Greeks Explorer

Options are contracts that give the right to buy (call) or sell (put) a stock at a fixed price (the strike) before a deadline (expiry). The Greeks measure how an option's value changes as the market moves — delta tracks the stock price, gamma tracks how fast delta itself changes, theta measures daily time decay, and vega measures sensitivity to volatility.

Drag the sliders. Watch how each Greek changes shape across stock price. Build intuition for what hedgers actually feel as the market moves.

Option type

Stock price S 100.00

Strike K 100.00

Time to expiry T (years) 0.50

Volatility σ 0.200

Risk-free rate r 0.050

Not sure what ATM / ITM / OTM means? Expand for the 30-second glossary.

ATM (at the money): Stock price S ≈ strike K. The option is right on the fence between paying off and not.
ITM (in the money): Call: S > K (already profitable if exercised). Put: S < K.
OTM (out of the money): Call: S < K. Put: S > K. The option only pays off if the stock moves significantly.
Implied volatility (σ): The market’s consensus forecast of how much the stock will move annually, expressed as an annualized standard deviation. A σ of 0.20 means ±20% moves in a year are considered “one sigma.”
Greeks: The partial derivatives of the option price with respect to each input (S, σ, T, r). They tell a hedger exactly how much exposure they have to each risk factor.

Δ Delta

How much the option price moves per $1 move in the stock.

ATM call ≈ 0.5 (option copies half the stock move); deep ITM → 1 (option tracks the stock dollar-for-dollar). Traders use it to stay neutral: short 1 call + hold Δ shares = hedged position.

Γ Gamma

How fast Delta itself changes as the stock price moves.

How fast delta itself changes when the stock moves — the “bang per buck” of a stock move. Peaks ATM; same for calls and puts. Traders use it: long gamma = profits from big moves; short gamma = bleeds if the stock gaps. Gamma-theta is the core tension in options market-making.

ν Vega

How much the option price moves per 1% increase in implied volatility. (Not an actual Greek letter — named after the star Vega.)

How much the option price moves per full-unit change in implied vol (σ). Peaks ATM; scales with sqrt(T). Traders use it: if you expect vol to spike (VIX crush), buy high-vega options. Vol desks PnL is largely marked-to-vega each day.

Θ Theta

How much the option price decays each day from time passing alone.

Daily time decay — the option loses value just by sitting there. Most negative ATM, accelerates near expiry. Traders use it: short-options strategies (covered calls, cash-secured puts) collect theta daily; long options pay it. Theta ≈ −Gamma × S² × σ² / 2 at ATM.

At S = 100.00

Price

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ν (per 1.00 σ)

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Θ (per year)

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Units: vega is quoted per 1.00 unit of σ (one full unit = 100 vol-points; divide by 100 for per-1% change). Theta is per calendar year (divide by 365 for per-day decay).

Intuition cards

Why is gamma maximized at ATM?

Think of delta as a slider that goes from 0 (option is worthless and stock moves don't affect it) to 1 (option tracks the stock dollar-for-dollar). Gamma is how fast that slider moves when the stock price changes.

The slider moves fastest in the middle — right around the strike price — because that's where the option is right on the edge of being worth something or not. A small stock move there can flip the option from "almost certainly expiring worthless" to "almost certainly worth exercising." Far away from the strike (deep OTM or deep ITM) the slider is already stuck at one extreme and barely budges, so gamma is near zero.

That's why gamma peaks at ATM and falls off on either side like a bell curve. Near-expiry options have a sharper, taller peak; longer-dated options have a flatter, wider one.

Why does theta accelerate near expiry?

Time value ≈ volatility × sqrt(T). The square root means most of the value bleeds out in the last stretch of life, not evenly across it. With one year left, going from T=1.00 to T=0.99 costs you almost nothing (sqrt(0.99) ≈ 0.995). With one day left, going from T=1/365 to T=0/365 costs you the entire remaining time premium. ATM options feel this hardest because they’re pure time value (no intrinsic). Pull the T slider toward 0 and watch the theta curve sharpen into a deep V.

Why is vega highest for long-dated, near-ATM options?

Vega = S · φ(d₁) · sqrt(T). It scales with sqrt(T): more time = more chance for vol to matter. The φ(d₁) factor is a normal density, peaking when d₁ ≈ 0 — i.e. near ATM. Deep ITM/OTM options are already “decided” — extra vol can’t change their fate much, so vega collapses. ATM options are maximally undecided, so a small change in σ shifts the probability of ending up ITM the most. Crank T up and slide S to K: the vega chart bulges most under those conditions.

How do traders use all four Greeks together?

A market-maker selling options faces a gamma-theta trade-off: they are short gamma (they lose money on big moves) and long theta (they collect time premium each day). The daily theta income is their compensation for taking on the gamma risk. The Black-Scholes relationship is exact ATM: Theta = −½ Γ · S² · σ².

A delta-neutral position is constructed by holding Δ shares against each short call — so the position has zero first-order sensitivity to S. But it still has gamma and vega exposure. Rehedging every time S moves costs transaction fees; the market-maker’s PnL is the difference between the implied vol they sold at and the realised vol over the option life.

Vega hedging requires trading other options (you can’t hedge vol risk with the underlying). Vol desks build vega-neutral books by offsetting long and short positions across maturities and strikes, keeping their net vega near zero while running large delta and gamma books.