Black-Scholes Calculator
The Black-Scholes-Merton model gives a theoretical price for a European call or put from five inputs plus an optional dividend yield. Enter the underlying price, strike, time to expiration, volatility and interest rate to get both option prices, all the Greeks, d1 and d2, a put-call parity check and a value-vs-price chart. Add a market price to solve for implied volatility.
Inputs Example values The numbers below are made up. Replace them with your own. Use a dot for decimals.
- Time to expiration, T
- –
- d1
- –
- d2 = d1 − σ√T
- –
- N(d1) · N(d2)
- –
- N(−d1) · N(−d2)
- –
- S·e−qT (price less dividends)
- –
- K·e−rT (present value of strike)
- –
- Forward price S·e(r−q)T
- –
- Put-call parity: C − P
- –
Prices and Greeks
| Per share | Call | Put |
|---|---|---|
| Theoretical value | – | – |
| Intrinsic value max(S − K, 0) / max(K − S, 0) | – | – |
| Time value | – | – |
| Delta per $1 move in the underlying | – | – |
| Gamma change in delta per $1 | – | – |
| Vega per 1 volatility point | – | – |
| Theta per calendar day (÷ 365) | – | – |
| Theta per year | – | – |
| Rho per 1% change in r | – | – |
Values are per share. A standard US equity option contract generally covers 100 shares, so multiply by 100 for one contract.
Implied volatility
–
- Black-Scholes value today
- Value at expiration (payoff)
- Current underlying price
Value at other underlying prices
| Underlying price | Call today | Call at expiry | Put today | Put at expiry |
|---|
“Today” keeps your time, volatility, rate and yield and changes only the underlying price. “At expiry” is the intrinsic value when time runs out.
Everything runs in your browser. Nothing is saved or sent.
How to use the Black-Scholes calculator
- Enter the underlying price and the strike. Use the current price of the stock or index and the option’s exercise price.
- Enter the time to expiration. In days, the calculator divides by 365. Pick “years” to type T directly, for example 0.5 for six months.
- Enter the volatility. This is the annualised standard deviation of returns, in percent. It is the one input that cannot be looked up directly: use an estimate from past prices, or the implied volatility of similar options.
- Enter the risk-free rate and the dividend yield. Both are annual, continuously compounded rates. Use a government bill or bond yield with a maturity close to the option’s. Leave the yield blank for a stock with no dividend before expiry.
- Optional: enter a market price for the call or put you selected. The calculator then solves for the volatility that makes the model price equal to that market price.
Black-Scholes formulas
Fischer Black and Myron Scholes published the call formula in the Journal of Political Economy in 1973. Robert Merton’s paper “Theory of Rational Option Pricing” in the same year extended the theory, including the effect of dividends. The calculator uses the standard form with a continuous dividend yield q (with q = 0 it is the original formula). S is the underlying price, K the strike, T the time in years, σ the volatility, r the risk-free rate and N(·) the standard normal cumulative distribution.
The Greeks are the partial derivatives of those prices. n(x) = e^(−x²/2)/√(2π) is the standard normal density.
| Greek | Call | Put | Units shown |
|---|---|---|---|
| Delta | e^(−qT)·N(d1) | −e^(−qT)·N(−d1) | $ per $1 in S |
| Gamma | e^(−qT)·n(d1) / (S·σ·√T), same for both | delta per $1 in S | |
| Vega | S·e^(−qT)·n(d1)·√T, same for both | ÷ 100: $ per 1 vol point | |
| Theta | −S·e^(−qT)·n(d1)·σ/(2√T) − r·K·e^(−rT)·N(d2) + q·S·e^(−qT)·N(d1) | −S·e^(−qT)·n(d1)·σ/(2√T) + r·K·e^(−rT)·N(−d2) − q·S·e^(−qT)·N(−d1) | $ per year; ÷ 365 per day |
| Rho | K·T·e^(−rT)·N(d2) | −K·T·e^(−rT)·N(−d2) | ÷ 100: $ per 1% in r |
Put-call parity. For European options with the same strike and expiry, C − P = S·e^(−qT) − K·e^(−rT). The calculator shows both sides. They match to rounding, which is a quick check that the two prices are consistent.
Implied volatility. There is no closed formula for σ given a price, so the calculator solves for it. It starts from an estimate and takes Newton steps (price error ÷ vega). Every step keeps a bracket around the answer between 0.01% and 500% volatility; if a Newton step leaves the bracket, or vega is too small to divide by, it bisects the bracket instead. It stops when the model price is within $0.000000000001 × max(1, price) of the market price. A price at or below the option’s lower bound (discounted intrinsic value) or at or above its upper bound (S·e^(−qT) for a call, K·e^(−rT) for a put) has no implied volatility, and the calculator says so.
