Analytical Greeks for European Options

Setup

What Greeks Measure

A Greek is the partial derivative of an option's price with respect to one of its inputs. Greeks serve two purposes simultaneously:

1. Hedging: each Greek tells you how much of a hedging instrument to hold to neutralise the corresponding risk.
2. P&L attribution: the Taylor expansion of option P&L, decomposed by Greeks, explains where the desk's daily P&L came from.

Neither purpose is decorative. A trading desk that cannot decompose its P&L into Greeks contributions has no risk management — it has guessing.

Assumptions and Notation

All results in this module derive from the Black-Scholes formula. The assumptions are those of the Black-Scholes model: GBM dynamics, constant volatility $\sigma$, constant risk-free rate $r$, no dividends.

Call price: $C = S\,N(d_1) - Ke^{-r\tau}\,N(d_2),$ $d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)\tau}{\sigma\sqrt{\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{\tau}, \qquad \tau = T - t.$

Put price: by put-call parity, $P = Ke^{-r\tau}N(-d_2) - S\,N(-d_1)$.

Core identity (used throughout): $S\,n(d_1) = Ke^{-r\tau}\,n(d_2),$ where $n(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ is the standard normal pdf. This identity is proved by substituting the definitions of $d_1$ and $d_2$ and simplifying the exponentials.

Useful partial derivatives of $d_1$: $\frac{\partial d_1}{\partial S} = \frac{1}{S\sigma\sqrt{\tau}}, \qquad \frac{\partial d_1}{\partial \sigma} = -\frac{d_2}{\sigma}, \qquad \frac{\partial d_1}{\partial \tau} = -\frac{d_2}{2\tau} + \frac{r}{\sigma\sqrt{\tau}}.$

Note: $\partial d_2 / \partial S = \partial d_1 / \partial S$ (since $d_2 = d_1 - \sigma\sqrt{\tau}$, independent of $S$).

Delta: $\Delta = \partial C / \partial S$

$\Delta_{\mathrm{call}} = N(d_1), \qquad \Delta_{\mathrm{put}} = N(d_1) - 1 = -N(-d_1).$

Derivation: $\frac{\partial C}{\partial S} = N(d_1) + S\,n(d_1)\frac{\partial d_1}{\partial S} - Ke^{-r\tau}n(d_2)\frac{\partial d_2}{\partial S} = N(d_1) + \underbrace{S\,n(d_1) - Ke^{-r\tau}n(d_2)}_{=\,0} \cdot \frac{1}{S\sigma\sqrt{\tau}} = N(d_1).$

The core identity eliminates the $n(d_1)$ and $n(d_2)$ terms.

Sign conventions:

• Long call: $\Delta \in (0, 1)$. At-the-money: $\Delta \approx 0.5$.
• Long put: $\Delta \in (-1, 0)$. At-the-money: $\Delta \approx -0.5$.
• $\Delta \to 1$ as call goes deep ITM; $\Delta \to 0$ as call goes deep OTM.

Hedge interpretation: hold $\Delta_{\mathrm{call}}$ shares per long call to be instantaneously delta-neutral.

Note: $\Delta_{\mathrm{call}} = N(d_1)$ is not the risk-neutral probability $\mathbb{Q}(S_T > K) = N(d_2)$. The difference $N(d_1) - N(d_2) > 0$ is the "probability gap" due to the lognormality of $S_T$.

Gamma: $\Gamma = \partial^2 C / \partial S^2$

$\Gamma = \frac{n(d_1)}{S\sigma\sqrt{\tau}}.$

Derivation: $\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial N(d_1)}{\partial S} = n(d_1)\frac{\partial d_1}{\partial S} = n(d_1) \cdot \frac{1}{S\sigma\sqrt{\tau}}.$

Key properties:

• $\Gamma_{\mathrm{call}} = \Gamma_{\mathrm{put}}$: identical for calls and puts with the same inputs. This follows from put-call parity: $\partial^2(C-P)/\partial S^2 = \partial^2(S - Ke^{-r\tau})/\partial S^2 = 0$.
• $\Gamma > 0$ always: option value is convex in $S$. Long options are long gamma.
• $\Gamma$ is maximised at-the-money and decays away from the money.
• $\Gamma \to \infty$ as $\tau \to 0$ with $S = K$: gamma spikes near expiry at-the-money (short-dated ATM options are extremely gamma-sensitive).

Trading interpretation: gamma income is the profit from rebalancing the delta hedge. A long gamma position earns $\frac{1}{2}\Gamma (dS)^2$ per unit time from spot moves — but pays theta to fund it.

