3.3.4 The Greeks & Dynamic Hedging (Full Sensitivity Lookup)

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and 03 · The Pricing Formulas.

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1. Intuition & Practical Objective

The Greeks are the derivatives of the option price with respect to its inputs - they quantify risk and drive hedging. The practical objective of this page is the lookup table: the exact first-order Greeks (delta, theta, vega, rho) that a desk quotes and hedges every day, with the scaling conventions that trip up everyone who reads a formula book.

The single most important relationship is the gamma–theta trade: a delta-hedged option position bleeds value through time (theta) but gains when the market moves (gamma). In the risk-neutral world these exactly offset in expectation:

$$\frac{1}{2}\Gamma S^2{\sigma }^2=-{\Theta }_{\text{driftless}},$$

i.e. the option earns its carry from the variance of the underlying (Haug §2.15; the residual of discrete hedging is governed by this identity).

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2. Mathematical Ground Truth & Derivations

The Greek table (Haug §2, verified).

Notation as the index page; $n(d_1)=\frac{1}{\sqrt{2\pi }}{e}^{-d_1^2/2}$. All formulas are $\partial /\partial$ of the generalized BSM.

Greek	Definition	Formula	Sign
Delta (call)	$\frac{\partial c}{\partial S}$	$e^{(b-r)T}N(d_1)$	$>0$
Delta (put)	$\frac{\partial p}{\partial S}$	$-e^{(b-r)T}N(-d_1)$	$<0$
Gamma (call=put)	$\frac{{\partial }^2}{\partial S^2}$	$\frac{e^{(b-r)T}n(d_1)}{S\sigma \sqrt{T}}$	$>0$
Theta (call)	$-\frac{\partial c}{\partial T}$	$-\frac{S{e}^{(b-r)T}n(d_1)\sigma }{2\sqrt{T}}-(b-r)S{e}^{(b-r)T}N(d_1)-rX{e}^{-rT}N(d_2)$	usually $<0$
Theta (put)	$-\frac{\partial p}{\partial T}$	$-\frac{S{e}^{(b-r)T}n(d_1)\sigma }{2\sqrt{T}}+(b-r)S{e}^{(b-r)T}N(-d_1)+rX{e}^{-rT}N(-d_2)$	ambiguous
Vega (call=put)	$\frac{\partial }{\partial \sigma }$	$S{e}^{(b-r)T}n(d_1)\sqrt{T}$	$>0$
Rho (call)	$\frac{\partial c}{\partial r}$	$TX{e}^{-rT}N(d_2)$	$>0$
Rho (put)	$\frac{\partial p}{\partial r}$	$-TX{e}^{-rT}N(-d_2)$	$<0$

Caveat - the generalized-BSM rho and cost-of-carry $b\ne r$. The compact form $TX{e}^{-rT}N(d_2)$ (and the worked example below, $\rho =0.109656$/pt) is the rho of the standard Black–Scholes where $b=r$; it is the convention Haug prints in his table (§2.16), and the numbers here reproduce it exactly. Strictly, however, $\partial c/\partial r$ under the generalized formula (where $d_1,d_2$ and the $e^{(b-r)T}$ factor also depend on $r$) is

$${\rho }_{\text{call}}=\frac{\partial c}{\partial r}=-TS{e}^{(b-r)T}N(d_1)+TX{e}^{-rT}N(d_2),$$

which collapses to the compact form only when $b=r$ and $d_1$ is allowed to move with $r$ (the standard-BS case, where the two normal terms cancel via $Sn(d_1)=X{e}^{-rT}n(d_2)$); holding $b$ fixed - the generalized-model convention (futures, FX, dividend yield) - the extra term survives, and at the worked example’s parameters ($b=0.05\ne r=0.10$) the full derivative is negative. The distinction only bites when the drift $b$ is treated as an independent input (futures, FX, dividend yield); for textbook equity BSM with $b=r$ the two coincide.

