Options Greeks Calculator

Calculate delta, gamma, theta, vega and rho for a single option and for a whole position in contracts.

A Greek quoted on an option chain describes one option on one unit of the underlying, which is almost never the thing actually being held. A theta of −0.0444 is not a figure anyone reacts to. The same theta on five contracts covering 100 units each is −22.20 a day, every day, whether or not the market moves — and that is a number people react to. This options Greeks calculator performs that conversion, turning the delta, gamma, theta and vega quoted per unit into position Greeks for the contracts actually held.

Formula: position greek = per-share greek x contracts x contract multiplier

Your figures

Currency

Option type

Days to expiry

Calendar days. The model converts to years internally.

Underlying price

Strike price

Implied volatility

Annualised, as a percentage.

Risk-free rate

Annualised. Roughly the yield on a short government bond.

Dividend yield

Optional — annualised continuous yield on the underlying.

Number of contracts

How many contracts you hold. Every position figure scales with this.

Shares per contract

Units of the underlying one contract controls. It differs by product and by venue — take it from your own contract specification.

Results

Equivalent holding in the underlying

539.9636

position delta on 10 call contracts

Cash per 1.00 move in the underlyingThe equivalent holding above, read as money rather than as a unit count.: +US$539.96
Theta — cash per calendar dayWhat one day passing is worth to the position, all else equal.: -US$44.99
Vega — cash per 1% of implied volatility: +US$113.80
Rho — cash per 1% of interest rate: +US$42.33
Change in the equivalent holding per 1.00 movePosition gamma, read as units: how far the equivalent holding above moves as the underlying does.: 69.2276
Units of the underlying controlled10 contracts x 100 per contract.: 1,000
Model value of the position: US$2,493.38
Delta, quoted per unitThe figure on the chain: option price change per 1.00 move in the underlying.: +0.5400
Gamma, quoted per unit: 0.06923
Theta, quoted per unit per day: -US$0.0450
Vega, quoted per unit per 1%: US$0.1138
Rho, quoted per unit per 1%: +US$0.0423
Option price, per unitUS$0.00 intrinsic, US$2.49 time value.: US$2.49
Break-even at expiry: US$102.49
Probability of finishing in the moneyA risk-neutral model probability, not a real-world forecast.: 51.72%

What this means

10 call contracts on a 100 multiplier control 1,000 units, and every quoted Greek is scaled by that figure. A quoted delta of 0.5400 becomes a position delta of 539.9636 units — the position moves like holding that many units of the underlying, which is US$539.96 for each 1.00 the price moves. The quoted theta of -US$0.0450 scales the same way to -US$44.99 a day, and vega to US$113.80 for each percentage point of implied volatility. These are rates of change measured at the price entered: they describe a small move, and the market recomputes them continuously.

Every figure is for the position held long. Held short, each Greek keeps its magnitude and reverses its sign — a written call carries negative delta and positive theta.
Greeks are derivatives, so they describe a small move. Over a large or fast one, delta multiplied by the distance travelled is a poor estimate of what happens — the size of that error is what gamma is measuring.
The multiplier is yours to enter. It differs by product and by venue, so an assumed 100 is wrong on plenty of contracts and would scale every cash figure by the same error.
Prices European exercise. American options can be exercised early, which adds value the model does not capture — most noticeably on puts and on calls over a dividend.

The link reopens this calculator with the same numbers. Nothing you enter is sent to a server.

Continue with these figures

About the Options Greeks calculation

Scaling is the whole of the arithmetic on this page, and it is the step most often skipped. Multiply each per-unit Greek by the number of contracts and by the units one contract controls, and the five sensitivities stop being properties of a contract specification and start being properties of an account: money per point of movement, money per day of waiting, money per point of implied volatility.

The most useful figure the scaling produces is the equivalent holding. Position delta, expressed in units of the underlying rather than as a decimal, says what the position behaves like for a small move. An exposure of 174 units is directly comparable with a share position, a futures position or another option position; the 0.3480 printed on the chain is comparable with nothing.

Everything here is computed from the contract details entered. No chain is fetched, no volatility is assumed and the contract multiplier is typed in rather than presumed, because it belongs to the contract specification and differs by product and by venue.