Theta: $\Theta = \partial C / \partial t = -\partial C / \partial \tau$

$\Theta_{\mathrm{call}} = -\frac{S\,n(d_1)\,\sigma}{2\sqrt{\tau}} - rKe^{-r\tau}N(d_2).$

Derivation (using $\tau = T - t$, so $\partial C/\partial t = -\partial C/\partial\tau$): $\frac{\partial C}{\partial \tau} = S\,n(d_1)\frac{\partial d_1}{\partial \tau} - Ke^{-r\tau}\left(-r\,N(d_2) + n(d_2)\frac{\partial d_2}{\partial \tau}\right).$ Using the core identity $S\,n(d_1) = Ke^{-r\tau}n(d_2)$, the $\partial d/\partial\tau$ terms cancel, leaving: $\Theta_{\mathrm{call}} = -\frac{\partial C}{\partial \tau} = -\frac{S\,n(d_1)\sigma}{2\sqrt{\tau}} - rKe^{-r\tau}N(d_2).$

Sign conventions:

• $\Theta_{\mathrm{call}} < 0$ always (for $r > 0$): long calls lose time value.
• $\Theta_{\mathrm{put}} < 0$ for near-the-money puts; can be positive for deep ITM puts (reflecting that a very deep ITM put is almost equivalent to a bond position with positive carry).
• $|\Theta|$ increases as $\tau \to 0$: time decay accelerates near expiry.

The BS PDE as a P&L Identity

The Black-Scholes PDE can be written in Greek notation: $\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + rS\Delta - rC = 0.$

Rearranged: $\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma = r(C - S\Delta) = r \cdot (-Ke^{-r\tau}N(d_2)) < 0$.

Interpretation: theta and gamma income sum to the risk-free financing cost of the position. A long gamma position (positive $\Gamma$) must pay theta to hold it — this is the gamma-theta tradeoff.

Vega: $\nu = \partial C / \partial \sigma$

$\nu = S\sqrt{\tau}\,n(d_1).$

Derivation: $\frac{\partial C}{\partial \sigma} = S\,n(d_1)\frac{\partial d_1}{\partial \sigma} - Ke^{-r\tau}n(d_2)\frac{\partial d_2}{\partial \sigma}.$ Using $\partial d_1/\partial\sigma = -d_2/\sigma$ and $\partial d_2/\partial\sigma = \partial d_1/\partial\sigma - \sqrt{\tau} = -d_2/\sigma - \sqrt{\tau}$: $\frac{\partial C}{\partial \sigma} = S\,n(d_1)\left(-\frac{d_2}{\sigma}\right) - Ke^{-r\tau}n(d_2)\left(-\frac{d_2}{\sigma} - \sqrt{\tau}\right).$ Applying the core identity to the $-d_2/\sigma$ terms (they cancel), the surviving term is $Ke^{-r\tau}n(d_2)\sqrt{\tau} = S\,n(d_1)\sqrt{\tau}$. Hence: $\nu = S\sqrt{\tau}\,n(d_1).$

Properties:

• $\nu_{\mathrm{call}} = \nu_{\mathrm{put}}$: same for both (from put-call parity).
• $\nu > 0$: long options are long vega — they benefit from rising implied vol.
• $\nu$ is maximised at-the-money and decreases for deep ITM/OTM options.
• $\nu$ grows with $\sqrt{\tau}$: longer-dated options have more vega per unit spot.

Note on units: vega is usually quoted in price change per 1 vol point (i.e., per 0.01 change in $\sigma$ in decimal, or equivalently per 1% change in annualised vol). Check units carefully — mixing percent and decimal conventions is a frequent source of desk errors.

Rho: $\varrho = \partial C / \partial r$

$\varrho_{\mathrm{call}} = K\tau e^{-r\tau}N(d_2), \qquad \varrho_{\mathrm{put}} = -K\tau e^{-r\tau}N(-d_2).$

Derivation: $\frac{\partial C}{\partial r} = S\,n(d_1)\frac{\partial d_1}{\partial r} + K\tau e^{-r\tau}N(d_2) - Ke^{-r\tau}n(d_2)\frac{\partial d_2}{\partial r}.$ Since $\partial d_1/\partial r = \partial d_2/\partial r = \sqrt{\tau}/\sigma$, the $n$ terms cancel by the core identity, leaving $\varrho_{\mathrm{call}} = K\tau e^{-r\tau}N(d_2)$.

Materiality: rho is small for short-dated equity options (where $\tau \ll 1$) but material for long-dated options and for interest rate derivatives. For equity exotics with $\tau = 5$ years and $r = 4\%$, rho can dominate the Greeks P&L on an absolute basis.

Higher-Order Greeks

Beyond the five first-order Greeks, second-order cross-sensitivities are essential for managing options books, particularly for barrier options and structured products.

Vanna: $\partial^2 C / \partial S \partial \sigma$

$\mathrm{Vanna} = \frac{\partial^2 C}{\partial S \partial \sigma} = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial \nu}{\partial S} = -\frac{n(d_1)\,d_2}{\sigma}.$

Derivation: $\mathrm{Vanna} = \partial(N(d_1))/\partial\sigma = n(d_1)\cdot\partial d_1/\partial\sigma = n(d_1)\cdot(-d_2/\sigma)$.

