Option Greeks: delta, gamma, theta, vega and rho with examples

The Greeks are a set of numbers that describe how an option's price responds when one of its inputs changes. Delta measures sensitivity to the stock price, gamma how fast delta itself changes, theta the effect of one day passing, vega the effect of a change in implied volatility, and rho the effect of interest rates. Together they turn a vague sense of "this option is risky" into specific dollar amounts.

They matter most once you hold more than one option, or options alongside stock. Two positions that cost the same can behave completely differently: one might lose $40 a day to time decay and barely react to the stock, another might act like 80 shares of stock and ignore the calendar. The Greeks are how traders, market makers and risk managers see that difference before it shows up in the account.

Every number on this page was computed with the Black-Scholes model for a non-dividend stock, using the same inputs throughout so you can follow how each Greek is derived and check it against a re-priced option. The inputs are illustrative.

The worked example used on this page

All the examples below refer to one pair of options. Keep these inputs in mind:

Black-Scholes values them and their Greeks as follows (per share; multiply by 100 for one contract):

The rest of this page explains each row and checks it against a full re-pricing.

Delta: exposure to the stock

Delta is the change in the option's price for a $1 change in the stock, with all other inputs held still. Calls have deltas between 0 and 1; puts between −1 and 0.

There are three useful ways to read delta.

1. As a hedge ratio. A delta of 0.546 means one call moves like 54.6 shares. A market maker who sells that call can offset the stock exposure by buying about 55 shares. This is delta hedging, and it is how dealers who sell options manage their risk.
2. As share equivalence. Add up delta × 100 × contracts across a portfolio, plus any shares, and you get the position delta: how many shares the whole book behaves like.
3. As a rough probability. Traders often treat a call's delta as an approximate chance of finishing in the money. A 0.20-delta call is commonly described as having about a one in five chance. This is a shorthand, not an exact probability, and it is a little high for calls in the model.

The chart shows how delta depends on moneyness and time. Deep out-of-the-money calls have deltas near 0; deep in-the-money calls approach 1. At the strike, delta is near 0.5. The time dimension changes the shape. With 180 days left the curve is gentle: a $110 stock gives a delta of about 0.74. With 7 days left it is almost a step: the same $110 stock gives about 0.99, because there is little time for the stock to fall back below the strike.

Put deltas follow the same curve shifted down by one: for a non-dividend European option, put delta = call delta − 1, which is why the example put has a delta of −0.454.

Gamma: how fast delta changes

Gamma is the change in delta for a $1 move in the stock. A call and a put with the same strike and expiration have the same gamma. Buyers of options have positive gamma; sellers have negative gamma.

Gamma is the reason delta is only a local estimate. With the example call's gamma of 0.039, a move from $100 to $101 lifts delta from 0.546 to about 0.585, and the re-priced option confirms it (0.5846). After a $5 rally the call behaves like roughly 72 shares instead of 55.

Gamma concentrates where the outcome is most in doubt: at the money, near expiration. With 30% volatility, an at-the-money $100 call has a gamma of about 0.019 with 180 days left, 0.046 with 30 days and 0.096 with 7 days. Near expiration, small moves across the strike can swing a position from nearly no exposure to nearly full exposure.

Theta: the cost of a day

Theta is the change in the option's price from one day passing, with the stock price, volatility and rate unchanged. Models report it per year or per day; the figures here are per calendar day.

Theta and gamma are two faces of the same trade-off. A position with positive gamma (bought options) pays for it with negative theta: it gains from large moves and bleeds slowly while waiting. A position with negative gamma (sold options) collects theta but is hurt by large moves. For at-the-money options the two are tied together in the model, so there is no free way to have positive gamma and positive theta at once.

Theta also grows as expiration nears. The Option premium and time value: decay, volatility and Black-Scholes page shows that curve in full.

Vega: exposure to implied volatility

Vega is the change in the option's price for a 1 percentage point change in implied volatility (for example, from 25% to 26%). Calls and puts with the same strike and expiration have the same vega. Buyers have positive vega; sellers have negative vega.

