The Options Greeks, Explained: Delta, Gamma, Theta, Vega (and Rho)

The Greeks are how options traders measure risk. Each one answers a single question: how much does this option's price move when the stock moves, when time passes, when volatility changes, or when interest rates shift? Delta, gamma, theta, vega, and rho between them describe almost everything an option can do — and once you can read them, you can see exactly which forces are working for you and which are working against you.

This matters most when you're selling options, because the Greeks flip sign. The same time decay that bleeds an option buyer pays the seller. Here's each Greek in plain terms, then what they look like when you sell a cash-secured put.

What the Greeks actually are

An option's price depends on a handful of inputs: the stock price, the strike, time to expiry, volatility, and interest rates. The Greeks are the sensitivities — the rate of change of the option's value with respect to each input, holding the others fixed.

Two things to keep in mind throughout. They're local: accurate for small moves, but they drift as conditions change, so for a big move you re-price rather than extrapolate. And they're additive: the net Greeks of a multi-leg position are just the sum of the legs (times position size and the ×100 contract multiplier).

Delta (Δ) — exposure to the stock

Delta is the option's sensitivity to a $1 move in the underlying. A call's delta sits between 0 and 1; a put's between −1 and 0.

The cleanest intuition is share-equivalent exposure: a 0.60-delta call behaves like owning about 60 shares per contract. At the money, delta is roughly ±0.5; deep in the money it approaches ±1 (the option moves dollar-for-dollar with the stock); far out of the money it approaches 0.

Delta has a second, very common reading: it's used as a rough proxy for the risk-neutral probability that the option finishes in the money. A 0.10-delta option is treated as having loosely a 10% chance of expiring in the money — which is why a put sold at 0.10 delta is one you'd expect to keep, premium and all, most of the time.

Treat that as a convention rather than a fact. Delta is N(−d₁) and the actual probability of finishing in the money is N(−d₂), and the second is always the larger. On a quiet stock the difference is under a percentage point. On a high-volatility name it can be ten points or more — and it always runs against the seller. Delta is not your chance of assignment works through why, and how wide the gap gets.

Gamma (Γ) — how fast delta changes

Gamma is the rate of change of delta as the stock moves — the option's second derivative with respect to price. It's always positive for a long option, peaks at the money, and explodes as expiry approaches for at-the-money options (the origin of "gamma week").

Gamma is convexity. If you're long gamma, big moves in either direction help you, because your delta grows in your favour as the stock runs. If you're short gamma, the opposite: your losses accelerate as the move goes against you. Gamma is where the asymmetry of an options position lives, and it's the Greek most likely to surprise someone who only watched delta.

Theta (Θ) — the passage of time

Theta is time decay — how much value the option loses for each day that passes, all else equal. It's negative for a long option (the buyer pays for time) and accelerates as expiry nears for at-the-money options.

By convention theta is often quoted per year; divide by 365 for a per-day figure. The sign is the whole story for income traders: if you own options, time is a cost; if you sold them, time is your paycheck. Every day the position you sold gets a little cheaper to close, and that drift is yours.

Vega (𝜈) — sensitivity to volatility

Vega is how much the option's price changes when implied volatility changes. It's the same sign for calls and puts — always positive for a long option — and it's largest for at-the-money, longer-dated options.

Vega is quoted per 1.00 (i.e. 100 volatility points) change in implied vol; divide by 100 for the more practical "per vol-point" number. If you're long options you're long vega — you want implied volatility to rise. If you sold options, you're short vega: a spike in implied volatility (which tends to come exactly when markets get scary) increases the value of what you sold, and works against you.

Two things about vega are easy to miss and matter a great deal to a seller: it grows as the underlying falls toward your strike, and it barely diversifies across a book, because implied vol on unrelated names moves together in a sell-off. Option vega and portfolio vega works through both with the numbers.

Rho (ρ) — sensitivity to interest rates

Rho measures sensitivity to the risk-free rate. Calls have positive rho, puts negative. It's usually the least important Greek — it only really matters for long-dated options (LEAPS) or in a high-rate environment. Worth knowing it exists; rarely the thing that decides a trade.

There are higher-order Greeks too — vanna (how delta shifts as volatility moves), vomma (the convexity of vega), charm (delta decay over time) — but for most positions, delta, gamma, theta, and vega carry the load.

The five at a glance

Greek	Measures sensitivity to	Sign if you own the option	Sign if you sold it
Delta (Δ)	Stock price	Call +, Put −	Call −, Put +
Gamma (Γ)	Delta's rate of change	Positive	Negative
Theta (Θ)	Time passing	Negative (a cost)	Positive (income)
Vega (𝜈)	Implied volatility	Positive	Negative
Rho (ρ)	Interest rates	Call +, Put −	Call −, Put +

The central tension: time versus convexity

Almost every options position lives on one trade-off: long gamma and short theta, or short gamma and long theta.

Buy options and you own convexity — big moves pay you — but you bleed time decay every day waiting for them. Sell options and you collect time decay every day, but you're short convexity: a large adverse move hurts disproportionately. There's no free lunch. You're either paying theta to own gamma, or collecting theta to be short it. Every income strategy is a version of the second choice.

What the Greeks look like when you sell a cash-secured put

Selling a cash-secured put — the first half of the wheel — puts you on the income side of every one of these:

• Positive delta. You profit if the stock rises or simply holds. Sell a put at 0.10 delta and you're carrying about +10 deltas — like being long ~10 shares, plus the premium.
• Positive theta. Time decay is your income engine. Each day the put you sold is worth a little less to buy back, and that's the return you're harvesting.
• Negative vega. A jump in implied volatility raises the value of the put you're short — it hurts. This is why selling into already-elevated volatility, then watching it fall, is the seller's friend, and why a volatility spike after you've sold is the headwind.
• Negative gamma. This is the one to respect. As the stock falls toward your strike, your losses accelerate — the position gets more negative-delta the worse it gets. Short gamma is precisely why a put seller's downside isn't linear, and why the premium feels small next to a bad week. It's also why, on a portfolio-margin account, your margin requirement rises in a sell-off just as your account value falls — the same short vega and short gamma, priced across your whole book at once.

Read together, the Greeks tell you the honest shape of the trade: you're being paid by time (theta) to accept a capped reward and a convex, accelerating downside (short gamma) on a stock you've agreed to buy. That's a perfectly sensible bargain — on a business you'd want to own at the strike. It's a poor one on a name you're only there for the premium, because short gamma will find you.

This is the layer levelbox.ai is built to make legible: for each cash-secured-put candidate we show the delta (your rough odds of assignment), the premium and annualised yield (your theta, capitalised), the breakeven, and the maximum loss — so the Greeks aren't abstract letters but the actual risk and reward of the position in front of you. Analytical, educational tooling — not investment advice.