How to use the Black-Scholes calculator
- Enter the stock price and the strike. Use the current share price or the bid-ask midpoint.
- Enter the days to expiry. Count calendar days to the expiration date. The calculator converts them to years by dividing by 365.
- Enter the volatility as an annual percentage. This is your forecast of how much the stock will move. Our implied volatility calculator backs it out of a market price; the stock volatility calculator measures it from past prices.
- Enter the risk-free rate and dividend yield. Use a Treasury bill yield for a maturity close to the expiry and the stock’s annual dividend yield. Both are treated as continuously compounded; set the dividend yield to 0 for a stock that pays none.
The results show both option values, the intermediate terms d₁, d₂, N(d₁) and N(d₂), the split between intrinsic and time value, and a put-call parity check. The chart plots today’s call and put values against the stock price, and the table repeats the calculation for strikes from 80% to 120% of the stock price.
The Black-Scholes-Merton formula
P = K·e−rT·N(−d₂) − S·e−qT·N(−d₁)C and P are the call and put values per share, S the stock price, K the strike, T the time to expiry in years, r the risk-free rate, q the dividend yield and N(·) the standard normal cumulative distribution function.
d₂ = d₁ − σ·√Tσ is the annual volatility. Robert Merton's extension adds the dividend yield q; with q = 0 the formula is the original 1973 Black-Scholes model.
The terms have practical meanings. e−qT·N(d₁) is the call’s delta, the number of shares that hedge one call. N(d₂) is the risk-neutral probability that the call finishes in the money, so K·e−rT·N(d₂) is the present value of the strike you expect to pay.
A worked example
A stock trades at $100 and pays a 1.5% dividend yield. You want to value options with a $105 strike expiring in 90 days, using 25% volatility and a 4% risk-free rate. Time to expiry is 0.2466 years.
The formula gives d₁ = -0.2813 and d₂ = -0.4054, so N(d₁) = 0.3892 and N(d₂) = 0.3426. The call is worth $3.16 per share and the put $7.50.
The call is out of the money, so all of its value is time value. The put is $5.00 in the money and carries $2.50 of time value. Parity checks out: C − P = -$4.34, the same as the discounted stock price minus the discounted strike.
What moves the option value
Each row changes one input from the example and keeps the rest:
| Change | Call value | Put value |
|---|---|---|
| Example (25% vol, 90 days) | $3.16 | $7.50 |
| Strike $100 (at the money) | $5.23 | $4.62 |
| Volatility 35% | $5.09 | $9.43 |
| 30 days to expiry | $1.13 | $5.91 |
| 365 days to expiry | $8.78 | $11.15 |
| Risk-free rate 6% | $3.34 | $7.17 |
| Dividend yield 4% | $2.93 | $7.88 |
Volatility and time raise both calls and puts, because they widen the range of outcomes while the loss on each option stays capped at the premium. Rates and dividends push in opposite directions: a higher rate lowers the present value of the strike, which helps calls and hurts puts, while a higher dividend yield lowers the expected future stock price, which helps puts and hurts calls.
For an at-the-money option with low rates, a quick sanity check is C ≈ 0.4 × S × σ × √T (0.4 ≈ 1 ÷ √(2π)), which shows why value grows with the square root of time.
The assumptions behind the model
Black-Scholes is exact only in an idealized market:
- European exercise. The option can be exercised only at expiry.
- Lognormal prices. The stock follows geometric Brownian motion, so log returns are normally distributed and the price cannot jump.
- Constant volatility and rates over the life of the option.
- Continuous, costless trading. The replicating hedge can be adjusted at every instant with no commissions, spreads or borrowing limits.
- Dividends as a continuous yield. A real stock pays discrete dividends on known dates.
Real markets break each assumption: stocks jump on earnings, volatility changes, and hedging is discrete and costly. Traders still use the model, mainly as a translation device: they quote options in implied volatility, and the shape of implied volatility across strikes (the smile or skew) absorbs what the model leaves out.
Why American options can be worth more
US equity options are American: they can be exercised on any trading day. That right can only add value, so an American option is worth at least as much as the European one.
For a call on a stock that pays no dividend, the extra right is worthless. Exercising early gives up the time value and pays the strike sooner, so the call is always worth more alive than exercised, and the Black-Scholes value is also the American value.
The right becomes valuable in two cases. Deep in-the-money puts: exercising early receives the strike now and earns interest on it. With the stock at $70, a $100 strike and a 6% rate, the European put is worth $25.34, less than its $30.00 intrinsic value; an American put would never trade below $30.00. Calls on dividend-paying stocks: exercising just before the ex-dividend date captures the dividend. A deep in-the-money European call with a 6% yield, stock at $150 and strike $100, is worth $47.09 against $50.00 intrinsic. Where these effects matter, a binomial tree gives the American value.
What this calculator does not include
- Discrete dividends. A continuous yield is a good approximation over long horizons but can misprice short-dated options that span a large ex-dividend date. A common fix is to subtract the present value of the expected dividends from the stock price and set the yield to 0.
- Early exercise. The values are European; see above for when American options differ.
- Volatility smile and term structure. One volatility is used for every strike in the table, while real markets price different volatilities by strike and expiry.
- Rate conventions. The model uses continuous compounding. For an annual yield y, the continuous equivalent is ln(1 + y), slightly lower; at typical rates the difference changes option values by cents.
- Costs and margin. Commissions, the bid-ask spread and margin requirements are not included. For the profit on a trade at expiry, use the options profit calculator; for sensitivities, the option Greeks calculator.