About the Black-Scholes Calculator
The Black-Scholes model gives a theoretical fair value for a European-style option from five market inputs: the underlying price, the strike, time to expiry, volatility and the risk-free interest rate, plus an optional continuous dividend yield (the Merton extension). This Black-Scholes calculator prices both the call and the put and reports the main Greeks, which describe how the option value reacts to changes in each input.
Options traders use it to check whether a quoted premium looks cheap or expensive relative to their volatility view, to estimate how much an option will lose to time decay each day, and to size hedges using delta. Students use it to learn how each input drives option prices; the chart shows call and put values across a range of underlying prices.
The model assumes constant volatility and interest rates, lognormally distributed prices, no transaction costs and exercise only at expiry. Most US single-stock options are American-style and can be exercised early, so real prices can differ, especially for deep in-the-money puts or calls on dividend-paying stocks.
With the default inputs, the call option value is $5.4934. Change any value above to recalculate instantly.
How to use the black-scholes calculator
- 1Enter the current underlying price and the option’s strike.
- 2Enter the number of calendar days until expiration.
- 3Enter volatility — the option’s implied volatility if you have it.
- 4Enter the risk-free rate and any dividend yield.
- 5Compare the model value with the market premium and review the Greeks.
Formula and method
N(·) is the cumulative standard normal distribution. d1 and d2 measure how far in or out of the money the option is, scaled by volatility over the remaining time. The call value is the discounted expected value of receiving the stock minus paying the strike when the option finishes in the money; the put is its mirror image, and the two satisfy put-call parity.
Greeks are the partial derivatives of the price. Delta is N(d1) (call) or N(d1) − 1 (put), scaled by e^(−qT). Gamma is the change in delta per $1 move. Vega is shown per 1 percentage point change in volatility, theta per calendar day (annual theta ÷ 365) and rho per 1 percentage point change in the interest rate. Time is measured as days ÷ 365.
- S
- Current price of the underlying
- K
- Strike price
- T
- Time to expiry in years (days ÷ 365)
- σ
- Annualised volatility
- r
- Continuously compounded risk-free rate
- q
- Continuous dividend yield
Worked examples
At-the-money, 90 days, 25% volatility
With the stock and strike both at $100, three months to expiry, 25% volatility and a 4.5% rate, the call is worth about $5.49 and the put $4.39. The call is worth more than the put because, by put-call parity, their difference equals the stock price minus the strike discounted at the risk-free rate (about $1.10 here). Delta of 0.56 means the call gains about 56 cents for a $1 rise in the stock.
Out-of-the-money call, 30 days, 35% volatility
A $260 call on a $250 stock with a month left is worth about $6.26, all of it time value. It loses roughly 17 cents a day to time decay, and has a delta of about 0.38.
In-the-money, one year, with 2% dividend yield
A one-year $45 call on a $50 stock with a 2% dividend yield is worth about $7.56 — $5 of intrinsic value plus $2.56 of time value. The dividend lowers the call value and raises the put value compared with a non-dividend stock.
Frequently asked questions
What does the Black-Scholes model calculate?+
It estimates the fair value of a European call or put option given the underlying price, strike, time to expiry, volatility, interest rate and dividend yield. It also gives the Greeks, which show how sensitive that value is to each input.
Which volatility should I use?+
To check a market price, use the option’s implied volatility from your broker. To form your own view, use historical volatility of the underlying or your forecast. Volatility is the input that moves option prices the most.
Does Black-Scholes work for American options?+
It is exact only for European options, which can be exercised only at expiry. For American calls on non-dividend stocks the result is the same; for American puts and calls on dividend-paying stocks, early exercise can make the true value slightly higher.
Why is theta negative?+
Theta measures time decay. As expiry approaches there is less time for the option to move into the money, so its time value shrinks. Theta is usually most negative for at-the-money options close to expiration.
How do I get an option’s implied volatility?+
Implied volatility is the volatility that makes the Black-Scholes value equal the market price. Use the implied volatility calculator, which solves for it from a quoted premium.
Results are estimates for educational purposes and are not financial advice. Rates, fees and terms vary — confirm with your lender or a licensed advisor before making decisions.