.15, §2.3.3).

$$\Gamma =-\frac{2{\Theta }_{\text{driftless}}}{S^2{\sigma }^2},\nu =\Gamma \sigma S^{2}T,\text{Vanna}=\frac{\partial \Delta }{\partial \sigma }=-en(d_1)\frac{d_2}{\sigma },\text{Volga}=\frac{{\partial }^{2}V}{\partial {\sigma }^2}=\nu \frac{d_1{d}_2}{\sigma }.$$

Vanna is the sensitivity of delta to vol - the reason a delta-hedged book acquires delta when the vol surface shifts. Volga is the convexity in vol - the reason long-dated wings pay so well in a vol spike. Both are cross-Greeks, and both live in the second-order matrix $H$ of 05 · Failure Modes.

Greeks of forwards and futures (Hull eq. 19.5/19.6). For a forward contract on an asset with yield $q$: $\Delta =e^{-qT}$; for a futures contract: $\Delta =e^{(r-q)T}$, and the hedge ratio is $H_F=e^{-(r-q)T}{H}_A$. The futures delta is not 1 - the daily settlement makes a futures position a levered spot position, which is why “hedging with futures” needs a scaling that “hedging with stock” does not.

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3. Computational Implementation - the Haug cross-check and a real book

Two blocks: first reproduce the Haug Table 2-3 numbers that Pillar 3 verified, so the shared engine is proven; then aggregate a two-leg book and derive the delta hedge. Stdlib only.

Read the aggregation. The book is long gamma (+1.38) and short theta (−1.27/day) - the classic long-volatility posture: it makes money when the market moves and bleeds when it does not, and the gamma–theta identity says those are the same trade viewed from two sides. Note the sign discipline: put gamma is positive and equals call gamma, so it is the position that sets the sign - the short put leg contributes $(-50)\times 0.027086=-1.3543$ of gamma, i.e. it reduces the book’s long-gamma, while its delta contribution $(-50)\times (-0.283374)=+14.17$ adds to the book’s delta. Netting delta is arithmetic; netting gamma requires getting both the instrument sign and the position sign right, and they are different questions.

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4. Failure Modes & First-Principles Breakdowns

1. A delta-hedged book is not a risk-free book - it is a gamma book. The hedge removes the first-order term only. Over one step the residual P&L is $\frac{1}{2}\Gamma S^2[(\Delta S/S{)}^2-{\sigma }^2\Delta t]$, zero in expectation under $\mathbb{Q}$, but realised with standard deviation proportional to $|\Gamma |S^2\sigma \sqrt{\Delta t}$. Selling a delta-hedged book is selling insurance.
2. Delta is regime-dependent through the moneyness. $\Delta \to 1$ deep ITM, $\Delta \to 0$ deep OTM, and $\Gamma \to 0$ in both limits - so the same notional can be a delta position or a gamma position depending on where spot sits. A book near the money can flip its gamma sign on a $2\%$ move.
3. Vega is a surface sensitivity, not a scalar one. $\nu$ assumes one $\sigma$; real books have a smile, so what is actually held is a bucket vega (per expiry) and skew/smile vega. Hedging a single aggregated vega number against a single ATM vol leaves the surface shape unhedged (see Volatility Surfaces & Smiles).
4. Rho sign changes with the instrument. Call rho $>0$, put rho $<0$ for stock options - but for futures options both are negative ($\rho =-Tc$, Haug §2.16). A book quoted in futures options gets its rate risk backwards if it uses equity-option intuition.
5. Theta must be reported in the right time unit. Theta per calendar day ($/365$) and per trading day ($/252$) differ by $45\%$; a desk that hedges to a per-trading-day theta limit while reporting per-calendar-day will always look comfortably inside the limit.

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5. References

• Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
• Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed., 2006)
• Taleb, Nassim Nicholas: Dynamic Hedging (Wiley, 1997)
• J.P. Morgan / RiskMetrics: Technical Document, 4th ed. (1996)

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6. Connected Graph Bridges

• Back: Pillar 3 · The Greeks & Dynamic Hedging (where these formulas are derived) · 01 · From Zero
• Continue: 03 · Rates & Key-Rate Duration · Index Hub
• Sibling: Volatility Surfaces & Smiles (why a scalar vega is not enough) · 06 · Delta–Gamma VaR