Interpretation: vanna measures how the delta changes as implied vol moves. If $d_2 < 0$ (OTM call), vanna is positive: as vol rises, the call's delta increases (the option becomes more likely to expire ITM). For a delta-hedged book, a simultaneous move in $S$ and $\sigma$ creates a vanna P&L of vanna $\cdot \Delta S \cdot \Delta\sigma$.

Barrier options: for a knock-out barrier near the current spot, vanna can be very large — a small spot move near the barrier changes the option's delta dramatically, and simultaneously, implied vol changes amplify this.

Volga: $\partial^2 C / \partial \sigma^2$

$\mathrm{Volga} = \frac{\partial^2 C}{\partial \sigma^2} = \frac{\partial \nu}{\partial \sigma} = \frac{\nu \cdot d_1 d_2}{\sigma}.$

Derivation: $\mathrm{Volga} = \partial(S\sqrt{\tau}\,n(d_1))/\partial\sigma = S\sqrt{\tau}\,(-d_1\,n(d_1))\cdot\partial d_1/\partial\sigma = S\sqrt{\tau}\,n(d_1)\cdot d_1 d_2/\sigma = \nu \cdot d_1 d_2/\sigma$.

Interpretation: volga is the convexity of the option price in volatility. Long volga positions benefit from large vol moves in either direction. OTM options (where $|d_1|, |d_2|$ are large and of the same sign) have higher volga than ATM options (where $d_1, d_2 \approx 0$).

Charm: $\partial^2 C / \partial S \partial t = \partial \Delta / \partial t$

Charm (delta decay) measures how the delta changes with time. Near expiry, ATM options have rapidly changing delta — a delta-hedged position must be rebalanced more frequently.

P&L Decomposition

A complete P&L decomposition for a single option position over a time interval $[t, t+dt]$, including both first-order and second-order contributions, uses Itô's lemma:

$dC \approx \Delta\,dS + \frac{1}{2}\Gamma\,(dS)^2 + \Theta\,dt + \nu\,d\sigma + \mathrm{Vanna}\,dS\,d\sigma + \frac{1}{2}\mathrm{Volga}\,(d\sigma)^2.$

For a delta-hedged portfolio ($\Pi = C - \Delta \cdot S$, where the stock position is funded at rate $r$):

d\Pi = \underbrace{\frac{1}{2}\Gamma\,(dS)^2}_{\text{gamma P&L}} + \underbrace{\Theta\,dt}_{\text{time decay}} + \underbrace{\nu\,d\sigma}_{\text{vega P&L}} + \underbrace{\mathrm{Vanna}\,dS\,d\sigma + \frac{1}{2}\mathrm{Volga}\,(d\sigma)^2}_{\text{cross-Greek P&L}}.

Under Black-Scholes assumptions: $(dS)^2 = \sigma^2 S^2\,dt$, $d\sigma = 0$. The portfolio earns zero P&L on average — the gamma income exactly offsets theta decay. In practice:

$d\Pi_{\mathrm{actual}} = \frac{1}{2}S^2\Gamma\left(\sigma_R^2 - \hat{\sigma}^2\right)dt + \nu\,d\sigma + \mathrm{Vanna}\,dS\,d\sigma + \cdots,$

where $\hat{\sigma}$ is implied vol and $\sigma_R$ is realised vol. The first term is the gamma trading P&L (earn when realised vol exceeds implied), the second is the vega P&L (earn when implied vol moves in your favour), and the cross-Greek terms are the second-order vol P&L — significant for exotic options and barrier-heavy books.

Limitations

Constant vol assumption. Black-Scholes Greeks assume $\sigma$ is constant. In practice, $\Delta_{\mathrm{BS}}$ with a flat-smile implied vol is not the correct hedge ratio — a sticky-strike or sticky-delta delta adjustment is needed, depending on which smile dynamics are assumed. The correct hedge ratio in a local vol model differs from $N(d_1)$.

Discrete hedging. Analytical Greeks are instantaneous rates. Real hedging occurs at discrete intervals, creating a replication error proportional to $\frac{1}{2}S^2\Gamma((\Delta S)^2 - \sigma^2 S^2\,\Delta t)$ per step.

Jump risk. Under Black-Scholes, $\Delta$-hedging is exact (in the continuous limit). With jumps, a gap in $S$ creates an unhedgeable P&L of $\frac{1}{2}\Gamma J^2$ for a jump of size $J$. No hedge using only the underlying eliminates jump risk.

Model dependence. Greeks are model outputs. A position that is delta-neutral under Black-Scholes is not delta-neutral under Heston or SABR. The hedge ratio is model-dependent; using the wrong model produces systematic hedging errors.