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Monte Carlo option simulation
Estimate option value through simulated scenarios. Running a simulation needs sign-in; the explanation below does not.
About this tool
What it does
The tool generates many possible prices of the underlying at expiry, works out what a European call or put would pay in each, and averages those payoffs, discounted to today. The average estimates the option's value under the model.
The simulation uses the same assumptions as the Black–Scholes–Merton formula (prices grow on average at the risk-free rate minus the dividend yield, with the volatility you enter), so the tool also shows the exact formula value for comparison.
When it is useful
- Seeing how simulation-based pricing, used for problems with no formula, arrives at a value.
- Seeing precision improve with more simulations: four times as many paths roughly halves the sampling error.
- Seeing what variance-reduction techniques do: with them, far fewer paths reach the same precision.
Inputs
| Input | Unit | What it means |
|---|---|---|
| Option inputs | as in Option pricing | Type, spot price, strike, time to expiry (years), volatility, risk-free rate and dividend yield (annual decimals). |
| Paths | 1,000 to 2,000,000 | How many scenarios to simulate. More paths, smaller sampling error. |
| Seed | whole number, 0 to 4,294,967,295 | Starts the random-number generator. The same seed and inputs reproduce the same result exactly; a different seed gives a different sample. |
| Antithetic variates | on or off | Uses each random draw together with its mirror image, which cancels some of the noise. |
| Control variate | on or off | Corrects the estimate using the simulated price at expiry, whose true average is known exactly, which removes much of the noise. |
Reading the results
- Estimate ± standard error
- The simulated value and the size of its sampling noise. The standard error describes how precise this number is, not how uncertain the future is.
- 95% confidence interval
- Estimate ± 1.96 standard errors: across repeated runs, an interval built this way contains the exact model value about 95% of the time. It is not a range for future market prices or for what the option will be worth later.
- Comparison with the formula
- The exact Black–Scholes–Merton value, the difference, and that difference in standard errors (usually within about ±2).
- Runs on this page
- Each run you make, newest first, with whether its interval contains the exact value: about 19 runs in 20 should.
Educational example (not market data)
- Inputs: a European call, spot 100, strike 100, one year, volatility 0.20, rate 0.05, no dividend; the exact formula value is 10.4506.
- 10,000 paths, seed 1, no variance reduction: estimate 10.3043 ± 0.1460, 95% interval 10.0181 to 10.5905. The interval contains 10.4506.
- 1,000,000 paths, same seed, no variance reduction: estimate 10.4276 ± 0.0147. A hundred times the paths gives one-tenth of the standard error.
- 200,000 paths with both antithetic variates and the control variate (the starting settings): estimate 10.4464 ± 0.0062, interval 10.4343 to 10.4585.
Assumptions and limitations
- More simulations reduce sampling error only. They do not fix a wrong model: if real prices jump or volatility changes, the simulated value is still wrong in the same way as the formula.
- European exercise only, a constant volatility and rate, and a continuous dividend yield.
- Simulations run on FinSavoir's server: sign-in is required, paths are capped, and when another simulation is running you are asked to try again a few seconds later.
Methodology and validation
- Risk-neutral geometric Brownian motion, simulated directly at expiry; NumPy PCG64 random generator seeded by you; antithetic pairs and a control variate on the discounted terminal price.
- The interval is estimate ± 1.96 × standard error. The comparison with the closed form is the tool's built-in check; validation results are on Model performance ("Options and Monte Carlo").
Validation results: Model performance · formulas and rules: Methodology
Simulator
Educational example values are filled in (not market data). Replace them with your own; the seed makes a run reproducible.
Company revenue simulation: not available yet. It will appear once its results have been checked on this site's data.