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