What this calculator does
The Black-Scholes model gives a theoretical price for a European option from six inputs. Five of them are observable; the sixth, volatility, is not, which is why in practice the model is often run backwards to infer what volatility the market price implies.
Volatility dominates the result. An at-the-money one year call on a $100 stock is worth $6.80 at 10 per cent volatility and $18.02 at 40 per cent, with nothing else changed. Uncertainty itself is what an option sells.
The formula
Two intermediate terms, d1 and d2, combine the price ratio, rate, volatility and time. The call price is the discounted expected payoff, and the put follows from the same terms, with put-call parity holding exactly between them.
| Term | Meaning |
|---|---|
| d1 and d2 | Standardised terms feeding the cumulative normal distribution. |
| Volatility (σ) | The annualised standard deviation of returns, the only unobservable input. |
| Put-call parity | The fixed relationship between call and put prices at the same strike and expiry. |
The inputs explained
| Field | What to enter |
|---|---|
| Current share price ($) | The current share price. |
| Strike price ($) | The strike price of the option. |
| Risk-free interest rate (%) | The risk-free interest rate, annualised. |
| Dividend yield (%) | The dividend yield on the underlying share. |
| Volatility (annualised) (%) | Annualised volatility as a percentage. This is the input the model cannot observe. |
| Time to expiry (years) | Time to expiry in years. Three months is 0.25. |
When to use it
Pricing an option position
The theoretical value is the starting point for judging whether a quoted price is reasonable.
Backing out implied volatility
Running the model in reverse from a market price gives the volatility the market is assuming.
Valuing employee share options
Accounting standards require a model of this kind, usually with adjustments for the vesting terms.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How much does volatility change an option price?
The same at-the-money option at three volatility levels.
Questions
What are the model's main assumptions?
Constant volatility, constant interest rates, lognormally distributed returns, no transaction costs and European exercise. Real markets violate every one of these to some degree, most obviously constant volatility.
What is the volatility smile?
The observation that options at different strikes imply different volatilities, when the model says they should all imply the same one. It is direct evidence that real return distributions have fatter tails than the model assumes.
Does this work for American options?
Not exactly. American options can be exercised early, which adds value the formula does not capture. For non-dividend-paying stocks the early exercise premium on a call is zero, so the formula still applies; puts need a numerical method.
Why do call and put prices differ by a fixed amount?
Because of put-call parity, which follows from arbitrage rather than from any model. A call minus a put at the same strike must equal the discounted share price less the discounted strike, and the model respects that exactly.
For the parity relationship on its own, see the put-call parity calculator. For volatility as a risk measure, see the Sharpe ratio calculator.