Black-Scholes Calculator (Call Option)
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
The Black-Scholes model gives the theoretical price of a European call option with C = S×N(d1) − K×e^(−rT)×N(d2). For the canonical example S=K=€100, risk-free rate 5%, volatility 20%, one-year expiry, the theoretical price is about €10.45, the value most cited in quantitative finance textbooks to illustrate this model.
Explanation
The Black-Scholes-Merton model, published in 1973 (and awarded the Nobel Memorial Prize in Economic Sciences in 1997), revolutionized quantitative finance by providing a closed-form formula to value a European option — a contract that gives the right, but not the obligation, to buy a stock at a price fixed in advance (the strike price K), on a given expiry date. The formula combines five ingredients: the current stock price S, the strike price K, the risk-free rate r (often approximated by the government bond yield), the volatility σ (a measure of how much the stock price is expected to fluctuate) and the time remaining until expiry T. The terms N(d1) and N(d2) represent risk-adjusted probabilities, computed from the cumulative distribution function of the standard normal distribution — a function with no simple algebraic expression, which this calculator approximates with extremely fine precision (error below 10⁻⁷) using a classic polynomial formula from the statistical literature. This model rests on several strong simplifying assumptions: constant and known volatility, no transaction costs, the ability to trade continuously, and log-normally distributed stock returns — assumptions that always deviate somewhat from real financial markets, which is why an option's real market price often diverges slightly from the theoretical price computed here, notably through the well-documented "volatility smile" phenomenon. The risk-free rate r used here plays a role similar to the rate used by our compound interest calculator, and reflects the same idea of the time value of money that underlies discounting K by e^(−rT).
Example: an at-the-money option over one year
Inputs
Stock price S = €100. Strike price K = €100 (an "at-the-money" option). Risk-free rate r = 5%. Volatility σ = 20%. Time to expiry T = 1 year.
Calculation
d1 = [ln(100/100) + (0.05+0.02)×1] ÷ (0.20×1) = 0.35. d2 = 0.35 − 0.20 = 0.15. N(0.35)≈0.6368, N(0.15)≈0.5596. C = 100×0.6368 − 100×e⁻⁰·⁰⁵×0.5596 ≈ €10.4506.
Result
This call option is theoretically worth about €10.45 — roughly 10.45% of the stock price itself, a premium that reflects both the time remaining until expiry and the uncertainty (the volatility) about the stock's future price.
Frequently asked questions
Why does an option's price increase with the volatility of the underlying stock?
The more volatile a stock is, the more scenarios exist where its final price ends up well above the strike price, making the option very profitable — and since the holder of a call option never loses more than the premium paid (unlike a shareholder, who fully bears any decline), higher volatility only increases the potential upside without increasing the downside risk beyond the initial premium. It's this fundamental asymmetry between unlimited potential gain and limited loss that explains why an option's price consistently increases with volatility — a very different kind of risk-taking from what our present value calculator captures, which instead values a fixed, known future cash flow with no such asymmetry.
Can this model also price a put option?
This calculator focuses on the call option, but the price of a put option with the same characteristics can be derived directly through put-call parity: P = C − S + K×e^(−rT). This relationship, itself a consequence of an arbitrage argument independent of the Black-Scholes model, recovers the put price from the call price already computed here, without redoing the whole calculation from scratch.
Why does an option's real market price often differ from the theoretical Black-Scholes price?
The model's weakest assumption is that of a constant volatility, identical across all expiries and strike prices, which doesn't match what's actually observed on markets (the well-known "volatility smile", where the market's implied volatility varies with the strike price). In practice, traders often use the model backwards: they observe the option's real market price and back out the implied volatility that, fed back into Black-Scholes, would reproduce that price — a volatility that then becomes a market data point in its own right, rather than a simple input parameter.