Free Tool

Options Greeks Calculator

Enter the spot, strike, days to expiry, implied volatility and a representative rate, choose a call or a put, and see the theoretical Black-Scholes price and all five Greeks with correct signs and conventions: delta and gamma per point of the underlying, theta per day, vega per one percentage point of volatility and rho per one percentage point of rate. A lot-size helper turns each Greek into per-lot rupee exposure, and a live chart plots any Greek across a spot range. Every number is a model value computed from the inputs you enter.

The Greeks do not tell you where the option is going. They tell you how it will move when the underlying, time and volatility move, and where that movement turns violent.

Load an example

Presets

Presets load example inputs to show how the Greeks behave, including how gamma and theta spike near expiry. They are not recommendations to trade.

Index

Spot / underlying (S) The current level of the underlying index.

Strike (K) The option's strike price. Equal to spot is at the money.

Days to expiry Calendar days. Converted to years as T = days / 365.

Implied volatility (IV, % per year) Annualised volatility assumption. This is an input you choose, not a fact; the market's IV differs by strike.

Risk-free rate (r, %) Representative short rate, editable. Near the RBI repo rate of 5.25 percent and the 91-day treasury bill yield of about 5.26 percent, as of June 2026. It barely moves a short-dated price.

Dividend yield (q, %) Continuous dividend yield of the underlying. Default 0; a broad index yield is small.

Option type

Lot size (units per lot) Jan 2026 revision: Nifty 65, Bank Nifty 30, FinNifty 60. Verify the current NSE circular.

Theoretical price

Delta

Gamma

Theta / day

Vega / 1% IV

! Theoretical model output. Every figure is a Black-Scholes value computed from the inputs you entered, using one constant implied volatility and a lognormal underlying. Real option prices, implied volatilities and fills differ, and the model breaks where the market gaps or the volatility skew bites. It is illustrative, not a prediction and not advice.

The one thing this tool teaches

Theta per lot, per day

Gamma per lot

Moneyness

Plot

Delta across the spot range

The selected Greek plotted against the underlying level, holding your strike, expiry, volatility and rate fixed. The dashed gold line marks the current spot and the dot marks the Greek's value there; the faint line marks the strike.

Delta curve Current spot Strike Zero line

The five Greeks in full theoretical

Greek	Value (per unit)	Convention	Per lot exposure

Read before you use these numbers

The Greeks are the honest description of how a position moves. They are not the whole of trading it. They do not size the position to a loss you can survive, they do not tell you the margin a short leg ties up and marks to market daily, and they assume a model the market does not obey. Reading the Greeks and then reading everything the model leaves out is the discipline the method we teach is built around.

Take the free diagnostic See the payoff at expiry

The one principle

The Greeks are partial derivatives of the option price, one per thing that moves it. Read two relationships first. Theta and gamma are joined: theta is the daily bleed that the holder pays and the writer receives, and it is the price of being short gamma, the curvature that turns a small move into an outsized one. Near an at-the-money expiry both spike together, so a writer collecting a little decay each day carries a tail that can give back many days of it in a single move. Vega is the reminder that implied volatility is an assumption, not a constant, and it is the input you are really guessing. The model's clean number is where the option is worth if the world matched its assumptions; the gap to the screen is where they break.

A desk reads an option by its Greeks before it reads its price, because the Greeks are the risk and the price is only today's snapshot of it. The SEBI study of individual traders in the equity derivatives segment found over 91 percent were net loss-making in FY25, with aggregate net losses near 1,05,603 crore rupees, up roughly 41 percent on the prior year. A recognisable route into that number is short-gamma writing that felt like steady income until one gap, because the theta looked like a wage and the gamma was never priced by the person collecting it. Everything below builds the Greeks honestly from the Black-Scholes model, states every convention, and is explicit about where the model lies.

The Black-Scholes math, derived

Black-Scholes-Merton prices a European option in closed form from six inputs. It starts with two standardised distances, d1 and d2, that measure how far in the money the option is in units of volatility, then weighs the spot and the discounted strike by the cumulative normal at those distances. The Greeks fall out as its derivatives. Here q is a continuous dividend yield, N is the cumulative standard normal and N' is its density.

