The Option Greeks

The Greeks are the partial derivatives of an option's value with respect to its inputs, spot, volatility, time, and rates. They are how a trading desk actually thinks: not "what is this option worth" but "what happens to my book when the underlying moves 1%, or vol jumps a point, or a day passes." Mastering their signs, their formulas, and above all the gamma-theta tradeoff is the difference between pricing options and risk-managing them.

Delta, sensitivity to spot

$\Delta = \frac{\partial V}{\partial S}, \qquad \Delta_{\text{call}} = N(d_1),\quad \Delta_{\text{put}} = N(d_1) - 1.$

Delta is the hedge ratio: hold $-\Delta$ shares against one long option to be first-order neutral. A call's delta runs from $0$ (deep OTM) to $1$ (deep ITM), crossing $\approx 0.5$ near the forward. It doubles as the risk-neutral, forward-measure probability of finishing in the money and as the number of shares the replicating portfolio holds, the same $N(d_1)$ that appears in the Black-Scholes price.

Gamma, curvature

$\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{n(d_1)}{S\sigma\sqrt T},$

where $n(\cdot)$ is the standard normal density. Gamma measures how fast delta changes as spot moves, so it is the convexity of the position. It is positive for long options (calls and puts alike), peaks near the money, and spikes sharply as a short-dated option approaches expiry ATM. High gamma means your delta hedge goes stale fast, you must rebalance often. Gamma is where the real action of a hedged book lives (see delta-hedging P&L).

Vega, sensitivity to volatility

$\mathcal{V} = \frac{\partial V}{\partial \sigma} = S\,n(d_1)\sqrt T.$

Vega (not a Greek letter, but universal) is the P&L per one-point change in volatility. It is positive for all long options, largest for at-the-money and long-dated options (it scales with $\sqrt T$), and it is the sensitivity a vol trader is really taking a view on. A long straddle is a pure long-vega, long-gamma position.

Theta, time decay

$\Theta = \frac{\partial V}{\partial t} = -\frac{S\,n(d_1)\sigma}{2\sqrt T} - rKe^{-rT}N(d_2)\quad(\text{call}).$

Theta is the value bled per unit time. It is negative for long options (owned options decay), most punishing for short-dated ATM options, exactly where gamma is highest. Theta is the rent you pay to be long gamma.

Rho, sensitivity to rates

$\rho = \frac{\partial V}{\partial r} = KT e^{-rT}N(d_2)\quad(\text{call}),\qquad -KTe^{-rT}N(-d_2)\quad(\text{put}).$

Usually the least important Greek for short-dated equity options, but it dominates for long-dated options and rates products.

The gamma-theta tradeoff, read off the PDE

The deepest relationship among the Greeks is not a coincidence; it is the Black-Scholes PDE rewritten in Greek notation. Starting from

$V_t + \tfrac12\sigma^2 S^2 V_{SS} + rS V_S - rV = 0,$

substitute $V_t = \Theta$, $V_{SS} = \Gamma$, $V_S = \Delta$:

$\boxed{\,\Theta + \tfrac12\sigma^2 S^2\Gamma + rS\Delta - rV = 0.\,}$

For a delta-hedged position ($\Delta = 0$) with negligible rates, this collapses to

$\Theta \approx -\tfrac12\sigma^2 S^2 \Gamma.$

Theta and gamma have opposite signs and are locked together: you cannot be long gamma without paying theta, and short gamma earns theta. A long-option book profits from realized moves (gamma) but bleeds with time (theta); the two exactly offset when the stock moves at implied volatility. This is the mechanical heart of option trading, see the hedging P&L equation.

Second-order Greeks

Real desks watch the cross-derivatives, especially where the smile matters:

• Vanna $= \dfrac{\partial^2 V}{\partial S\,\partial\sigma}$, how delta moves as vol changes (or vega as spot changes). Central to skew hedging and risk-reversal pricing.
• Volga (vomma) $= \dfrac{\partial^2 V}{\partial\sigma^2}$, convexity in vol; long for OTM options and the reason the smile has curvature. Wing options are long volga.
• Charm $= \dfrac{\partial^2 V}{\partial S\,\partial t}$, delta decay; matters for hedges held over weekends and into expiry.

Worked example

One-year ATM call, $S = 100$, $\sigma = 20\%$, $r = 0$: $d_1 = 0.1$, $n(d_1) = 0.397$. Then $\Delta = N(0.1) = 0.54$, $\Gamma = 0.397/(100\times0.2\times1) = 0.0199$, $\mathcal V = 100\times0.397\times1 = 39.7$ (per unit $\sigma$, so $\approx 0.40$ per vol point), and $\Theta \approx -\tfrac12(0.2)^2(100)^2(0.0199) = -3.97$ per year $\approx -0.016$ per calendar day. Check the tradeoff: $\Theta = -\tfrac12\sigma^2 S^2\Gamma$ holds exactly by construction. If the stock realizes a 1% move ($\Delta S = 1$), the gamma P&L is $\tfrac12\Gamma(\Delta S)^2 = \tfrac12(0.0199)(1) = 0.010$, you need about a 1% daily move to cover the theta.

What breaks in practice

• Greeks are model outputs. They are computed in the Black-Scholes world at a chosen $\sigma$. With a smile, the "true" delta includes a vanna term (delta shifts because vol shifts with spot), so sticky-strike vs sticky-delta assumptions change your hedge. Naive BS delta systematically mis-hedges skewed books.
• Discrete hedging and gaps. Gamma P&L assumes small moves; a jump delivers a loss no delta hedge catches, and short-gamma books blow up on gaps.
• Aggregation illusions. A book can be delta-neutral yet dangerously short gamma or vega; net Greeks hide the tail. Pin risk near expiry (gamma exploding at the strike) is a classic hazard.

In interviews

Know the signs cold: long options are long gamma, long vega, short theta; the tradeoff is $\Theta \approx -\tfrac12\sigma^2 S^2\Gamma$ for a delta-hedged book, which you should derive from the PDE. Expect "what's the delta of an ATM call?" ($\approx 0.5$, slightly above because of $d_1$), "where is gamma/vega largest?" (ATM; gamma for short-dated, vega for long-dated), and "why does a delta-hedged long option still make or lose money?", the answer is gamma versus theta, i.e. realized versus implied volatility. A strong candidate connects $\Delta = N(d_1)$ back to its dual role in the Black-Scholes formula.