Gamma–theta / vega–gamma relations (Haug §2.15, §2.3.3):

$$\Gamma =-\frac{2{\Theta }_{\text{driftless}}}{S^2{\sigma }^2},\nu =\Gamma \sigma S^{2}T,{\Theta }_{\text{driftless}}=-\frac{\nu \sigma }{2T}.$$

Higher-order cross-Greeks (Haug §2.3.3; the sensitivity of Delta to vol, and of Vega to vol - the vega-convexity of a book):

$$\text{Vanna}=\frac{{\partial }^{2}V}{\partial S\partial \sigma }=-e^{(b-r)T}{N}^{\prime }(d_1)\frac{d_2}{\sigma },\text{Volga}=\frac{{\partial }^{2}V}{\partial {\sigma }^2}=\nu \frac{d_1{d}_2}{\sigma }.$$

Vanna matters for vol-skew risk (Delta changes as vol moves) and Volga for vega-convexity (the $d_1{d}_2$ sign flips across the strike, so a book’s volga changes sign with moneyness) - both are first-order in the skew-stickiness / vanna-volga pricing corrections.

Critical scaling convention (Haug §2 - read before using any number): raw derivatives are per unit; screen/lookup values quote Vega, Rho, Phi, Carry, Vanna, Zomma per 1 vol/rate point $=$ raw $/100$; Vomma $/10^4$; Ultima $/10^6$; Theta per day $=\frac{1}{365}\Theta$.

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3. Computational Implementation - Greeks vs the verified table

Haug’s Table 2-3 is reproduced (inputs $S=98,X=100,T=.25,r=.10,b=.05,\sigma =.30$). Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. Scaling - the #1 lookup error. Quoting raw Vega/Rho (19.3, 10.97) when the market convention is per-point (0.193, 0.1097), or theta per-year instead of per-day, changes a hedge size by $100\times$ or $365\times$. Always restate the convention alongside the number.
2. Discrete hedging leaves gamma risk. Delta is only locally exact; between rebalances the position is exposed to $d{S}^2$. The residual P&L over one rebalance step is $\sim \frac{1}{2}\Gamma S^2\left[(\Delta S/S{)}^2-{\sigma }^2\Delta t\right]$ - zero in expectation under $\mathbb{Q}$, nonzero in reality (see 05 · Failure Modes).
3. Vega is the most fragile assumption. BSM vega assumes $\sigma$ is a single constant; real markets have a surface (skew/smile). A delta-hedged book hedges delta and gamma but is systematically short/long the higher moments the constant-vol model ignores.
4. Rho sign confusion. Call rho $>0$, put rho $<0$; but for futures options both are $<0$ ($\rho =-Tc$). Haug §2.16.
5. Pin risk at expiration. When spot lingers near strike at expiry, $\Gamma \to \infty$ and Delta flips violently between 0 and 1, so a delta-hedger is whipsawed into large trades hedging a binary outcome - the classic expiry-day failure (Taleb, Dynamic Hedging). Mitigation: close or roll the position before expiry rather than hedging the knife-edge.
6. Discrete-rebalancing gamma bleed (variance of hedging error). Rebalancing at intervals $\Delta t$ leaves unhedged residual variance $\approx \frac{1}{2}{S}^4{\sigma }^4{\Gamma }^2\Delta t$; in fast crashes discrete hedging systematically sells lows and buys highs. This is the practical cost of the “continuous” delta-hedge ideal (see also failure mode 2).

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5. References

• Haug, The Complete Guide to Option Pricing Formulas
• Shreve, Stochastic Calculus for Finance II
• Hull, Options, Futures, and Other Derivatives
• Taleb, Nassim Nicholas: Dynamic Hedging: Managing Vanilla and Exotic Options, Wiley

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6. Connected Graph Bridges

• Back: 03 · Pricing Formulas
• Forward: 05 · Failure Modes · Index Hub
• Sibling: Index Hub · Heston & SABR