A Black-Scholes calculator with Greeks and implied volatility
This Black-Scholes calculator prices European call and put options and shows all five Greeks — delta, gamma, theta, vega and rho — with their units spelled out. It handles a continuous dividend yield (the Merton extension, used for index options), checks put-call parity, and solves for implied volatility from market prices.
Every number is shown with the working: d1, d2, N(d1), N(d2) and the discount factors, so students can follow a textbook problem line by line and traders can sanity-check a broker’s option chain for Nifty, Bank Nifty, US stocks or the S&P 500. Nothing is sent anywhere — the maths runs in your browser.
How to use the Black-Scholes Calculator
- Enter the underlying price, the strike and the time to expiry in years or calendar days.
- Enter the annual volatility, the risk-free rate and, for an index or dividend stock, the dividend yield (all in %).
- Read the call and put prices, then the Greeks table — theta is shown per calendar day, per trading day and per year.
- To find implied volatility, type the market price of a call and/or put; the parity check also tests those prices.
- Copy the results for your notes or spreadsheet.
Black-Scholes Calculator features
Black-Scholes-Merton
Call and put prices with a continuous dividend yield q.
Greeks with clear units
Theta per day and per year, vega and rho per 1 percentage point, delta and gamma per 1 point of the underlying.
Implied volatility solver
Newton-Raphson from a Manaster-Koehler starting point, with a bisection fallback that cannot diverge.
Put-call parity
Checks C − P = S·e^(−qT) − K·e^(−rT) for the model and for market prices.
Verified
Matches textbook and published software examples to four decimals (listed below).
The Black-Scholes formula
d1 = [ln(S/K) + (r − q + σ²/2)T] / (σ√T), d2 = d1 − σ√T. Then call = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2) and put = K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1), where N is the standard normal distribution (see the normal distribution calculator). Rates are continuously compounded; T is in years, so 30 calendar days is 30 ÷ 365.
The Greeks follow by differentiation. Delta is e^(−qT)·N(d1) for a call; gamma and vega are the same for calls and puts. Textbooks quote theta per year and vega and rho per 1.00 (100%) change — divide theta by 365 for a calendar day (some platforms use 252 trading days) and vega and rho by 100 for one percentage point.
Checked against published examples
Examples checked 19 October 2026; the selftest of this page repeats them in the browser.
| Source | Inputs | Published | This calculator |
|---|---|---|---|
| Hull, Options, Futures and Other Derivatives, Example 15.6 | S 42, K 40, r 10%, σ 20%, T 0.5 | call 4.76, put 0.81 | 4.7594, 0.8086 |
| Hull, Greek letters example | S 49, K 50, r 5%, σ 20%, T 20 weeks | Δ 0.522, Γ 0.066, Θ −4.31/yr, vega 12.1, rho 8.91 | 0.5216, 0.0655, −4.3053, 12.1055, 8.9070 |
MathWorks blsprice (index, dividend yield) | S 910, K 980, r 2%, T 0.25, σ 25%, q 2.5% | call 19.6863, put 90.4683 | 19.6863, 90.4683 |
MathWorks blsimpv | S 100, K 95, r 7.5%, T 0.25, call 10 | σ 31.30% | 31.30% |
| Hull, implied volatility illustration | S 21, K 20, r 10%, T 0.25, call 1.875 | about 23.5% | 23.45% |
Limits
- Black-Scholes values European options. Most US single-stock options are American, and early exercise can make them worth more — especially puts and calls before a dividend.
- It assumes constant volatility and rates; real markets show a volatility smile, so each strike has its own implied volatility.
- Discrete cash dividends are approximated by a continuous yield.
- To see a whole strategy’s payoff, use the options profit calculator.
Black-Scholes Calculator FAQ
What does a Black-Scholes calculator do?
It gives the theoretical value of a European call and put from five inputs — price, strike, time, volatility and interest rate (plus dividend yield) — and the Greeks that measure how that value changes.
Is theta per day or per year?
The formula gives theta per year. This option Greeks calculator shows per calendar day (÷ 365), per trading day (÷ 252) and per year, labelled.
How is implied volatility calculated?
There is no closed formula, so the implied volatility calculator searches for the σ that makes the Black-Scholes price equal the market price — Newton-Raphson steps using vega, falling back to bisection if a step would leave the bracket.
Why does my broker show a different price?
Brokers use market implied volatility for each strike, their own rate and dividend assumptions, and sometimes a model for American exercise. Bid-ask spreads matter too.
What rate should I use for Indian options?
A short-term risk-free rate such as the 91-day treasury bill yield, entered as a percentage. The effect on short-dated options is small.
Is this investment advice?
No — calculations only. Options carry high risk of loss.