The Black-Scholes model gives a closed-form theoretical price for a European option directly from five inputs — no simulation or lookup table required.
How it works
Combining spot price, strike price, risk-free rate, volatility, and time to expiry into the model’s two intermediate terms (d1 and d2) and applying the cumulative normal distribution to each gives both the theoretical call and put price.
What this does not include
This does not include dividends on the underlying (the classic model assumes none), American-style early exercise, or the reality that implied volatility varies by strike and expiry in real markets — all simplifications the original 1973 model makes.
How to use this calculator
- Enter spot price, strike price, risk-free rate, volatility, and time to expiry.
Frequently asked questions
Why does higher volatility raise both the call and put price?
A wider range of possible future prices raises the value of the optionality itself in both directions — more upside potential for a call, more downside protection value for a put.
Does this work for American-style options?
Not precisely — American options can be exercised early, which can add value the pure Black-Scholes model doesn’t capture; more complex models (like binomial trees) are typically used for American-style pricing.
What is N(d1) and N(d2) actually measuring?
Roughly speaking, N(d2) approximates the risk-neutral probability the option expires in the money, while N(d1) relates to the option’s sensitivity to the underlying’s price (its delta).