Normal distribution. N(x) is computed with the double-precision rational approximation of Hart (1968) as published by Graeme West (2005), checked against SciPy’s norm.cdf to within 1e-15 on this page’s tests.
Worked examples
1. The textbook example: S = 42, K = 40, six months
John Hull’s Options, Futures, and Other Derivatives uses this example in its Black-Scholes-Merton chapter (Example 15.6 in the edition cited by the R package QFRM): a stock at $42, a strike of $40, a risk-free rate of 10% a year, volatility of 20% a year and six months to expiration (T = 0.5), no dividend. HP’s 12C calculator handbook works the same numbers. To repeat it here, choose “years”, enter 0.5, and leave the dividend yield and market price blank.
| Step | Arithmetic | Result |
|---|---|---|
| d1 | [ln(42/40) + (0.10 + 0.2²/2) × 0.5] ÷ (0.2 × √0.5) | 0.7693 |
| d2 | 0.7693 − 0.2 × √0.5 | 0.6278 |
| N(d1), N(d2) | standard normal CDF | 0.7791, 0.7349 |
| K·e−rT | 40 × e−0.05 | 38.0492 |
| Call | 42 × 0.7791 − 38.0492 × 0.7349 | $4.7594 |
| Put | 38.0492 × N(−0.6278) − 42 × N(−0.7693) | $0.8086 |
| Parity check | 4.7594 − 0.8086 = 42 − 38.0492 | 3.9508 |
Hull rounds the result to a call of $4.76 and a put of $0.81. The Greeks for the same inputs: call delta 0.7791, gamma 0.0500, vega 0.0881 per volatility point, call theta −0.0125 a day (−4.5591 a year) and call rho 0.1398 per 1%.
2. Days, a dividend yield and implied volatility (the calculator’s example)
A hypothetical stock at $100, a $105 strike, 45 days to expiration, 25% volatility, a 4% risk-free rate and a 1.5% dividend yield. T = 45 ÷ 365 = 0.1233 years.
| Step | Arithmetic | Result |
|---|---|---|
| d1, d2 | [ln(100/105) + (0.04 − 0.015 + 0.25²/2) × 0.1233] ÷ (0.25 × √0.1233) | −0.4768, −0.5646 |
| S·e−qT, K·e−rT | 100 × e−0.015 × 0.1233, 105 × e−0.04 × 0.1233 | 99.8152, 104.4835 |
| Call, put | formulas above | $1.7157, $6.3839 |
| Call delta, put delta | e−qT·N(d1), −e−qT·N(−d1) | 0.3162, −0.6820 |
| Call theta | per year ÷ 365 | −0.0366 a day |
| Implied volatility | call trades at $2.00: solve price(σ) = 2.00 | 27.25% |
The market price of $2.00 is above the $1.7157 model value at 25%, so the market is pricing in more volatility than the input: 27.25%. That is what implied volatility measures. It does not say whether the option is cheap or expensive, only which volatility makes the model agree with the price.
Assumptions and limits
Black and Scholes list the “ideal conditions” behind their formula on page 640 of the 1973 paper. Among them: “The short-term interest rate is known and is constant through time”; the variance rate of the stock’s return is constant, so prices at expiry are lognormal; “The stock pays no dividends or other distributions”; the option is “European,” that is, it can only be exercised at maturity; and there are no transaction costs or penalties to short selling. Real markets break each of these to some degree.
- European exercise only. The formula prices options that can be exercised only at expiry. FINRA explains that “American-style options allow the buyer of a contract to exercise at any time during the life of the contract”, and that in US markets “options on stocks and exchange-traded funds (ETF) are American-style”. An American option is worth at least as much as the European one with the same terms, because the right to exercise early cannot have a negative value.
- Early exercise and dividends. For calls, the Options Industry Council says that “in general it is not optimal to exercise a call option until the last day before expiration”, with an exception “the day before a stock goes ex-dividend”. So for a call on a stock with no dividend before expiry, the European price here is a reasonable stand-in. American puts, and calls with a dividend ahead, can be worth more than this calculator shows; brokers often use binomial or other models for them.
- Dividends as a yield. The continuous yield q suits an index or a stock with many small payouts. A single known cash dividend is often handled by subtracting its present value from S and setting q to 0. That gives a different number from a yield, especially near the ex-date.
- Constant volatility and rate. One σ for the whole life of the option is a simplification. In practice implied volatility differs by strike and expiry (the “smile” or “skew”), so one input will not match every option on the same stock.
- Theoretical value, not a quote. Real prices have a bid-ask spread, and fees apply. The model also ignores taxes and margin.
The Options Clearing Corporation’s Characteristics and Risks of Standardized Options is the disclosure document investors receive before trading listed options. It explains exercise styles and the risks of options positions.
How to read the Greeks
- Delta is the approximate change in the option price for a $1 move in the underlying. The Options Industry Council notes that delta is derived “from the Black-Scholes formula for pricing options”. Call delta runs from 0 to e−qT, put delta from −e−qT to 0.
