Calculator Inputs
Use annual percentages for rates and volatility. The responsive form uses 3 columns on large screens, 2 on smaller screens, and 1 on mobile.
Example Data Table
These rows are illustrative sample scenarios for testing. They help users verify the interface and compare how changing inputs affects output behavior.
| Scenario | Spot | Strike | Rate % | Dividend % | Vol % | Years | Type | Simulations | Illustrative Price |
|---|---|---|---|---|---|---|---|---|---|
| Base At-The-Money | 100 | 100 | 5 | 2 | 20 | 1.00 | Call | 30000 | About 9.00 to 9.50 |
| Higher Volatility | 100 | 100 | 5 | 2 | 35 | 1.00 | Call | 30000 | About 14.50 to 15.80 |
| In-The-Money Put | 90 | 100 | 4 | 1 | 25 | 0.75 | Put | 30000 | About 12.00 to 13.80 |
| Longer Maturity | 100 | 105 | 5 | 0 | 22 | 2.00 | Call | 50000 | About 12.50 to 14.20 |
Formula Used
This calculator simulates terminal prices under a risk-neutral geometric Brownian motion model and discounts the average payoff back to today.
S(T) = S(0) × exp((r - q - 0.5 × σ²) × T + σ × sqrt(T) × Z)
Call payoff = max(S(T) - K, 0)
Put payoff = max(K - S(T), 0)
Option price ≈ exp(-r × T) × average(payoff)
Standard error = sample standard deviation / sqrt(number of simulations)
Confidence interval = estimated price ± z-critical × standard error
Control variate adjustment:
Adjusted payoff = Raw payoff - β × (discounted S(T) - expected discounted S(T))
Here, S(0) is spot price, K is strike, r is the risk-free rate, q is the dividend yield, σ is volatility, T is time in years, and Z is a standard normal random draw.
How to Use This Calculator
- Enter the current spot price and strike price.
- Select whether you want to price a call or a put.
- Enter annual risk-free rate, dividend yield, volatility, and time to maturity.
- Choose the number of simulations and the confidence level.
- Enable antithetic variates or the control variate option for lower estimator variance.
- Optionally enter a random seed if you want repeatable results.
- Click Price Option to place the result panel above the form.
- Review the convergence graph, confidence interval, and benchmark comparison.
- Use the CSV or PDF buttons to export the result summary.
FAQs
1) What does this calculator price?
It prices European call and put options using Monte Carlo simulation under a risk-neutral lognormal asset model. It also compares the estimate with a Black-Scholes benchmark.
2) Why do more simulations matter?
More simulations usually reduce random noise. The estimate becomes more stable, and the confidence interval usually narrows, though computation time also increases.
3) What are antithetic variates?
They pair each random draw with its negative counterpart. This often reduces variance by balancing extreme outcomes and can improve convergence without changing the expected price.
4) What is the control variate option doing?
It uses discounted terminal stock value as a known-reference control. The adjustment can reduce estimator noise when the payoff and terminal stock value move together strongly.
5) Why compare against Black-Scholes?
For a standard European option, Black-Scholes gives a closed-form benchmark. Comparing both values helps you judge whether the simulation is converging reasonably.
6) Does the dividend yield affect the result?
Yes. A higher dividend yield lowers the expected growth of the stock under the pricing measure, which usually reduces call values and can support put values.
7) Is the Monte Carlo result exact?
No. It is a statistical estimate. The confidence interval shows the likely range of the true simulated price given the chosen confidence level and sample size.
8) Can this be extended to exotic options?
Yes. Monte Carlo methods are flexible. With extra path logic, the same framework can be extended to barrier, basket, rainbow, or Asian-style structures.