The Greeks: What Option Risk Actually Feels Like

The Greeks are partial derivatives of an option's price — but memorizing that gets you nothing. What interviews (and desks) require is the system: how the Greeks trade off against each other, and what a position "feels like" through them.

Delta: the hedge ratio

$\Delta = \partial C / \partial S$: how much the option price moves per $1 of stock move. For calls, between 0 and 1 — near 0 deep out-of-the-money, near 1 deep in-the-money, about 0.5 at-the-money. Working facts:

• Delta is the replicating share position: a market maker short 100 calls at $\Delta = 0.4$ buys 40 shares (per contract-share) to neutralize small moves.
• Delta doubles as a rough probability of expiring in the money — imprecise but the standard desk shorthand ("a 25-delta put").
• Portfolio deltas add; "delta-neutral" means first-order immune to spot.

Gamma: how fast your hedge rots

$\Gamma = \partial \Delta / \partial S$: the curvature. Gamma is largest at-the-money near expiry — where delta lurches between 0 and 1 on small moves — and it determines how often a delta hedger must rebalance.

The crucial sign convention: long options = long gamma. Your delta increases as the market rallies and decreases as it falls — meaning your hedge adjustments systematically buy low and sell high. Rebalancing a long-gamma book manufactures money from movement. Short gamma is the mirror: hedging forces you to buy rallies and sell dips, bleeding money whenever the market moves.

Theta: rent

$\Theta = \partial C / \partial t$: price decay per day, negative for long options. An at-the-money option's value shrinks like $\sigma\sqrt{T}$ (time value ∝ square root of remaining time), so decay accelerates into expiry.

Theta is not a flaw; it's rent paid for gamma. That's the relationship that organizes everything:

$$\text{long options} = \text{long gamma, pay theta} \qquad \text{short options} = \text{short gamma, collect theta}$$

The Black–Scholes PDE makes this exact: for a delta-hedged position, $\Theta + \tfrac{1}{2}\sigma^2 S^2 \Gamma = rC - rS\Delta \approx 0$, i.e. expected gamma gains = theta paid when realized volatility equals implied. Trading options while delta-hedged is therefore a bet on realized vs. implied volatility, not on direction. Say that sentence in an interview and the interviewer relaxes.

Vega: the exposure to the price of risk itself

$\text{Vega} = \partial C / \partial \sigma$: sensitivity to implied volatility. Largest at-the-money and — unlike gamma — for longer-dated options ($\text{vega} \propto S\sqrt{T}\,\phi(d_1)$). The gamma/vega contrast matters: a short-dated ATM option is a gamma instrument (bet on movement now); a long-dated one is a vega instrument (bet on the market's repricing of future movement). Vol traders structure books along exactly this axis — e.g. calendar spreads isolate vega from gamma.

What a position feels like

The Greeks turn any structure into an experience you can narrate:

• Long straddle: delta ≈ 0, long gamma, long vega, paying heavy theta. You want a big move or a vol spike, soon. Every quiet day costs rent.
• Short OTM puts: collect theta, short gamma, short vega, positive delta. Feels like steady income — until a selloff, when your delta grows into the fall and vol spikes against you simultaneously. This "picking up pennies in front of a steamroller" profile is the standard example of hidden tail risk.
• Covered call: long stock, short call = short gamma with a return ceiling; you've sold the upside to fund income.

The interview version

Classics, in the order they usually escalate: "You're delta-hedged and the market gaps 5% — long or short gamma, which do you want to be?" (Long: your hedge adjustments profit from the gap; the short-gamma hedger locks in a loss.) "Why is ATM gamma huge near expiry?" (Delta must travel 0→1 over a shrinking price band.) "Implied vol is 20, you think realized will be 25 — trade?" (Buy options, delta-hedge; collect gamma P&L exceeding theta.) All three are the same idea — the gamma–theta engine — approached from different doors.