How this calculation works
An option-pricing model connects a specified payoff with assumptions about price dynamics, discounting and time. Black–Scholes–Merton gives a theoretical European option value under constant volatility and continuous rate assumptions. This page also reports delta, gamma and vega so the local sensitivities are distinguishable from the option value itself. It uses entered scenarios and makes no live market-data request.
d1=[ln(S/K)+(r−q+sigma²/2)T]/(sigma√T); call=Se^(−qT)N(d1)−Ke^(−rT)N(d2)A worked example
For spot and strike one hundred, one year, volatility twenty percent, rate five percent and zero dividend yield, the textbook European call value is about 10.45058 per unit. The corresponding put is about 5.57353. Their difference equals one hundred minus one hundred discounted at five percent continuously, which checks the implemented payoff relationship.
Good to know
The normal CDF uses a polynomial approximation with absolute probability error around 1e−7. Deep-tail option prices can therefore lose relative accuracy even when absolute CDF error is small. American exercise, stochastic volatility, jumps, discrete dividends, fees and trading execution are absent. The result is an educational model value, not a forecast, recommendation or promised obtainable premium.
Reference: Options Industry Council — Black Scholes formula (opens in a new tab) ↗
What this calculation answers
An option-pricing model connects a specified payoff with assumptions about price dynamics, discounting and time. Black–Scholes–Merton gives a theoretical European option value under constant volatility and continuous rate assumptions.
This page also reports delta, gamma and vega so the local sensitivities are distinguishable from the option value itself. It uses entered scenarios and makes no live market-data request.
The Covered interest FX forward scenario calculator uses entered funding-parity assumptions. A theoretical option premium also depends on volatility and payoff, not just relative interest rates.
Use the right input conventions
All prices are per underlying unit in one currency. Remaining time is positive years, volatility is annualized percent, and the rate and dividend yield are continuously compounded annual percentages.
Vega is quoted for one volatility percentage point, not a full volatility fraction of one. Results do not include an option contract multiplier or assume that an American-style option can only be exercised at expiry.
Interpret the result
Delta is the model’s local value sensitivity to spot; gamma describes the change in delta. These are derivatives at the entered point, not guarantees about a large or discontinuous price move.
A put-call parity residual is shown as an algebraic consistency check. It does not validate the model assumptions, identify an arbitrage opportunity or compare the output with actual bid and ask prices.
Work through the example
For spot and strike one hundred, one year, volatility twenty percent, rate five percent and zero dividend yield, the textbook European call value is about 10.45058 per unit.
The corresponding put is about 5.57353. Their difference equals one hundred minus one hundred discounted at five percent continuously, which checks the implemented payoff relationship.
Where the model stops
The normal CDF uses a polynomial approximation with absolute probability error around 1e−7. Deep-tail option prices can therefore lose relative accuracy even when absolute CDF error is small.
American exercise, stochastic volatility, jumps, discrete dividends, fees and trading execution are absent. The result is an educational model value, not a forecast, recommendation or promised obtainable premium.
A few questions, answered
Is vega reported for a one hundred percentage point change?
No. It is scaled to one volatility percentage point, such as twenty to twenty-one percent.
What changes in the comparison?
Switch payoff type at the same inputs and inspect put-call parity.