Vega is largest for at-the-money options and grows with time to expiration: long-dated options are far more sensitive to volatility than short-dated ones, while short-dated options are more sensitive to the stock (gamma) and the clock (theta). Strictly, vega is not a Greek letter, but the name has stuck.

Rho: exposure to interest rates

Rho is the change in the option's price for a 1 percentage point change in the interest rate. Calls have positive rho and puts negative, because owning a call instead of the stock lets you keep the cash earning interest until you exercise.

For short-dated options rho is minor, because rates rarely move a full point in weeks. It matters more for long-dated contracts, such as options that run for a year or more, and for large books of them.

Putting the Greeks together

Because each Greek is a sensitivity per unit of change, you can combine them to estimate what happens when several things change at once. This is a first-order estimate, good for small moves.

The same arithmetic works for a whole portfolio: multiply each Greek by 100 and by the number of contracts (negative for short positions) and add them up. Shares have a delta of 1 each and no other Greeks. A covered call, for example, has the stock's delta of +100 minus the short call's delta, positive theta and negative vega.

A less sensitive option for contrast

The same model gives very different Greeks for an out-of-the-money option. A $110 call with the stock at $100, 60 days left and 25% volatility is worth $1.11 and has:

• delta 0.205, so it acts like about 20 shares per contract;
• gamma 0.028;
• theta −0.026, so it loses about $2.61 a day;
• vega 0.115.

Measured against its price, that option is far more fragile than the at-the-money one. Its daily decay is 2.3% of its value ($0.026 ÷ $1.11), against 0.9% for the at-the-money call ($0.039 ÷ $4.36). And a one-point drop in IV takes 10% of its value, against 3.7% for the at-the-money call.

Pitfalls

Treating Greeks as constants. Every Greek changes as the stock, time and volatility change. A delta of 0.30 today can be 0.70 after a rally; a theta of −$2 a day can be −$8 in the final week.

Mixing units. Some platforms quote theta per year, vega per 1.00 of volatility (100 points) or Greeks per contract instead of per share. Check the units before adding positions together.

Using delta as a promise. Delta's probability reading is a model-based approximation. It says nothing about what a particular stock will do.

Ignoring model choice. The figures here come from Black-Scholes. Brokers using models that handle early exercise and dividends can report slightly different numbers for the same option.

How Finabase helps

Finabase option chains, for example AAPL options, list each contract's bid, ask, last price, change, volume, open interest and implied volatility, with calls on the left, strikes in the center and puts on the right, and every listed expiration one click away. The chain does not display Greeks. Its implied volatility column is the input you would feed a calculator like the one used on this page, together with the stock price shown on the chain, the strike, the days to the expiration you selected and an interest rate. Quotes come from a data provider and may be delayed.

Frequently asked questions

What is a good delta for buying a call?

There is no correct delta; it is a trade-off. Higher-delta calls cost more and behave more like the stock; lower-delta calls are cheaper, need a larger move and expire worthless more often.

Why is gamma highest just before expiration?

Because an at-the-money option's fate is decided by small moves at the end. With little time left, a few cents above or below the strike separates an option that becomes 100 shares from one that expires worthless, so delta can swing from near 0 to near 1 very quickly.

Is theta the same every day?

No. For an at-the-money option theta grows larger as expiration approaches, and it changes when the stock moves away from the strike or volatility shifts.

Do the Greeks work for American options?

Yes, as concepts. The exact numbers come from the model used. For options where early exercise can matter, such as puts or calls before a dividend, brokers use models that allow for it, and their Greeks will differ slightly from Black-Scholes values.

Summary

The Greeks measure how an option's price responds to each input: delta to the stock, gamma to changes in delta, theta to the passage of a day, vega to implied volatility and rho to interest rates. They are local, per-unit estimates that shift as markets move, but they add up across positions, which makes them the standard tool for seeing what a mix of options and shares will do before the market does it.