Black-Scholes Option Pricing Calculator
Use this Black-Scholes calculator to price European call and put options from spot, strike, time, rate and volatility, and see each option's fair value.
What it does
The Black-Scholes Calculator prices a European call and put option on a non-dividend-paying asset. Enter the spot price, strike, time to expiry, risk-free rate and volatility to see the theoretical fair value of each contract.
Data sources & parameters
This calculator uses standard financial formulas and runs entirely in your browser. It needs no external rate data — your result is computed only from the values you enter above.
We always strive to use the latest data for every variable so our results and reports are as accurate as possible. Even so, errors are possible — you should not rely on this tool or its report alone, and we accept no liability for any decision made based on it.
The formula
call = S·N(d1) − K·e^(−rT)·N(d2), with d1 = (ln(S/K) + (r + v²/2)T) / (v√T) and d2 = d1 − v√T
How it works
The model discounts the strike at the risk-free rate and weighs the probability that the option finishes in the money, captured by the normal distribution terms N(d1) and N(d2). The call value is the spot times N(d1) minus the discounted strike times N(d2); the put follows from put-call parity.
Where it is used
- Estimating the fair premium of an equity option before trading
- Seeing how volatility and time to expiry change an option's value
- Comparing quoted market prices against a theoretical benchmark
Worked example
A stock trades at $100.00 with a strike of $100.00, one year to expiry, a 5% risk-free rate and 20% volatility. The model prices the call at … and the put at …, with d1 of ….
What the Black-Scholes calculator does
The calculator prices a European call and put on a non-dividend-paying asset. Enter the spot price, strike price, time to expiry, risk-free rate and volatility, and it returns the theoretical fair value of each contract, which you can compare with the price quoted in the market.
Background
Fischer Black and Myron Scholes published the model in 1973, with key contributions from Robert Merton. It gave traders a closed-form equation for an option's value where before there had been judgement and rules of thumb. Merton and Scholes received the 1997 Nobel Memorial Prize in Economics for the work; Black had died two years earlier and could not share it.
The model rests on the idea that an option's payoff can be replicated by continuously trading the underlying asset and a risk-free bond. The option must then be worth the cost of that replicating portfolio, or a risk-free profit would exist. Because of this no-arbitrage argument, the stock's expected return does not appear in the formula.
How the formula works
The calculator first computes two terms, d1 and d2. d1 measures how far the option is in the money, adjusted for drift at the risk-free rate and for volatility over the remaining life. d2 is d1 minus the volatility multiplied by the square root of time; N(d2) is the risk-neutral probability that a call finishes in the money.
Both are passed through the standard normal cumulative distribution, N(d1) and N(d2), which turns them into probabilities between zero and one. The call value is the spot price times N(d1), minus the strike discounted at the risk-free rate times N(d2). The put value follows from put-call parity. The calculator also shows d1.
The five inputs
- Spot price: a higher spot raises the call and lowers the put.
- Strike price: a higher strike lowers the call and raises the put.
- Time to expiry: more time gives the price more room to move, raising both calls and puts.
- Risk-free rate: a higher rate raises the call slightly and lowers the put, through the discounting of the strike.
- Volatility: higher volatility raises both the call and the put, because a wider spread of outcomes is valuable when the downside is limited to the premium.
Why volatility matters most
Spot, strike, time and rate are known or quoted; volatility has to be estimated, and the price is very sensitive to it. Traders often run the model in reverse, solving for the volatility that would justify an option's market price. That implied volatility is one of the most closely watched figures in options markets. When the model price differs from a quoted price, the gap usually reflects different volatility assumptions more than a real mispricing.
Using the results
The model assumes constant volatility, no dividends, no transaction costs and continuous trading, which real markets only approximate, so treat its output as a benchmark. Change the volatility to see how much of an option's price is time value from uncertainty. Shorten the time to expiry and watch the price fall, a preview of the time decay that erodes long positions. Move the strike above and below the spot to see how an option shifts between mostly intrinsic value and mostly time value.
Common mistakes
A volatility guessed from one day's move is unreliable; use an estimate covering the option's remaining life. This form assumes no dividends, so applying it directly to a high-dividend stock or index overprices the call and underprices the put. And Black-Scholes is a model, useful mainly as a consistent reference point that everyone can compute the same way.
This article is general information only and is not financial advice.
About This Black-Scholes Calculator
Use this Black-Scholes calculator to get an instant estimate based on the figures you enter. Adjust the inputs above to match your situation, or browse all our calculators to plan your next step.
Frequently asked questions
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