THE TWO DISTANCES
d1 = ( ln( S / K ) + ( r − q + sigma² / 2 ) × T ) / ( sigma × √T )
d2 = d1 − sigma × √T T = days / 365, sigma = IV / 100

PRICE
call = S × e^(−qT) × N(d1) − K × e^(−rT) × N(d2)
put = K × e^(−rT) × N(−d2) − S × e^(−qT) × N(−d1)

THE FIVE GREEKS
delta call = e^(−qT) × N(d1) per 1 point of S
delta put = e^(−qT) × ( N(d1) − 1 )
gamma = e^(−qT) × N'(d1) / ( S × sigma × √T ) same for call and put
vega = S × e^(−qT) × N'(d1) × √T / 100 reported per 1% of IV
theta = ( full Black-Scholes theta per year ) / 365 reported per calendar day
rho = ( ∂price / ∂r ) / 100 reported per 1% of rate; +for calls, −for puts

Worked on a textbook benchmark, so you can check the engine: S = 100, K = 100, T = 1 year, r = 5 percent, q = 0, IV = 20 percent. The model returns a call price of about 10.4506, a delta of about 0.6368, a gamma of about 0.0188 and a vega of about 0.375 per one percentage point of volatility. This calculator reproduces those figures, which is the standard way to confirm a Black-Scholes implementation is correct before trusting its Greeks. On the tool's own default, an at-the-money 30-day Nifty-style call at 15 percent volatility, the price is about 467 points and delta about 0.55, near the 0.5 that an at-the-money option sits at.

The conventions, stated once. Delta and gamma are per one point of the underlying. Vega is per one percentage point of implied volatility, that is the raw derivative divided by 100. Theta is the full Black-Scholes theta for a year divided by 365, so it is per calendar day, matching the days-to-expiry input; some platforms divide by trading days instead, which makes their theta look larger. Rho is per one percentage point of the risk-free rate. The lot-size helper multiplies each by the contract lot to give rupee exposure per lot.

Delta is the slope of the price curve

Delta is a slope, and the slope is not constant. Where the price curve is steep, delta is high; where it is flat, delta is near zero. An at-the-money option sits on the steepest part of the bend, which is why its delta is around one half and why it changes fastest as the underlying moves. That rate of change is gamma, the curvature of the price line. Reading delta as a static hedge ratio misses the point the diagram makes: the ratio itself moves, and it moves fastest exactly where most short-dated options are written.

Gamma and theta both spike into expiry

The two Greeks a writer lives and dies by move together, and they move fastest at the end. As expiry nears, an at-the-money option's gamma rises and concentrates into a narrow band around the strike, while its theta, the daily decay, deepens. The writer earns the deeper theta and carries the taller gamma at the same time. That is the mechanism, not a judgement: a short-gamma position collects more decay per day precisely when a single move through the strike can flip its delta and hand back many days of that decay at once. The days-to-expiry table below puts real model numbers on both curves.

Vega: the price moves when the assumption moves

Vega is why an option can lose money while the underlying goes your way. Implied volatility is the market's assumed future volatility, backed out of the traded price, and it changes on its own. Raise it and every option is worth more; drop it and every option is worth less, with the underlying unmoved. A long option is long vega and a short option is short vega, so a volatility collapse after entry can hurt a holder who was right about direction and help a writer who was wrong about it. The single constant volatility this model uses is itself the fiction the next section takes apart.

Reference: Greek signs by position

The sign is the first thing to read, because it says which way each force pushes your position. A long option and its short mirror have opposite signs on every Greek. This table is the full set for the four single-leg positions; every structure is a sum of these.

Sign of each Greek by position. Positive means the position gains as that input rises; negative means it loses. Descriptive of the model, not a recommendation for any position.
Position	Delta	Gamma	Theta	Vega	Rho
Long call	+	+	−	+	+
Long put	−	+	−	+	−
Short call	−	−	+	−	−
Short put	+	−	+	−	+

Read the pairs. Every long option is long gamma and pays theta, the two signs that always disagree: you own the curvature and you rent it by the day. Every short option is short gamma and earns theta, the same trade reversed. That is the whole tension of an option position in one row: the side that collects time decay is the side exposed to the sharp move, and the calculator prints both so neither is hidden behind the other.

