Black-Scholes Calculator

• Enter the stock price, the strike and the days to expiration.
• Enter the volatility, the risk-free rate and the dividend yield. Use the implied volatility from your broker's option chain if you have it.
• Read the call and put values and the Greeks. To work backward from a market price, switch to Find implied volatility.

The Black-Scholes model, also called Black-Scholes-Merton or BSM, values a European option from six inputs. This calculator uses the version with a continuous dividend yield:

d1 = [ln(S / K) + (r - q + σ² / 2) × T] / (σ × √T)

d2 = d1 - σ × √T

Call value = S × e^(-qT) × N(d1) - K × e^(-rT) × N(d2)

Put value = K × e^(-rT) × N(-d2) - S × e^(-qT) × N(-d1)

Here S is the stock price, K the strike, T the time to expiration in years (days divided by 365), r the risk-free interest rate, q the dividend yield, σ the volatility, ln the natural log, e the number 2.718 and N the standard normal distribution.

In plain words, a call is worth the stock price you would receive, adjusted for dividends and weighted by N(d1), minus the strike you would pay, discounted to today and weighted by N(d2), the model's chance that the call finishes in the money. The put is the same idea from the other side. The Black-Scholes formula calculator above does every step, so you can check a textbook problem or a broker's model price in seconds.

Example: a stock at $100, a $100 strike, 30 days to expiration, 25% volatility, a 4% interest rate and a 2% dividend yield give a call value of $2.93 and a put value of $2.77.

The call's delta is 0.52, so it gains about $0.52 for each $1 rise in the stock. Theta is about -$0.05 a day, vega about $0.11 for each 1 point of volatility. Without the dividend yield the call would be $3.02 and the put $2.69.

Stock price is the current price of the shares. Strike is the price at which the option lets you buy, for a call, or sell, for a put.

Days to expiration counts calendar days until the option expires. The model converts them to years by dividing by 365, so 30 days is about 0.082 years.

Volatility is how much the stock is expected to move, as a yearly percentage. Historical volatility measures past moves. Implied volatility is the figure that makes the model match the option's market price, and it is the one traders use in a Black-Scholes model calculator.

The risk-free rate is the yield on a short-term US Treasury bill. The button next to it fills the latest 3-month Treasury yield, the rate GNG's Options Screener uses.

Dividend yield is the stock's yearly dividend divided by its price. Enter 0 for a stock that pays none.

Read about volatility in the GNG glossary.

Read about dividend yield in the GNG glossary.

Find a stock on GNG stock pages to read its free option chain and end-of-day prices.

Compare price moves in Advanced Charting, free with a GNG account.

The Greeks show how the option's value responds when one input changes and the others stay put. This option Greeks calculator shows all five for both the call and the put.

Delta is the change in the option's value for a $1 move in the stock. Calls run from 0 to 1 and puts from -1 to 0. A delta of 0.52 means the call gains about $0.52 when the stock rises $1.

Gamma is the change in delta for a $1 move in the stock. It is largest for options near the money and close to expiration.

Theta is the value an option loses each calendar day as time passes, with everything else unchanged. It is usually negative for options you own, and it speeds up as expiration nears.

Vega is the change in value for a 1 point change in implied volatility, for example from 25% to 26%. Longer-dated options have more vega.

Rho is the change in value for a 1 point change in the interest rate. Calls gain when rates rise and puts lose.

The calculator also shows the model's chance that each option finishes in the money, N(d2) for the call and N(-d2) for the put. It is a model probability under constant volatility, not a forecast.

Implied volatility is the volatility that makes the Black-Scholes value equal the option's market price. Switch to Find implied volatility, enter the market price, choose call or put, and the implied volatility calculator searches for the volatility that matches it to the cent.

For example, with the default inputs a call that trades at $3.50 instead of $2.93 implies a volatility of about 30%, not 25%. A higher implied volatility means the market expects bigger moves, or is paying up for protection.

Use the midpoint between the bid and ask as the market price when you can. Prices from thinly traded contracts can give odd results.

A price outside the model's range for these inputs has no implied volatility within the 1% to 500% search range.

Dividends lower call values and raise put values, because the share price drops by about the dividend on each ex-dividend date. This calculator uses a continuous dividend yield, the standard Black-Scholes-Merton form, so it suits steady payers and index funds. For a single large dividend before expiration, check the early assignment section of the covered call calculator.

In the default example, the 2% dividend yield cuts the call from $3.02 to $2.93 and lifts the put from $2.69 to $2.77. The gap grows with the yield and with time to expiration, which is why a Black-Scholes calculator with dividends matters for high-yield stocks and longer-dated options.

For ex-dividend assignment risk, use the covered call calculator.

The model assumes the option can be exercised only at expiration, that volatility and interest rates stay constant, that the stock's returns follow a lognormal distribution, and that trading is continuous with no costs. Real markets break each of these a little. Volatility changes, and options at lower strikes often trade at higher implied volatility than at-the-money options.

Most US stock options are American, so they can be exercised early. For a call on a stock that pays no dividend before expiration, early exercise never pays, so the Black-Scholes value is right. Puts deep in the money, and calls just before an ex-dividend date, can be worth more than the model shows. Employee stock options and warrants use the same formula with longer terms, but their valuation rules go beyond this calculator.

The calculator uses the Black-Scholes-Merton formula with a continuous dividend yield. Time is days to expiration divided by 365. Theta is shown per calendar day, vega per 1 point of volatility and rho per 1 point of the interest rate, the same units GNG's Options Screener uses. The risk-free rate button fills the 3-month Treasury yield, the rate the screener uses for its Black-Scholes values.

The implied volatility mode searches for the volatility that reproduces your market price to the cent.

Values are model estimates for European options, not quotes, and they do not include fees or taxes.

Read the free Options Series Part 2 for a guide to hedging with options.

The series is written by Connor Graham.

Read more about GNG's methodology.

To see what a trade could make or lose at any stock price and date, with a payoff chart, use the options profit calculator, which also works as an option price calculator before expiration. Selling calls on shares you own? The covered call calculator adds the ex-dividend date and an early assignment check.

Model calls, puts and spreads with the options profit calculator.

Model income and assignment risk with the covered call calculator.

Find contracts in the Options Screener with Greeks and IV rank, available with GNG Pro.