What the Black-Scholes model does
An option gives the right, but not the obligation, to buy (a call) or sell (a put) a stock at a fixed strike price before or at a set date. In 1973, Fischer Black and Myron Scholes, with Robert Merton, published a formula for the fair price of such an option. It earned a Nobel Prize in economics in 1997 and is still the reference model for options pricing.
The formula
With S the stock price, K the strike, t the time to expiry in years, σ the volatility, r the risk-free rate and q the dividend yield:
- 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)
N() is the cumulative standard normal distribution.
Worked example
A stock trades at $100. You look at options with a $105 strike, 90 days to expiry, 25% volatility, a 4.5% risk-free rate and no dividend.
| Result | Value |
|---|---|
| Call price | $3.35 |
| Put price | $7.20 |
| Call delta | 0.405 |
| Put delta | −0.595 |
| Gamma | 0.0312 |
| Vega | $0.19 per volatility point |
| Call theta | −$0.031 per day |
| Risk-neutral chance the call ends in the money | 35.7% |
Call minus put equals S − K e^(−rt) = −$3.84, which is put-call parity: a quick check that both prices are consistent.
The Greeks
- Delta: how much the option moves for a $1 move in the stock. It is also a rough hedge ratio.
- Gamma: how fast delta changes; highest for at-the-money options near expiry.
- Vega: the price change for a one-point rise in volatility.
- Theta: the time decay, usually negative for option buyers and faster in the last weeks.
Assumptions and limits
The model assumes constant volatility and interest rates, prices that follow a lognormal random walk with no jumps, no transaction costs and European exercise only. Real markets show a volatility smile: options far from the money trade at higher implied volatilities than the model predicts. Use the result as a benchmark, not as a guaranteed fair value, and remember that options can expire worthless.
This calculator gives estimates for education and planning. It is not investment, tax or legal advice; for decisions about your own situation, talk to a qualified financial professional.