Options Greeks formula

per-unit Greeks, from Black-Scholes-Merton:
  delta(call) = e^(−qT) × N(d1)
  delta(put)  = e^(−qT) × [N(d1) − 1]
  gamma       = e^(−qT) × φ(d1) ÷ (S × σ × √T)
  vega        = S × e^(−qT) × φ(d1) × √T

scaling, which is the whole of this calculator:
  units controlled = contracts × contract multiplier
  position Greek   = per-unit Greek × units controlled

the readings taken from that:
  equivalent holding  = position delta, in units of the underlying
  cash per 1.00 move  = position delta, read as money
  theta cash per day  = per-unit theta × units controlled
  vega cash per point = per-unit vega × units controlled

S, K, T: Underlying price, strike, and time to expiry in years — 45 days is 45 ÷ 365.
σ (sigma): Annualised implied volatility as a decimal, so 28% enters the formulae as 0.28.
q: Annualised continuous dividend yield on the underlying, zero where none is paid.
N(x): Cumulative standard normal distribution.
φ(x): Standard normal density, which is what puts a peak in gamma and vega at the money.
contract multiplier: Units of the underlying one contract controls. Taken from the contract specification, not assumed.
units controlled: Contracts × multiplier — the single factor every position figure is scaled by.
equivalent holding: Position delta read as a quantity of the underlying rather than as a decimal per unit.

The scaling is a single multiplication because a position in identical contracts is a plain sum. Its value is the value of one option added up N times, and differentiation is linear, so the derivative of the sum is the sum of the derivatives. Nothing has to be re-priced to move from a chain figure to a position figure: multiply by contracts and by the multiplier and the result is exact, not an approximation.

The multiplier is where the arithmetic goes wrong most often, which is why it is an input rather than a constant. A quoted price of 2.1788 is 217.88 of cash for one 100-unit contract, and the same 100 sits inside every position Greek on the page. Index options, futures options and contracts adjusted after a corporate action all control something other than 100 units, and an assumed multiplier scales every cash figure on screen by exactly its own error.

Theta arrives per calendar day and vega and rho per percentage point because those are the units the figures are read in. The model produces theta per year and vega per whole unit of volatility — a move from 0.28 to 1.28 — which are not quantities anyone holds a position in. Dividing by 365 and by 100 respectively happens before display, so the numbers on the page can be compared directly against a day of waiting and a point of implied volatility.

Delta is reported twice on purpose, as an equivalent holding and as cash per 1.00 move. They are numerically identical — one unit moving 1.00 is 1.00 of money — but they answer different questions. The unit count is what you compare against a share holding when working out net exposure; the cash figure is what you compare against the size of a typical daily range. An option delta calculator that merged them into one row would quietly lose one of the two readings.

Gamma is presented as the change in that equivalent holding per 1.00 move, which is the reading that makes it concrete. A position gamma of 18.80 says the exposure is not fixed at 174 units: a point higher in the underlying and it is about 193, a point lower and about 155. That drift in delta is precisely the error you make by multiplying delta by a large move, and it is why the Greeks describe a small move rather than a projection.

Every figure is computed for the position held long. A written position keeps every magnitude and reverses every sign, so a sold call carries negative delta, negative gamma, positive theta and negative vega. That is a reading of the same numbers rather than a separate calculation, which is why the calculator does not ask which side you are on.

Options Greeks: a worked example

Five 105-strike calls held with the underlying at 100, 45 days to expiry, 28% implied volatility, a 4.5% risk-free rate, no dividend and 100 units per contract.

σ√T0.28 × √(45 ÷ 365) = 0.28 × 0.3511 = 0.0983
d1[ln(100 ÷ 105) + (0.045 + 0.0392) × 0.1233] ÷ 0.0983 = −0.3907
d2−0.3907 − 0.0983 = −0.4890
Option price per unit100 × 0.3480 − 105 × e^(−0.00555) × 0.3124 = 2.1788
Units controlled5 × 100 = 500
Equivalent holding0.3480 × 500 = 174.01 units
Theta cash per day−0.0444 × 500 = −22.20
Vega cash per point0.1298 × 500 = 64.89
Position gamma, as units0.0376 × 500 = 18.80 units per 1.00 move
Model value of the position2.1788 × 500 = 1,089.40

The five contracts behave like holding 174 units of the underlying, so a 1.00 move is worth 174.01 either way. They give up 22.20 to each calendar day that passes and gain or lose 64.89 to each point of implied volatility. The quoted theta of −0.0444 looked immaterial; scaled across 500 units it is the figure that decides whether the position is worth holding through a quiet week.

