Black-Scholes Option Pricer
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Enter the spot price, strike, time to expiry, risk-free rate, volatility and dividend yield to price a European call or put with the Black-Scholes-Merton model, complete with the full set of option Greeks. Every calculation runs on your device.
What the Black-Scholes-Merton model does
The Black-Scholes-Merton model gives a closed-form fair value for a European option, meaning one that can only be exercised on its expiry date. It assumes the underlying asset follows geometric Brownian motion with constant volatility, that markets are frictionless, and that a constant risk-free rate and dividend yield apply over the option's life. Under those assumptions it produces a single theoretical price for a call or a put.
This calculator implements the dividend-adjusted (Merton) version. It computes two intermediate quantities, d1 and d2, then combines them with the standard normal cumulative distribution function to weigh the discounted payoff of the option against the discounted cost of the strike.
The formulas used
The model first computes d1 = (ln(S/K) + (r - q + sigma^2/2) x T) / (sigma x sqrt(T)) and d2 = d1 - sigma x sqrt(T), where S is spot, K is strike, T is time to expiry in years, r is the risk-free rate, q is the dividend yield and sigma is volatility.
The call price is S x e^(-qT) x N(d1) - K x e^(-rT) x N(d2), and the put price is K x e^(-rT) x N(-d2) - S x e^(-qT) x N(-d1), where N() is the standard normal CDF. As a check, with S=100, K=100, T=1, r=5%, sigma=20% and no dividend, the call comes out to about 10.45.
- N(d1) and N(d2) are the risk-neutral probabilities that weight the payoff
- e^(-rT) discounts the strike back to today at the risk-free rate
- e^(-qT) discounts the spot for dividends paid over the option's life
Reading the Greeks
The Greeks measure how the option price responds to changes in the inputs. Delta is the change in option value per one-dollar move in the underlying; gamma is how fast delta itself changes. Vega measures sensitivity to volatility and is shown here per one percentage point of vol. Theta is the daily time decay, computed as the annual figure divided by 365, and rho is the sensitivity to a one percentage point change in interest rates.
Traders use the Greeks to hedge and to understand risk. A near-the-money option has the highest gamma and the steepest time decay, while deep in- or out-of-the-money options behave more like the stock or like nothing at all.
Limitations to keep in mind
Black-Scholes assumes constant volatility, but real markets show a volatility smile where implied vol varies by strike and expiry. It also ignores transaction costs, assumes continuous trading, and cannot value early exercise. For that reason the output is a theoretical benchmark, not a market quote.
Use it to sanity-check option prices, to explore how Greeks shift as inputs change, or to back out implied volatility by adjusting sigma until the model price matches the market. Treat the numbers as a model estimate rather than a guarantee.
Frequently asked questions
How is the normal distribution computed?
The standard normal cumulative distribution is evaluated with the Zelen and Severo approximation from Abramowitz and Stegun (26.2.17), which is accurate to roughly 7.5 x 10^-8, more than enough for pricing.
Why do call and put prices differ by a fixed amount?
They are linked by put-call parity: Call minus Put equals S x e^(-qT) minus K x e^(-rT). If you change spot or strike, both prices move together to preserve that relationship.
What units should time and rates be in?
Time to expiry is in years, so 90 days is about 0.2466. The risk-free rate, volatility and dividend yield are entered as annual percentages and the tool converts them to decimals for you.
What does a negative theta mean?
Theta is usually negative for long options because they lose value as expiry approaches, all else equal. The figure shown is the expected value lost per calendar day.
Can I use this to find implied volatility?
Indirectly. Adjust the volatility input until the model price matches the option's market price; the volatility that produces a match is the implied volatility for that option.
Does it handle zero volatility or zero time?
No. The formula requires positive volatility and positive time to expiry because it divides by sigma times the square root of time. Enter values greater than zero for a result.
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