Reference: how the Greeks behave across moneyness

Model values for a 30-day call on a Nifty-style underlying at 24,000, at 15 percent implied volatility, as the strike moves from deep in the money to deep out. Read down the gamma and vega columns: both peak at the money and fade at the extremes, while delta runs monotonically from near 1 to near 0.

Illustrative Black-Scholes Greeks for a 30-day call, spot 24,000, IV 15 percent, r 5.5 percent, q 0, by strike. Per unit of the underlying; theta per day; vega per one percentage point of IV. A model illustration, not a prediction.
Moneyness	Strike	Price (pts)	Delta	Gamma	Vega / 1%	Theta / day
Deep ITM	22,000	2,105	0.984	0.000038	2.72	−3.92
ITM	23,400	852	0.763	0.000299	21.25	−7.94
At the money	24,000	467	0.550	0.000383	27.23	−8.73
OTM	24,600	216	0.327	0.000350	24.83	−7.36
Deep OTM	26,000	17	0.041	0.000086	6.10	−1.67

Gamma and vega are at-the-money Greeks. A deep in-the-money option behaves almost like the underlying: delta near 1, and gamma, vega and theta all small, because there is little uncertainty left about whether it finishes in the money. A deep out-of-the-money option is nearly inert until it is not. All the sensitivity, and all the danger for a writer, concentrates at the money, which is exactly where liquidity and open interest concentrate too.

Reference: gamma and theta acceleration as expiry nears

The same at-the-money call, held at 24,000 with 15 percent volatility, as days to expiry fall. This is the single most important table on the page: it shows that gamma and theta are not stable numbers but accelerate together into the last days, the mechanism behind fast blow-ups on short-dated writing.

Illustrative Black-Scholes values for an at-the-money call, spot and strike 24,000, IV 15 percent, r 5.5 percent, q 0, lot 65, by days to expiry. Per lot exposure multiplies by the lot size. A model illustration, not a prediction.
Days to expiry	Price (pts)	Gamma / unit	Gamma / lot	Theta / day (pts)	Theta / lot / day
90	881	0.000218	0.0142	−5.86	−₹381
30	467	0.000383	0.0249	−8.73	−₹567
14	307	0.000564	0.0366	−11.89	−₹773
7	212	0.000799	0.0519	−16.04	−₹1,043
3	136	0.001221	0.0794	−23.53	−₹1,529
1	77	0.002117	0.1376	−39.41	−₹2,561

Gamma rises roughly tenfold from 90 days to 1, and theta with it. On the same strike, gamma per lot climbs from about 0.014 to about 0.138 and theta per lot per day deepens from about 381 rupees to about 2,561 rupees. The writer earning that deeper decay is also carrying that taller gamma: near expiry a move of a few tens of points through the strike swings delta hard, and the single-day loss can exceed a week of collected theta. This is a description of the model's mechanics on a short-dated at-the-money option, not a recommendation to trade any expiry.

Failure modes: where Black-Scholes lies

The Greeks are exact derivatives of the model. The model is an approximation of the market, and a known one. Six places the clean number departs from the traded one, each a documented assumption of Black-Scholes that reality violates.

1. Constant volatility is false: the skew and smile. Black-Scholes assumes one volatility for every strike. The market does not: out-of-the-money index puts almost always trade at a higher implied volatility than at-the-money or call strikes, the volatility skew, because crash protection is bid. Plot implied volatility against strike and you get a skew or a smile, not a flat line. A single sigma cannot fit the whole option chain at once, so a price computed with one number will disagree with the screen at strikes away from the money, and the disagreement is largest exactly where tail risk lives.

2. The lognormal assumption underprices the tails. The model assumes the underlying moves continuously along a lognormal random walk, so large sudden moves are treated as almost impossible. Real markets gap on news, on open, on policy. Those jumps are fatter and more frequent than the bell curve allows, which is why deep out-of-the-money options often trade above their Black-Scholes value: the market is pricing a tail the model says should barely exist. A writer relying on the model's tail probabilities is short exactly the event the model underweights.

