Black-Scholes Calculator
The Black-Scholes value of a European call and put from the share price, strike, time to expiry, risk-free rate, volatility and dividend yield, with d1, d2 and delta.
Result
Call option value: $4.76Estimate
A model value on the volatility and rate you enter. Market prices differ when those inputs, or the model’s assumptions, do.
- Put option value
- $0.81Estimate
- d1
- 0.7693
- d2
- 0.6278
- Call delta
- 0.7791
- Put delta
- -0.2209
- Note
- For European options, exercised only at expiry. An American option can be worth more.
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Results are arithmetic on the returns, rates and horizons you enter, not a forecast and not investment, tax or retirement advice. The site supplies no expected return, inflation rate, contribution limit, tax rate or withdrawal rate. Real returns vary from year to year, and fees, taxes and the order of returns all change the outcome. Check anything that matters with a qualified adviser. Read the full disclaimer
How to use the Black-Scholes Calculator
- Enter the share price, the option's strike price and the time left to expiry in years.
- Enter the risk-free rate and the volatility you want to use, and a dividend yield if the share pays one.
- Read the model value of a call and a put, with d1, d2 and each option's delta.
The formula
d1 = [ln(S ÷ K) + (r − q + σ² ÷ 2) × T] ÷ (σ × √T)d2 = d1 − σ × √Tcall = S × e^(−qT) × N(d1) − K × e^(−rT) × N(d2)put = K × e^(−rT) × N(−d2) − S × e^(−qT) × N(−d1)S is the share price, K the strike, T the years to expiry, r the risk-free rate, q the dividend yield and σ the volatility. N is the standard normal cumulative distribution.
What the model assumes
The share price moves randomly with constant volatility and no jumps, the rate and dividend yield stay fixed, trading is continuous and free, and the option is exercised only at expiry. Real markets break each of these a little, which is why traders quote different volatilities for different strikes.
What the diagram shows
The call and put values side by side. Raise the share price above the strike and the call grows while the put shrinks; add time or volatility and both grow.
With your numbers
- d1 = [ln($42.00 ÷ $40.00) + (r − q + σ² ÷ 2) × 0.5] ÷ (σ√0.5) = 0.7693
- d2 = d1 − σ√T = 0.6278
- Call = S e^(−qT) N(0.7693) − K e^(−rT) N(0.6278) = $4.76
- Put = K e^(−rT) N(-0.6278) − S e^(−qT) N(-0.7693) = $0.81
Quick reference
A model value on the volatility and rate you enter. Market prices differ when those inputs, or the model’s assumptions, do.
| Example | Call option value |
|---|---|
| 42, strike 40, half a year, 10%, 20% | $4.76 |
| At the money: 100, a year, 5%, 20% | $10.45 |
| Same with a 2% dividend yield | $9.23 |
| Deep in the money: 150 against 100 | $50.75 |
| Out of the money at a 0% rate | $6.54 |
| High volatility: 60% | $25.52 |
Common questions about the Black-Scholes Calculator
- Which volatility should I use?
- The model needs the volatility of the share over the life of the option, which nobody knows in advance. People use the historical volatility of past returns, or the implied volatility that makes the model match a quoted option price. The page takes whichever you enter and supplies none. The value is very sensitive to it: at the money, doubling volatility roughly doubles the option value.
- Does this work for American options?
- It values European options, which can only be exercised at expiry. An American call on a share that pays no dividend is worth the same, because exercising it early never pays. An American put, or a call on a share paying a dividend, can be worth more than this page shows.
- What do d1 and d2 mean?
- N(d2) is the probability, under the model's risk-neutral assumptions, that the call finishes in the money. N(d1), adjusted for dividends, is the call's delta: how much its value moves for a one-unit move in the share price.
- Why do the call and put differ by a fixed amount?
- Put-call parity: owning a call and selling a put with the same strike and expiry is the same as owning the share and owing the strike. So call − put = S e^(−qT) − K e^(−rT), whatever the volatility. The page is tested against that identity.