Updated
Using Black scholes calculator
d₁ = (ln(S/K) + (r − q + σ²/2)T)/(σ√T); d₂ = d₁ − σ√T. Call = S exp(−qT) Φ(d₁) − K exp(−rT) Φ(d₂). Put = K exp(−rT) Φ(−d₂) − S exp(−qT) Φ(−d₁). Rates and volatility in these formulas are decimal fractions.
European exercise only, continuous dividend yield, constant rates and volatility, and no transaction costs. The value is per underlying unit, not per contract. At expiry the answer is intrinsic value; at zero volatility it is the discounted deterministic payoff. This is a model value, not a market quote or trading recommendation.
A worked example
Example inputs: underlying price = 100, strike price = 100, years to expiry = 1, annual volatility (%) = 20, continuous risk-free rate (%) = 5, continuous dividend yield (%) = 0. European call value = 10.4505835722.
Before you use the result
European exercise only, continuous dividend yield, constant rates and volatility, and no transaction costs. The value is per underlying unit, not per contract. At expiry the answer is intrinsic value; at zero volatility it is the discounted deterministic payoff. This is a model value, not a market quote or trading recommendation.
Example results for underlying price
These examples use Strike price: 100; Years to expiry: 1; Annual volatility (%): 20; Continuous risk-free rate (%): 5; Continuous dividend yield (%): 0. They are reference calculations, not recommended settings.
| Underlying price | European call value |
|---|---|
| 50 | 0.00 |
| 100 | 10.45 |
| 200 | 104.88 |
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ToolOctopus. “Black scholes calculator.” Updated 2026-09-26. https://tooloctopus.com/black-scholes-calculator.
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