Black–Scholes Option Price & Greeks Calculator
The theoretical value of an option, its Greeks and its implied volatility.
The Greeks
Value against the price of the underlying
Prices and deltas now if the underlying were higher or lower
How this was calculated
Results are estimates for general information and planning, not financial advice. Banks and institutions may calculate differently (rounding, fees, rate changes). Confirm figures with your lender or a qualified adviser before deciding.
About the Black–Scholes Option Price & Greeks Calculator
The Black–Scholes model gives the theoretical value of a European option — one that can be exercised only at expiry — from five inputs: the price of the underlying, the strike, the time to expiry, the volatility and the risk-free rate. Robert Merton added a dividend yield, which also lets the same formula price options on stock indices, currencies and futures.
This calculator works out the call and put prices, all five Greeks (delta, gamma, vega, theta and rho) for both, the implied volatility from a market price, and checks put–call parity. It shows every formula with your numbers, explains how to count the days to expiry and which rate to use, and keeps everything in your browser.
How to use it
- Enter the price of the underlying now, the strike and the days to expiry — calendar days, or trading days if you choose that count.
- Enter the volatility a year in % (for example 25), the risk-free rate in % a year with how it is compounded, and the dividend yield (0 if none).
- Read the call and put prices and the Greeks table; copy the summary or download the numbers as CSV.
- To find the implied volatility, enter the market price of the call, the put or both. Use this volatility puts it into the model.
Examples
Underlying 42, strike 40, six months (182.5 calendar days), volatility 20%, rate 10% continuous, no dividend
d₁ 0.7693 · d₂ 0.6278 · call 4.7594 · put 0.8086
Underlying 49, strike 50, 20 weeks (0.3846 years), volatility 20%, rate 5% continuous
Call 2.4005 · delta 0.5216 · gamma 0.0655 · vega 0.1211 per 1% volatility · theta −4.3054 a year (−0.0118 a calendar day) · rho 0.0891 per 1% rate
Index 930, strike 900, two months, volatility 20%, rate 8%, dividend yield 3%
Call 51.8330 · put 14.5510
A call trading at 1.875 with the underlying at 21, strike 20, three months, rate 10%, no dividend
Implied volatility 23.45% a year
Common uses
- Check whether an option quote looks cheap or dear against your own volatility estimate.
- Turn a quoted option price into its implied volatility, and compare strikes or expiries.
- See how much an option loses each day (theta) or gains from a rise in volatility (vega).
- Hedge a position with the underlying using delta, and see how fast the hedge changes (gamma).
The formulas
With S the price of the underlying, K the strike, T the time to expiry in years, σ the volatility, r the risk-free rate and q the dividend yield (both continuously compounded) and N the standard normal distribution:
- d₁ = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T), d₂ = d₁ − σ√T
- Call = S e^(−qT) N(d₁) − K e^(−rT) N(d₂)
- Put = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁)
The Greeks are the formula’s sensitivities (Hull, Options, Futures, and Other Derivatives): delta e^(−qT) N(d₁) for a call and −e^(−qT) N(−d₁) for a put; gamma e^(−qT) φ(d₁) ÷ (Sσ√T); vega S e^(−qT) φ(d₁) √T; theta and rho as Hull gives them. Vega and rho are shown per percentage point (1% more volatility or interest), theta per day and per year.
Days to expiry and the rate
- Time: T is the days to expiry ÷ 365 when you count calendar days, or ÷ 252 when you count trading days (about 252 a year on most exchanges). Theta per day uses the same count. Calendar days are the usual choice for interest; some traders count trading days because prices only move when markets are open.
- Rate: the formula needs a continuously compounded rate. If your rate is quoted as a yearly yield (compounded once a year), choose that and the calculator uses ln(1 + rate). A government treasury-bill yield for a term close to the expiry is a common choice of risk-free rate.
- Volatility is the yearly standard deviation of the underlying’s returns, in %. Use your estimate, or the implied volatility of a similar option.
Dividends, futures and currencies
The dividend yield q covers a stock index or a share whose dividends are spread through the year. For an option on a futures price, enter the futures price as the underlying and set the dividend yield equal to the rate: this is Black’s model. For a currency option, the dividend yield is the risk-free rate of the foreign currency (the Garman–Kohlhagen model). A share with one known cash dividend before expiry is usually priced by taking the present value of that dividend off the share price.
Implied volatility and put–call parity
The implied volatility is the volatility that makes the model price equal to a market price. There is no formula for it: the calculator solves for it with Newton–Raphson steps from Manaster and Koehler’s starting point, and falls back to bisection when a step would leave the range. A price below the option’s value with no volatility at all, or above its upper bound, has no implied volatility, and the calculator says so.
Put–call parity links European calls and puts with the same strike and expiry: c − p = S e^(−qT) − K e^(−rT). Model prices always satisfy it; enter market prices for both to see how far they are from it.
Limitations
- The prices are for European options. An American option can be worth more, because it can be exercised early — a put deep in the money, or a call just before a dividend.
- The model assumes constant volatility and rates and no jumps in the price. Real markets show different implied volatilities at different strikes (the smile), so one volatility does not fit every strike.
- Dividends are a continuous yield; a known cash dividend is better handled by lowering the price of the underlying by its present value.
- A theoretical price is not a quote: bid–ask spreads, fees and taxes are not included, and nothing here is advice to trade.
Privacy
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
What volatility should I use?
The volatility the market expects until expiry. Traders usually take the implied volatility of options on the same underlying: enter a market price and the calculator finds it. A historical volatility (the standard deviation of past daily returns × √252) is a common alternative.
What does a delta of 0.54 mean?
The option’s price changes by about 0.54 for a change of 1 in the price of the underlying, so 100 such options move like 54 units of the underlying. A put’s delta is negative: it gains when the price falls.
Is theta per calendar day or per trading day?
It follows how you count the days to expiry: per calendar day (Θ ÷ 365) by default, per trading day (Θ ÷ 252) when you choose trading days. The yearly theta (Hull’s Θ) is shown too.
Is N(d₂) the chance that the option ends in the money?
Only in the model’s risk-neutral world, where every asset earns the risk-free rate. It is useful for comparing options, but it is not a forecast of what will happen.
Why is my broker’s price different?
Brokers show market prices (and sometimes Greeks from their own models): the market may expect a different volatility, the option may be American, dividends may be modelled as cash amounts, and the rate or the day count may differ. The implied volatility tells you which volatility the market price assumes.
Can I price index options with it?
Yes, if they are European-style: enter the index level, the strike, the days to expiry, a risk-free rate and the index’s dividend yield. For options on index futures, use the futures price and set the dividend yield equal to the rate.