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Option pricing
Explore what affects an option's theoretical value. No sign-in needed.
About this tool
What it does
An option gives its holder the right, but not the obligation, to buy (a call) or sell (a put) something at a fixed price (the strike) by a set date (expiry). This tool estimates what such an option is worth under a mathematical model, using inputs you choose.
It also shows how sensitive that estimate is to each input (the "Greeks"), and it can work backwards from an option price you have seen to the volatility that price implies.
When it is useful
- Learning how each input moves an option's value: more time or more volatility raises the value of both calls and puts; a higher underlying price raises a call and lowers a put.
- Comparing scenarios: change one input at a time and compare the results.
- Reading a quoted price in volatility terms: implied volatility translates a price into the volatility the model would need to produce it.
- Call / put
- A call is the right to buy at the strike; a put is the right to sell at the strike.
- European / American
- A European option can be exercised only at expiry; an American option at any time up to expiry.
- Volatility
- How much the underlying price typically moves in a year, as a percentage. It measures the size of moves, not their direction.
- Implied volatility
- The volatility that makes the model's value equal a price you enter. It is a translation of that price, not a forecast.
- Greeks
- How much the model value changes when one input changes a little, with everything else held fixed.
Inputs
| Input | Unit | What it means |
|---|---|---|
| Type | call or put | The right to buy (call) or to sell (put). |
| Spot price | a price, same currency as the strike | Today's price of the underlying asset, for example a share. |
| Strike | a price | The price at which the option lets you buy or sell. |
| Time to expiry | years | Time left until expiry: 0.5 is six months; 30 days is about 0.082. |
| Volatility | annual, decimal | 0.20 means 20% a year. Your assumption: the model treats it as constant. |
| Risk-free rate | annual, continuously compounded, decimal | 0.05 means 5%. The button fills in the U.S. Treasury par yield for the same maturity, converted to this convention, with its date. |
| Dividend yield | continuous annual yield, decimal | 0 when there is none. Fixed cash dividends are not supported and are never approximated. |
| Model | — | Black–Scholes–Merton (an exact formula for European options, with Greeks) or a binomial or trinomial tree (European or American, no Greeks). |
| Tree steps | 1 to 5,000 | Trees only. More steps give a finer approximation; the tool re-prices with twice the steps so you can see the difference. |
Reading the results
- Value
- The model's theoretical value of one option on one unit of the underlying (for a share option, per share), in the currency you entered. It is not a price you can trade at.
- Delta
- Change in value for a 1-unit rise in the spot price. A call's delta is between 0 and 1, a put's between −1 and 0.
- Gamma
- How quickly delta itself changes as the spot price moves.
- Vega
- Change in value if volatility rises by one percentage point (for example from 20% to 21%).
- Theta
- Change in value from one calendar day passing, everything else unchanged; usually negative.
- Rho
- Change in value if the interest rate rises by one percentage point.
- Implied volatility
- The constant volatility at which the Black–Scholes–Merton value equals the price you entered. The no-arbitrage bounds show the range a valid price must fall in.
Educational example (not market data)
- Inputs: a European call, spot 100, strike 100, one year to expiry, volatility 0.20 (20%), rate 0.05 (5%), no dividend. These are the values filled in when the page opens.
- Black–Scholes–Merton gives a value of 10.4506. The put with the same inputs is worth 5.5735.
- Delta 0.6368: if the spot rose from 100 to 101, the call's model value would rise by about 0.64. Vega 0.3752: raising volatility from 20% to 21% adds about 0.38. Theta −0.0176: one day passing takes about 0.018 off.
- Change volatility to 0.25 and the call's value rises to 12.3360: more uncertainty makes the right to buy more valuable.
- Enter an observed price of 12.00 under implied volatility: the tool solves a volatility of about 24.11%.
- Switch to a binomial tree with 500 steps and American exercise for the put: 6.0888, above the European 5.5735, because the right to exercise early has value for a put.
Assumptions and limitations
- The models assume volatility and interest rates stay constant and that prices move randomly without jumps. Real markets do not behave this way, and quoted options imply different volatilities for different strikes and expiries.
- A model value is not an executable quote. Market prices include bid–ask spreads, fees and supply and demand. A difference between a model value and a market price does not by itself show a trading opportunity.
- Dividends only as a continuous yield; fixed cash dividends are not supported. No exotic options. Greeks come from the exact formula only, not from the trees.
- Your inputs, especially volatility, drive the result. The tool cannot tell you which volatility is right.
Methodology and validation
- Black–Scholes–Merton closed form with a continuous dividend yield; Cox–Ross–Rubinstein binomial and trinomial trees, European or American exercise; implied volatility solved with Brent's method inside the no-arbitrage bounds.
- Treasury rate pre-fill: the U.S. Treasury daily par yield curve (public domain), interpolated to the maturity and converted to continuous compounding; the curve's date is shown.
- Each result names its model version. The pricing functions are checked against independent reference values (see Model performance, "Options and Monte Carlo").
Validation results: Model performance · formulas and rules: Methodology
Calculator
Educational example values are filled in: an example, not a quote or market data. Replace them with your own.
Discrete cash dividends are not supported and are never approximated by a yield: enter 0 and read the value as ex-dividend, or enter a yield only when the dividend really is proportional.
Implied volatility from a price you observed
Enter a price you have seen for the same option (type, spot, strike, expiry, rate and yield above). The tool finds the volatility at which the Black–Scholes–Merton value equals that price. It translates the price into volatility terms; it is not a forecast of future volatility.