What Are the Options Greeks? Delta, Gamma, Theta, Vega and Rho Explained

The options greeks measure how an option's price responds to its inputs: delta and gamma to the underlying, theta to time, vega to volatility and rho to rates.

An option's price moves for more reasons than the stock under it. Time passes, implied volatility shifts, interest rates change, and each of those reprices the contract on its own. The greeks put a number on each of those sensitivities, one input at a time, so they can be read and added up. This page covers the five main greeks, a worked example that checks each one by repricing the same contract, the second-order greeks that describe how the first ones change, how greeks become dealer exposure figures, and what the numbers leave out. It is descriptive throughout.

What the greeks measure

If calls and puts are new to you, start with what a call option and a put option are. A pricing model takes five inputs and returns a value: the underlying price, the strike, the time left to expiry, the volatility, and the interest rate. Each greek is the change in that value when one input moves and the other four hold still. In calculus terms each one is a partial derivative, which is why the formulas below come straight out of the model.

The house model on Kresmion's options pages is closed form Black-Scholes with a zero dividend yield, so the formulas here use the notation of the delta and theta explainers: S for spot, K for the strike, T for time to expiry in years, s for volatility as a fraction, r for the risk free rate, N for the standard normal cumulative function and phi for its density, with

``` d1 = (ln(S/K) + (r + s^2/2)T) / (s sqrt(T)) d2 = d1 - s sqrt(T) ```

Greeks are quoted per share. A standard US equity option covers 100 shares, so a contract's sensitivity is the per share figure times 100.

Delta: the response to the underlying

Delta is how much the option's price changes for a one dollar move in the underlying. In this model a call's delta is N(d1), which lands between 0 and 1, and a put's delta is N(d1) minus 1, which lands between -1 and 0. A call gains when the underlying rises and a put loses, so the two sides carry opposite signs.

Delta is also read as a share equivalent: a call with a delta of 0.50 behaves, for small moves, like 50 shares per contract. That is the quantity a market maker offsets to stay flat, and the full treatment, including how a book-wide figure is built and signed, is in dealer delta exposure.

Gamma: how fast delta changes

Gamma is the change in delta for a one dollar move in the underlying:

``` gamma = phi(d1) / (S s sqrt(T)) ```

The formula has no term that knows whether the contract is a call or a put, so a call and a put at the same strike and expiry carry the same gamma. It is largest for contracts near the money, and for those contracts it grows as expiry approaches. Gamma is the reason a delta hedge does not stay correct: after the underlying moves, every delta in the book has shifted, and gamma is the size of that shift. Aggregated across a chain it becomes dealer gamma exposure, and the case where that hedging feeds a price move is a gamma squeeze.

Theta: what one day does

Theta is the change in an option's value when time passes and nothing else moves. The closed form gives theta per year, so a per day figure is theta divided by 365, which is the convention Kresmion publishes. For contracts near the money theta is negative and grows as expiry nears, because the time value left in the contract has fewer days to run off.

Of the five greeks on this page, theta and rho are the ones apart from delta whose call and put sides differ, and for theta the gap is exactly the discounted financing term on the strike. That is why a deep in the money European put can carry positive theta when rates are positive, a case worked through in what theta is.

Vega: the response to volatility

Vega is the change in the option's value when implied volatility changes:

``` vega = S phi(d1) sqrt(T) ```

That expression is per 1.00 of volatility, which is a move of one hundred volatility points, so screens quote vega per one point: the formula times 0.01. The Options Industry Council defines it the same way, as the premium change for a 1% change in the implied volatility assumption. Like gamma, vega is the same for a call and a put at the same strike and expiry. Unlike gamma, for contracts near the money it grows with time to expiry, because a volatility change matters more when there is more time for it to act. Volatility itself is covered in what implied volatility is.

Rho: the response to interest rates

Rho is the change in the option's value when the risk free rate changes:

``` rho_call = K T e^{-rT} N(d2) rho_put = -K T e^{-rT} N(-d2) ```

As with vega, the closed form is per 1.00 of rate, so a per percentage point figure is the formula times 0.01. A higher rate raises a call's value and lowers a put's, which the Options Industry Council states as rho being positive for purchased calls and negative for purchased puts. The T in the formula is why rho matters mostly for long-dated contracts. With the inputs used in the worked example below, an at the money call has a rho of about 0.04 per point with 30 days left and about 0.49 with a year left. Kresmion's greeks suite does not compute rho exposure.

One contract, all five greeks

The table uses illustrative inputs, not market data: a stock at 100, a strike of 100, 30 calendar days to expiry over a 365-day year, implied volatility of 25%, a 4% rate and no dividend. Figures are per share.