How to use the Options Greeks calculator

Choose call or put, then enter the strike, the underlying price and the days to expiry. Days are calendar days and are converted to years internally.
Enter the implied volatility for that specific strike and expiry. If you only have a traded price, solve for it first with the implied volatility calculator and bring the figure back here.
Enter the risk-free rate, and a dividend yield if the underlying pays one — a yield lowers call values and raises put values.
Enter the number of contracts and the units each one controls, taken from your own contract specification rather than assumed.
Read the equivalent holding first. It states the position in units of the underlying, which is the only form in which it can be compared with the rest of an account.
Read the theta cash figure as the daily cost of time decay on the position, and compare it against the vega cash figure — on most option positions a one-point move in implied volatility outweighs several days of decay.
If the position is written rather than bought, keep every magnitude and reverse every sign.

Limitations of the Options Greeks calculator

Every calculation makes assumptions. These are the ones that matter here.

Differentiates a European pricing model, so every Greek here describes a contract that cannot be exercised before expiry. On an American contract the difference shows up in the sensitivities before it shows up in the price: delta on a deep in-the-money put approaches one earlier than this reports, because the holder's right to exercise now puts a floor under the option that the model has no term for, and theta on such a put is understated for the same reason. The prices are close; the Greeks near the exercise boundary are not.
Uses the one implied volatility entered. Real chains price a different volatility at every strike and expiry, so Greeks computed from a single figure will not match a broker's risk screen exactly.
Greeks are derivatives, so they hold for a small move. Over a large or fast one, delta multiplied by the distance travelled is a poor estimate of the outcome, and the size of that error is what gamma measures.
Theta assumes only the calendar changes. In practice implied volatility moves at the same time, and a one-point change in it frequently outweighs several days of decay.
Excludes commission, the bid-ask spread, financing and any margin a written position requires. Those are position costs rather than sensitivities, and none of them appears in a Greek.
The probability of finishing in the money is N(d2), a risk-neutral model probability computed under an artificial measure. It is not a real-world forecast and it takes no account of the premium paid.
Assumes the underlying moves continuously. Gaps over earnings and news are exactly what the model cannot represent, and gamma is at its least reliable precisely when a gap happens.

Frequently asked questions about Options Greeks

What is the difference between the Greeks quoted on a chain and position Greeks?

A quoted Greek is per unit of the underlying, for one option. A position Greek is that figure multiplied by the number of contracts and by the units each contract controls. The quoted delta of 0.3480 becomes a position delta of 174.01 units on five 100-unit contracts — the same sensitivity, expressed at the size actually held.

Why do I have to enter the shares per contract?

Because it varies. 100 units is common on listed equity options, but index options, futures options and contracts adjusted after a split or merger all control something else, and the number belongs to the contract specification rather than to a calculator. Since every cash figure here is scaled by it, an assumed multiplier would be wrong by exactly the same factor on every line.

What does the equivalent holding actually mean?

It is position delta expressed as a quantity of the underlying. An equivalent holding of 174 units means that for a small move the position gains and loses roughly what 174 units of the underlying would. That makes it directly comparable with shares, futures or another option position, which is what lets separate positions be added into a single net exposure.

How do the Greeks change if I have written the option instead?

Every magnitude stays the same and every sign flips. A written call carries negative delta, negative gamma, positive theta and negative vega, so it gains from the passage of time and from a fall in implied volatility, and loses from movement in either. The calculator prices the long position, and reading the same figures with reversed signs gives the short one.

Is the theta decay figure what the position will lose tomorrow?

Only if nothing else moves, which is rare. Theta is the partial derivative with respect to time holding the underlying, volatility and rates fixed. In a real day all three change, and a one-point shift in implied volatility is worth 64.89 on the example position against 22.20 for the day. Theta is also not constant — it accelerates into expiry for an at-the-money option.

Why is gamma shown in units rather than as a decimal?

Because the decimal is hard to act on. Position gamma of 18.80 units per 1.00 move says the 174-unit exposure becomes roughly 193 units a point higher and 155 a point lower. Stated that way it is obviously the rate at which the position's own size changes underneath you, which is the thing gamma is worth knowing for.

Does the dividend yield matter much?

It matters more than rho does on most equity options. A continuous yield discounts the underlying in the pricing formula, which lowers calls and raises puts, and it shows up in delta through the e^(−qT) factor. On a high-yielding underlying over a long expiry it moves the figures materially; on a non-payer it can be left at zero.

Last reviewed 29 August 2026. See our methodology and editorial policy.

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