3. Continuous, costless hedging does not exist. The derivation assumes you can rebalance a hedge continuously and for free. In practice hedging happens in discrete steps, each crossing a bid-ask spread and paying charges and taxes, and gamma is what makes discrete hedging expensive: the more curvature, the more the hedge drifts between adjustments. The tidy replication that justifies the price leaks money in the real world, and it leaks most for the short-dated, high-gamma options that are most heavily traded.

4. European exercise and cash settlement, only sometimes. The closed form here is for a European option that cannot be exercised early. Indian index options fit that assumption: they are European and cash-settled, so this model is appropriate for them. Single-stock options in India are American and physically settled, so they carry early-exercise value this formula does not capture, and pricing them with it understates a short position's risk. Use the model where its exercise assumption holds and know when it does not.

5. Implied volatility is an input you are guessing. Every Greek here depends on the volatility you type. Feed in the wrong volatility and the price and all five Greeks are wrong together, in the same direction. Because implied volatility is not observable directly but backed out of traded prices, the honest use of this tool is to enter the market's current implied volatility for that strike, not a guess or a historical figure, and to treat the output as conditional on that number. Vega is the size of your exposure to being wrong about it.

6. Model price is not market price. Even with the right volatility, the traded price carries a bid-ask spread, the supply and demand for that specific strike and expiry, and, near expiry, pin risk around the strike. The Greeks tell you how the model price responds to its inputs; they do not tell you where a real order fills. Treat the model value as the anchor and the screen as the anchor plus a spread and a crowd, and never size a position as though you can transact at the theoretical number.

The honest summary. This tool computes the Black-Scholes-Merton price and Greeks from your inputs, using one constant implied volatility, a lognormal underlying and European exercise, as of July 2026. It is correct for what it models and appropriate for Indian index options, which are European and cash-settled. It does not model the volatility skew, jumps and gaps, hedging costs, early exercise on American single-stock options, or the bid-ask spread. Use the Greeks to understand how a position moves, and read the gap to the screen as information about where the assumptions break.

The risk-manager's view: theta is a wage for a tail

The instinct the Greeks exist to correct is treating theta as income. Selling an option produces a steady positive theta, a little value accruing each day as the option decays, and on a screen that looks like a salary. It is not. Theta is the premium a writer is paid for being short gamma, and gamma is the risk of a sharp move. The two are a single trade seen from two sides: the daily credit is small and frequent, the gamma loss is large and rare, and the calculator prints both so the credit is never read without the tail attached. On a short-dated at-the-money option, as the days-to-expiry table shows, both grow together into expiry, so the wage rises at exactly the moment the tail gets fattest.

This is the line the SEBI FY25 numbers sit on. Over 91 percent of individual traders in the equity derivatives segment were net loss-making, with aggregate net losses near 1,05,603 crore rupees, up roughly 41 percent on the prior year. A well-worn route into that statistic is short-gamma writing that felt like collecting rent until a single gap through the strike gave back many weeks of decay at once. None of this argues for buying over selling or the reverse; both sides of an option have their Greeks and their failure modes. The discipline is to read the whole set, size to the loss you can survive rather than the decay you can collect, and remember that the model printing these numbers is a map with known holes. That reading is where this tool ends and the work begins.

Common Questions

Frequently Asked Questions

What are option Greeks and what does this calculator compute?

The Greeks are the sensitivities of an option's theoretical price to the things that move it: the underlying level, the passage of time, implied volatility and the interest rate. Each is a partial derivative of the Black-Scholes price. Delta is the change in the option price per one point move in the underlying; gamma is the change in delta per one point, the curvature; theta is the change in price per day as time passes; vega is the change per one percentage point of implied volatility; and rho is the change per one percentage point of the risk-free rate. This calculator takes the spot, the strike, days to expiry, implied volatility, a representative risk-free rate, an optional dividend yield and the option type, and returns the theoretical price and all five Greeks with correct signs and per-unit conventions, plus a lot-size helper that converts each into per-lot rupee exposure. Every figure is a model value computed from the inputs you enter, not a market quote.

