Black-Scholes Calculator
Theoretical call and put option prices plus delta, gamma, theta, vega, and rho
Black-Scholes Inputs
Enter as a percentage. 20 = 20% annualized volatility.
Use the yield on a government bond matching your expiry.
Set to 0 for non-dividend stocks, ETFs, or indices without distributions.
Theoretical Price & Greeks
Call Price
Put Price
At-the-Money (ATM) • Greeks below are for the selected call option
Delta
0.6368
Price change per $1 spot move
Gamma
0.0188
Delta change per $1 spot move
Theta (per day)
$-0.02
Daily time decay
Vega (per 1% vol)
$0.38
Price change per 1% IV
Rho (per 1% rate)
$0.53
Price change per 1% rate move
Model Internals
| d1 | 0.3500 |
| d2 | 0.1500 |
| Approx. probability ITM | 55.96% |
Note
Prices and Greeks are calculated using the Black-Scholes-Merton model with continuous dividend yield. The model assumes European-style exercise, constant volatility, and lognormal price distribution. Real-world American options and stocks with discrete dividends may deviate slightly from these values.
Complete Guide to the Black-Scholes Model
What is the Black-Scholes Model?
The Black-Scholes-Merton model, published in 1973, is the foundational formula for pricing European-style options. It derives a theoretical fair value for a call or put from six observable inputs: the underlying's spot price, the strike price, time to expiration, expected volatility, the risk-free interest rate, and any continuous dividend yield.
This tool computes both the call and put price from the same inputs, then breaks down all five Greeks for whichever side you are trading. Pair it with the Option Delta Calculator for a deeper look at sensitivity, or the Options Profit Calculator to model your actual trade P&L once you know the theoretical entry price.
Formula
Black-Scholes-Merton Pricing:
d1 = [ln(S/K) + (r - q + 0.5*sigma^2)*T] / (sigma*sqrt(T))
d2 = d1 - sigma*sqrt(T)
Call = S*e^(-qT)*N(d1) - K*e^(-rT)*N(d2)
Put = K*e^(-rT)*N(-d2) - S*e^(-qT)*N(-d1)
Where: S = spot price, K = strike price, T = time to expiry in years, r = risk-free rate, q = dividend yield, sigma = annualized volatility, and N() is the standard normal cumulative distribution function.
Benefits
Fair Value Benchmark
Compare a broker's quoted premium against a theoretical price to spot rich or cheap options.
Call and Put Together
Both sides of the same strike are computed at once, so you can compare them instantly.
Full Greeks Panel
Delta, gamma, theta, vega, and rho are all derived from the same model in one pass.
Dividend-Aware
A continuous dividend yield input keeps prices accurate for dividend-paying underlyings.
Tips
Tip 1: Use implied volatility from the option chain, not trailing historical volatility — the market's forward-looking estimate is what actually prices the option.
Tip 2: Match the risk-free rate to a government bond yield with a maturity close to your option's expiry.
Tip 3: Cross-check theoretical prices against real breakeven levels using the Options Breakeven Calculator before placing a trade.
Common Mistakes
Wrong Volatility Input
Plugging in historical volatility instead of implied volatility produces a theoretical price that can diverge sharply from the market.
Ignoring Dividends
Leaving dividend yield at 0% for a dividend-paying stock overstates call prices and understates put prices.
Applying It to American Options Blindly
Early-exercise value on American puts and dividend-sensitive calls is not captured by the European Black-Scholes formula.
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OpenFrequently Asked Questions
What is the Black-Scholes model?
The Black-Scholes (Black-Scholes-Merton) model is a mathematical formula for pricing European-style call and put options. It estimates a theoretical fair value using the underlying's spot price, strike price, time to expiry, volatility, the risk-free rate, and dividend yield, assuming lognormal price movement and constant volatility.
How is the Black-Scholes price calculated?
Call Price = S*e^(-qT)*N(d1) - K*e^(-rT)*N(d2), and Put Price = K*e^(-rT)*N(-d2) - S*e^(-qT)*N(-d1), where d1 = [ln(S/K) + (r - q + 0.5*sigma^2)*T] / (sigma*sqrt(T)) and d2 = d1 - sigma*sqrt(T). N() is the standard normal cumulative distribution function.
How does this compare to the Option Delta Calculator?
The Option Delta Calculator focuses purely on delta and the other Greeks for a single option. This tool starts one step earlier: it computes the full theoretical price for both the call and the put side-by-side from the same inputs, then shows all five Greeks (delta, gamma, theta, vega, rho) for whichever side you select.
What are common mistakes people make with Black-Scholes pricing?
The three most frequent errors are using historical volatility instead of implied volatility (the market's forward-looking estimate), ignoring dividends on dividend-paying stocks which inflates call prices and understates puts, and applying the model to American-style options, which can be exercised early and may trade above the European theoretical price.
Does Black-Scholes work for American options?
Not precisely. Black-Scholes assumes European exercise (only at expiry). American options that can be exercised early, especially dividend-paying stock puts and deep ITM calls before an ex-dividend date, can carry a small early-exercise premium that this model does not capture. It remains a very close approximation for most non-dividend and short-dated contracts.
Worked example?
With Spot = $100, Strike = $100, 365 days to expiry, 20% volatility, a 5% risk-free rate, and 0% dividend yield: d1 = 0.35 and d2 = 0.15, giving a Call Price of $10.45 and a Put Price of $5.57. The call's delta is 0.6368, gamma is 0.0188, and vega is $0.3752 per 1% change in volatility.