- Gamma is how fast delta changes. It is highest near the strike and close to expiry.
- Vega is the change in price for one volatility point (25% to 26%). Longer-dated options have more vega.
- Theta is time decay. The per-day figure divides the annual rate by 365 calendar days. Some platforms divide by 252 trading days instead, which gives a larger daily number.
- Rho is the change in price for a 1% change in the interest rate. It is usually small for short-dated options.
All Greeks are local, “all else equal” estimates. For larger moves, change the input and recalculate instead of multiplying the Greek.
FAQ
What is the Black-Scholes formula?
It is a formula for the theoretical price of a European option. For a call: C = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2), where d1 = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T) and d2 = d1 − σ√T. S is the underlying price, K the strike, T the time in years, r the risk-free rate, q the dividend yield, σ the volatility and N the standard normal distribution. Fischer Black and Myron Scholes published it in 1973, and Robert Merton extended the theory the same year. Merton and Scholes received the 1997 prize in economic sciences “for a new method to determine the value of derivatives”.
What inputs does a Black-Scholes calculator need?
Five: the underlying price, the strike, the time to expiration, the volatility and the risk-free interest rate, plus a dividend yield if the underlying pays one. Four can be looked up. Volatility cannot: it is an estimate of how much the underlying will move until expiry, so the result is only as good as that estimate. Many traders work the other way and use the market price to solve for the implied volatility.
How is implied volatility calculated?
Implied volatility is the volatility that makes the Black-Scholes price equal to the option's market price. There is no closed formula for it, so it is found numerically. This calculator uses Newton's method (price error divided by vega) inside a bracket from 0.01% to 500% and falls back to bisection when a Newton step would leave the bracket. A price below the option's discounted intrinsic value, or above its upper bound, has no implied volatility.
Can Black-Scholes price American options?
Not exactly. The formula assumes European exercise, at expiry only. FINRA notes that in US markets options on stocks and ETFs are American-style, so they can be exercised at any time. For a call on a stock that pays no dividend before expiry, early exercise is generally not optimal and the European price is a close guide. American puts, and calls just before an ex-dividend date, can be worth more than the Black-Scholes value; brokers often use binomial or other models for them.
How does the dividend yield change the price?
A continuous dividend yield q lowers the expected growth of the underlying from r to r − q. In the formula the underlying price is replaced by S·e^(−qT). A higher yield lowers call values and raises put values, all else equal. For a single known cash dividend, a common alternative is to subtract the dividend's present value from S and set q to zero.
Why do broker platforms show different Greeks?
Usually because of conventions or inputs, not the formula. Brokers may quote theta per trading day (÷ 252) instead of per calendar day (÷ 365), use a different volatility for each strike, a different interest rate or dividend assumption, an American-exercise model, or prices per contract (× 100) instead of per share. This calculator shows vega per 1 volatility point, theta per calendar day and per year, and rho per 1% change in the rate.
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Sources
- Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy 81(3), 1973, pp. 637–654, doi:10.1086/260062. The formula and the “ideal conditions” (p. 640).
- Robert C. Merton, “Theory of Rational Option Pricing”, Bell Journal of Economics and Management Science 4(1), 1973, pp. 141–183 (EconPapers record). Extensions of the theory, including dividends.
- The Royal Swedish Academy of Sciences, press release for the 1997 prize in economic sciences (Merton and Scholes), 14 Oct 1997: awarded “for a new method to determine the value of derivatives”.
- John C. Hull, Options, Futures, and Other Derivatives (Pearson), chapter “The Black-Scholes-Merton Model”, example with S = 42, K = 40, r = 10%, σ = 20%, T = 0.5. Also in the HP 12C owner’s handbook (call 4.76, put 0.81) and the R package QFRM documentation (“See Hull p.339, Ex.15.6”; call 4.759422, put 0.8085994).
- The Options Industry Council, Option Price Behavior (delta from the Black-Scholes formula) and Covered Call (Buy/Write) (early exercise of calls and the ex-dividend exception).
- FINRA, Trading Options: Understanding Assignment, 14 Dec 2020. American-style and European-style exercise.
- The Options Clearing Corporation, Characteristics and Risks of Standardized Options.
- U.S. SEC / Investor.gov, An Introduction to Options – Investor Bulletin. Contract size and the factors that drive premiums.
- Graeme West, “Better approximations to cumulative normal functions”, Wilmott Magazine, 2005, implementing Hart (1968) algorithm 5666. The normal CDF used here.
Web sources retrieved 5 Nov 2026.
Educational tool only, not financial or investment advice. Results are a theoretical model value from the numbers you enter, not a quote or a forecast. Options involve risk and are not suitable for all investors. All prices in the examples are hypothetical except the published textbook example.