What is the Black-Scholes model and what are its inputs?

Black-Scholes-Merton is a closed-form model that prices a European option from six inputs: the spot level S, the strike K, the time to expiry T in years, the risk-free rate r, the implied volatility sigma and, in the Merton extension, a continuous dividend yield q. It first computes d1 = ( ln(S over K) + (r minus q + half sigma squared) times T ) divided by ( sigma times the square root of T ), and d2 = d1 minus sigma root T. A call is then S e to the minus qT times N(d1) minus K e to the minus rT times N(d2), where N is the cumulative standard normal; a put is the mirror. The model assumes the underlying follows a lognormal random walk with constant volatility and that the option is held to expiry with continuous, costless hedging. Those assumptions are exactly where it departs from the market, which is why the number here is theoretical and the traded price differs.

What is delta and how do I read it?

Delta is the first derivative of the option price with respect to the underlying: how much the option's theoretical value changes for a one point move in spot. A call delta runs from 0 to 1 and a put delta from minus 1 to 0, so a call delta of 0.55 means the option gains about 0.55 points of value for each point the underlying rises, before the other Greeks act. Delta doubles as a hedge ratio, the number of units of the underlying that offset the option, and it is loosely read as the risk-neutral chance the option finishes in the money, which is near 0.5 for an at-the-money option. To turn it into money, multiply by the lot size: a delta of 0.55 on a lot of 65 is about 35.8 rupees of position value per one point move per lot, which this tool shows in the lot-size helper. Delta itself moves with the underlying, and the rate at which it moves is gamma.

What is gamma, and why does it matter most near expiry?

Gamma is the second derivative of the price and the first derivative of delta: how fast delta itself changes as the underlying moves. It is largest for an at-the-money option and it rises sharply as expiry approaches, because an at-the-money option near expiry flips from behaving like nothing to behaving like the underlying over a very small move. In the days-to-expiry table on this page, gamma per lot on an at-the-money Nifty-style option climbs from about 0.014 at ninety days to about 0.138 at one day, roughly a tenfold increase, on the same strike. High gamma means delta is unstable, so a position that looked balanced can become badly one-sided after a small move. A long option owner is long gamma, which helps them; an option writer is short gamma, and that short gamma is the mechanism behind a fast, outsized loss on a gap. It is described here as mechanics, not as advice for or against writing options.

What is theta, and does selling options earn it as income?

Theta is the change in the option's theoretical price per day from the passage of time alone, holding everything else fixed. For a long option it is negative, because the time value in the premium decays toward zero at expiry; for the writer it is the mirror image, the same decay accruing in their favour. This tool reports theta per calendar day, dividing the annual Black-Scholes theta by 365 to match the days-to-expiry input. Theta is often described as an income the writer collects, and that framing is misleading and is not how it is presented here. Theta does not arrive as free money: it is the price paid for being short gamma, the small, steady credit set against the large, rare loss when the underlying moves against an uncovered short. The two Greeks are joined, and near expiry both spike together, which is precisely why short-gamma positions can lose many days of decay in a single move. It is a description of the trade-off, not a recommendation to write options for income.

What is vega and how does implied volatility change the price?

Vega is the change in the option's theoretical price per one percentage point change in implied volatility. It is largest for at-the-money options with time left and falls toward expiry, the opposite of gamma's path. Implied volatility is not a fact you can look up; it is the market's assumed future volatility backed out of the traded price, and it is the single input here that you are really guessing. Feed a higher volatility in and every option is worth more, feed a lower one in and every option is worth less, with no change in spot at all. That is why an option can lose value even when the underlying moved your way: implied volatility fell. This tool reports vega per one percentage point so you can see the sensitivity directly, and the wider point it makes is that a single constant volatility is an assumption the market itself does not hold, which is the volatility skew discussed further down the page.

What is rho and why is it usually the smallest Greek for short-dated options?

