Black-Scholes Option Price Calculator
Price European calls and puts with Black Scholes: fair value plus delta.
What it does
Options have fair prices, and Black, Scholes, and Merton found them. Their 1973 formula values a European option from five inputs: stock, strike, time, rate, and volatility. Dealers still quote around it half a century later, bending volatility to fit while keeping the frame.
The tool above prices European calls and puts with no dividends. Enter the five inputs and it returns the call, the put by parity, and delta. Defaults price an at the money one year call at 20 percent volatility, so the page opens solved at $10.45 call, $5.57 put, delta 0.64.
The formula splits into two economic pieces. S x N(d1) is stock exposure weighted by exercise odds, while discounted strike times N(d2) is what you expect to pay, in today's dollars, when exercise happens. Deep in the money, N terms approach one and the call nears stock minus discounted strike; far out they fade toward zero.
Volatility is the only input nobody observes, which makes it the market's quote language. Implied volatility backs out of traded prices; historical volatility counts past wiggle. When the two disagree, traders call the gap edge, rightly or not.
Use the table to see moneyness move prices, then run the contract. Remember the frame assumes constant volatility and no early exercise, so American puts and dividend payers need wider models.
How to use
- Enter the stock price and strike.
- Enter years to expiry as a decimal.
- Enter the risk free rate in percent.
- Enter volatility in percent.
- Read call first, then put and delta.
- Stress volatility up and down before trading.
Formula
- S
- Current stock price.
- K
- Strike price.
- T
- Years to expiry.
- r
- Annual risk free rate as a decimal.
- vol
- Annual volatility as a decimal.
d1 = (ln(S/K) + (r + vol^2/2) x T) / (vol x root T); d2 = d1 - vol x root T. N is the standard normal CDF. European exercise, no dividends, constant rate and volatility, lognormal prices. Delta equals N(d1).
Worked example
Example: at the money call, 1 year, 20% vol
Stock at $100, strike $100, 1 year to expiry, 5% rate, 20% volatility. Price the European call and put.
- Convert percents: r = 0.05, volatility = 0.20, T = 1.
- d1 = (ln(100/100) + (0.05 + 0.02) x 1) / 0.20 = 0.35.
- d2 = 0.35 - 0.20 = 0.15. N(d1) = 0.6368, N(d2) = 0.5596.
- Discounted strike: 100 x e^-0.05 = 95.12.
- Call = 100 x 0.6368 - 95.12 x 0.5596 = 63.68 - 53.23 = $10.45.
- Put = 10.45 - 100 + 95.12 = $5.57. Delta = N(d1) = 0.64.
Reference table
| Contract | Call | Put | Delta |
|---|---|---|---|
| 100 strike at 100, 20% vol | $10.45 | $5.57 | 0.64 |
| 100 strike at 120, 25% vol, 0.5 yr | $22.76 | $1.27 | 0.89 |
| 100 strike at 80, 30% vol | $4.55 | $19.68 | 0.33 |
Limits
- European exercise only; no early exit math.
- No dividends; payouts need an adjusted model.
- Flat constant volatility assumed.
- Lognormal prices assumed; crashes break it.
Common mistakes
- Pricing American options as European.
- Using historical vol where implied belongs.
- Entering days instead of years for expiry.
- Forgetting dividends on high yield names.
- Reading delta as a probability.
Try it yourself
Quick checks. Answers sit tucked under each question.
Stock $120, strike $100, half year, 3%, 25% vol?
Call $22.76, put $1.27, delta 0.89. Parity: 22.76 - 120 + 98.51 = 1.27.
Stock $80, strike $100, 1 year, 5%, 30% vol?
Call $4.55, put $19.68, delta 0.33. Out of the money keeps little time value at 1 year.
Recompute d1 and d2 for the example?
d1 = (0 + (0.05 + 0.02)) / 0.20 = 0.35, d2 = 0.15, matching the worked steps.
Questions
What is delta here?
The hedge ratio: how many dollars the option gains per 1 dollar stock move. A 0.64 delta call gains about 64 cents on the next dollar.
How is the put found?
Call minus stock plus discounted strike. The page uses it so the put needs no second integral.
Does volatility lift puts too?
Yes. Higher volatility widens both tails, so both calls and puts gain. Time decay still eats value daily.
What does 95.12 mean?
The discount factor e^-rT. At 5% and 1 year the $100 strike costs $95.12 today.
Can I price American options here?
No. US options exercise early, dividends leak value, and volatility smiles bend the flat assumption. Treat the price as a fair value baseline.
How fast does time value grow?
Proportional to root time and volatility. Doubling maturity multiplies an at the money call by about root 2.
Sources
- MIT lecture notes: risk neutral probability and Black-Scholes
- Columbia lecture notes: Black-Scholes option pricing formula