Black-Scholes Option Price Calculator

The theoretical price of a European call and put under the Black-Scholes-Merton model, with d1, d2 and the Greeks: delta, gamma, theta per day and vega per 1% of volatility.

Last updated 24 Sep 2026

Option price

CurrencySpot + strike + time + volatility → price

Show the headline for

Spot price of the underlying ($)

Strike price ($)

Days to expiry (calendar days)

Volatility (% a year)Annualised volatility of the underlying. Implied volatility from the option chain, or your own estimate. The default is an illustration.

Risk-free rate (% a year)A government Treasury bill yield for a similar term is the usual choice. The default is an illustration; enter the current yield.

Dividend yield (% a year, 0 = none)Continuous dividend yield of the underlying over the option’s life. 0 if none is expected.

$230.21Example

Spot $24,000, strike $24,500, 30 days, volatility 14%, risk-free 6.5%, no dividend

Black-Scholes-Merton with a dividend yield

C = S e^−qT N(d₁) − K e^−rT N(d₂); P = K e^−rT N(−d₂) − S e^−qT N(−d₁)

d1: [ln(S ÷ K) + (r − q + σ² ÷ 2) T] ÷ (σ √T); d2 = d1 − σ √T
S, K: the spot price and the strike
T: time to expiry in years: calendar days ÷ 365
σ, r, q: volatility, the risk-free rate and the dividend yield, a year, continuously compounded
N: the standard normal cumulative distribution, by Abramowitz and Stegun 26.2.17

Worked example

Spot $24,000, strike $24,500, 30 days, volatility 14%, risk-free 6.5%, no dividend
T = 30 ÷ 365 = 0.08219; σ√T = 0.04014
d1 = [ln(24,000 ÷ 24,500) + (0.065 + 0.0098) × 0.08219] ÷ 0.04014 = -0.3606; d2 = -0.4007
N(d1) = 0.3592; N(d2) = 0.3443
Call = 24,000 × 0.3592 − 24,500 × e^{−0.065 × 0.08219} × 0.3443 = $230.21
Put = $599.67; check: call − put = -369.46 = S − K e^−rT = -369.46

Spot 24,000, strike 24,500, 30 days, risk-free 6.5%: effect of volatility

Both calls and puts are worth more when volatility is higher; an out-of-the-money option most of all, in proportion.
Volatility	Call	Put	Call delta	Vega per 1%
10%	$130.23	$499.69	0.302	$24.00
14%	$230.21	$599.67	0.359	$25.72
20%	$387.92	$757.38	0.406	$26.68
30%	$657.88	$1,027.34	0.446	$27.20

What the Black-Scholes model assumes

The Black-Scholes model, published by Fischer Black and Myron Scholes in 1973 and extended by Robert Merton the same year to cover a continuous dividend yield, prices a European option: one that can be exercised only at expiry. It assumes the underlying’s price moves in a lognormal random walk with constant volatility, that you can borrow and lend at a constant risk-free rate, and that there are no costs or taxes. The only input you cannot look up is volatility, and the price is very sensitive to it; traders usually work backwards from market prices to an implied volatility instead.

The Greeks. Delta is how much the option price moves for a one-rupee move in the underlying (and roughly the market’s odds of a call finishing in the money). Gamma is how fast delta itself changes. Theta is the value lost per calendar day with everything else unchanged, which is why bought options lose value as expiry nears. Vega is the change for one percentage point of volatility. The page also checks put–call parity, C − P = S e^−qT − K e^−rT, which any consistent European prices must satisfy.

Where it applies in India. Index options on NSE and BSE are European style. NSE’s stock options have been European since January 2011, so the model’s exercise assumption fits Indian exchange-traded options; American-style options elsewhere can be worth more, and a put especially. Days are counted as calendar days ÷ 365 here; some desks use trading days, which gives slightly different theta and volatility figures.

The normal distribution. The expression language on this site has no built-in normal CDF, so the page uses the polynomial approximation 26.2.17 from Abramowitz and Stegun’s Handbook of Mathematical Functions, whose error is below 7.5 × 10⁻⁸; it was checked against the exact error function at 16,001 points before publishing. A model price is not a forecast of profit. For the payoff at expiry, see the option payoff calculator. SEBI’s own studies of individual traders in equity futures and options found that 89% lost money in 2021–22 (study of January 2023), 93% over the three years 2021–22 to 2023–24 (September 2024), and nearly 91% in 2024–25 (July 2025), when their net losses came to ₹1,05,603 crore. This is arithmetic on the figures you enter, not financial advice.

Frequently asked questions

What inputs does the Black-Scholes formula need?

The spot price, the strike, the time to expiry, the volatility, the risk-free rate and, if the underlying pays one, the dividend yield. Only the volatility has to be estimated.

Can Black-Scholes be used for Indian options?

Its exercise assumption fits: index options in India are European, and NSE stock options have been European since January 2011. Its other assumptions, such as constant volatility and no costs, hold only approximately anywhere.

What does theta per day mean?

How much the option’s theoretical price falls in one calendar day if nothing else changes. In the example the call loses about $7.50 a day.

How accurate is the normal distribution used here?

It uses Abramowitz and Stegun formula 26.2.17, with an error under 7.5 × 10⁻⁸ in N(x). Across typical inputs that changes an option price by far less than a paisa.

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References

Black F, Scholes M. The Pricing of Options and Corporate Liabilities. Journal of Political Economy. 1973;81(3):637–654.
Merton RC. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science. 1973;4(1):141–183. (Continuous dividend yield.)
Abramowitz M, Stegun IA, eds. Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Series 55. 1964. Formula 26.2.17: normal probability integral, |ε(x)| < 7.5 × 10⁻⁸.
Hull JC. Options, Futures, and Other Derivatives. Pearson. Payoffs of long and short calls and puts at expiry; the Black-Scholes-Merton model and the Greeks, with theta quoted per calendar day.
National Stock Exchange of India: stock option contracts expiring on or after 27 January 2011 are European style, exercisable only at expiry (reported by Business Standard, 3 January 2011, “NSE to offer European-style stock options”). Index options were already European.
SEBI. Press release, 23 September 2024: Updated SEBI Study Reveals 93% of Individual Traders Incurred Losses in Equity F&O between FY22 and FY24; Aggregate Losses Exceed ₹1.8 Lakh Crores Over Three Years. Average loss about ₹2 lakh per trader; about ₹50,000 crore spent on transaction costs.
SEBI. Comparative study of growth in Equity Derivatives Segment vis-à-vis Cash Market after recent measures. Released 7 July 2025. Nearly 91% of individual traders incurred a net loss in equity derivatives in FY 2024–25; their net losses were ₹1,05,603 crore, against ₹74,812 crore in FY 2023–24.