Rho is the change in the option's theoretical price per one percentage point change in the risk-free rate. It is positive for calls and negative for puts, and it grows with time to expiry, because the interest rate discounts the strike over the life of the option and a longer life means more discounting. For the weekly and monthly index options that dominate Indian volumes, expiry is close, so the discounting window is short and rho is small relative to delta, gamma, theta and vega. That is why the risk-free rate is offered as a representative, editable default here and why a small error in it barely moves a short-dated price. Rho matters for long-dated options and in environments where rates move a great deal; for a two-day at-the-money option it is close to a rounding error, which the tool will show if you shorten the expiry.

Why is the model price different from the price on my screen?

Because the model is a clean map and the market is the territory. Black-Scholes assumes one constant volatility across all strikes, a lognormal underlying with no jumps, continuous and costless hedging, and a European option held to expiry. Real markets violate every one of these. Volatility is not constant across strikes: out-of-the-money puts usually price at a higher implied volatility than calls, the skew, so a single sigma cannot fit the whole chain at once. The underlying gaps rather than moving continuously, which the lognormal assumption underprices in the tails. Hedging costs money and happens in discrete steps. And the traded price also carries the bid-ask spread, supply and demand for that strike, and, near expiry, pin risk. The number this tool prints is what the option is worth if the world matched the model's assumptions. The gap between that and the screen is not an error; it is information about where the assumptions break, and reading that gap is a large part of what options traders actually do.

Is this options Greeks calculator financial advice?

No. It is an educational tool that computes the theoretical Black-Scholes price and Greeks of an option you describe, and nothing it shows is a recommendation to buy, sell, write or adjust any position. Every output is a model value derived from the inputs you enter, not a prediction of profit, a claim about how any option will perform, or a statement that a real trade would fill at these levels. The presets are example inputs loaded to illustrate how the Greeks behave, not suggestions to trade. Selling options and collecting theta is presented as a risk mechanism to understand, not as an income method. Bharath Shiksha is an educational publisher, not a SEBI-registered investment adviser or research analyst. For a decision about whether any option or strategy suits your circumstances and risk tolerance, consult a registered adviser.

Where the facts come from

Sources

The Black-Scholes-Merton model. Fischer Black and Myron Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 1973; Robert C. Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 1973. The pricing and Greek formulas used here, including the continuous dividend yield, are the standard textbook forms of that model.
The cumulative normal. The tool evaluates the cumulative standard normal N with an error-function approximation, and the standard normal density N' in closed form, the usual numerical method for a browser Black-Scholes implementation. It reproduces the textbook benchmark of a 10.4506 call price at S = K = 100, T = 1, r = 5 percent, IV = 20 percent.
The January 2026 lot sizes. NSE circular NSE/FAOP/70616 revising index derivative market lot sizes effective the January 2026 series, Nifty 50 from 75 to 65, Bank Nifty from 35 to 30 and Nifty Financial Services to 60, pursuant to SEBI circular SEBI/HO/MRD-PoD2/CIR/P/2024/00181 dated 30 December 2024. Verify the current lot size before you rely on it. nsearchives.nseindia.com
The representative risk-free rate. RBI repo rate at 5.25 percent and the 91-day treasury bill implicit yield at cut-off near 5.26 percent, as of June 2026. Used only as an editable default; a short-dated option's price is nearly insensitive to it. rbi.org.in
The FY25 loss base rate. SEBI study on the profit and loss of individual traders in the equity derivatives segment: over 91 percent net loss-making in FY25, with aggregate net losses of about 1,05,603 crore rupees, up roughly 41 percent from 74,812 crore rupees in FY24. sebi.gov.in

Educational note. This tool computes the theoretical Black-Scholes-Merton price and Greeks from the inputs you enter, using one constant implied volatility, as of July 2026; every output is illustrative and depends on your inputs. It does not model the volatility skew, jumps, hedging costs, early exercise on American single-stock options, or the bid-ask spread, and real prices and fills differ from the model. Nothing here is a recommendation to trade or to buy or sell any security, the presets are example inputs to illustrate the Greeks, and option writing and theta are presented as risk mechanics to understand, not as an income method. Bharath Shiksha is an educational publisher, not a SEBI-registered investment adviser or research analyst.