Each line can be checked by repricing the contract with one input changed. Move the stock from 100 to 101 and the model prices the call at 3.58, up 0.56: the delta of 0.533 plus half the gamma, 0.028, accounts for it to the cent. One day later, everything else unchanged, the call is worth 2.97, about 0.053 less. At 26% volatility instead of 25% it is worth 3.14, about 0.114 more. At a 5% rate instead of 4% it is worth 3.06, about 0.04 more. For one contract of 100 shares each figure is multiplied by 100, so the call's theta is about $5.30 a day.

The check also shows what the greeks are: local, linear readings taken at one point. Delta is 0.533 at a price of 100 and about 0.587 at 101, which is gamma at work. Over a large move, or a long stretch of time, the single numbers in the table stop describing the contract and have to be recomputed.

Second-order greeks

Each first-order greek moves too, and the greeks of the greeks have their own names. Gamma is the first of them, delta's response to spot. The others most often quoted:

• Vanna: the change in delta for a change in implied volatility, identically the change in vega for a change in spot. See vanna exposure.
• Charm: the change in delta as time passes, also called delta decay. See what charm is.
• Vomma: the change in vega for a change in implied volatility, covered with vanna.
• Speed and zomma: third-order terms, the change in gamma for a change in spot and in implied volatility respectively.

They matter most to anyone holding a hedged book, because they describe how the hedge drifts out of line without a trade being made.

From one contract to a dealer book

A single contract's greeks are small. Added up across every strike and expiry listed on one underlying, weighted by open interest, they become exposure figures: dealer gamma exposure (GEX), delta exposure (DEX), vega and vanna exposure, charm and theta exposure. The SPY panel on this page, when a recent capture is available, shows the gamma version, and Kresmion's greeks exposure tool shows the gamma snapshot to everyone, while the rest of the suite, for SPY and every other covered name, opens with a free account.

Those figures are modeled, not measured. Kresmion computes them nightly for a curated universe of US listed names from the end-of-day options chain and prior-session open interest, with closed form Black-Scholes and a stated assumption about which side dealers hold. Open interest does not say who bought and who sold, so the sign of each figure rests on that assumption, and the tool says so in its assumptions block.

What the greeks do not tell you

The greeks are outputs of a model, so they inherit its assumptions. Black-Scholes assumes one volatility for the life of the contract and European exercise, and this version assumes no dividend. Listed US equity options are American style and many underlyings pay dividends, so a broker's greeks for the same contract can differ from the numbers above. Two platforms can also quote the same contract with theta a factor of 365 apart, or vega a factor of 100 apart, purely through units.

They are also readings at an instant. A greek describes what happens for a small change in one input with the rest held still, and markets rarely move one input at a time: a sharp drop in a stock often comes with a rise in implied volatility, so the value moves through vega as well as delta, and delta itself moves through vanna as well as gamma. And no greek says which way the underlying will go. They describe how a price responds to inputs, not what the inputs will do.

Key takeaways

Frequently asked questions

What are the five main options greeks?

Delta, gamma, theta, vega and rho. Delta measures the option's response to the underlying price, gamma how fast delta changes, theta the effect of one day passing, vega the response to implied volatility, and rho the response to interest rates. Gamma is technically second order, but it is grouped with the other four because it governs how delta hedges have to change.

Which greek matters most?

It depends on the contract and the question. Delta dominates the day to day change in value of most options, gamma and theta dominate short-dated contracts near the money, and vega and rho matter more as time to expiry grows. Someone hedging a book watches all of them, because each one describes a different way the hedge goes out of line.

Are greeks the same on every platform?

Not always. Different models, dividend assumptions and exercise styles give slightly different values, and the units differ too: theta can be quoted per year or per day, and vega per 1.00 of volatility or per one point. Checking the unit is the first step when two sources disagree.

Do the greeks predict where an option's price is going?

No. The greeks describe how an option's value responds to a change in each input at a given moment. They say nothing about which way the underlying, volatility or rates will move, and they need to be recomputed as those inputs change.

This page is information, not investment advice.

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Source: closed form Black-Scholes greeks with illustrative inputs (Black and Scholes 1973; Merton 1973); Options Industry Council, vega and rho definitions, https://www.optionseducation.org/advancedconcepts/vega and https://www.optionseducation.org/advancedconcepts/rho ; Kresmion modeled dealer greeks exposure, with the assumptions stated on Kresmion's greeks exposure tool.

Kresmion Research.