Calculator Inputs
This page uses correlated geometric Brownian motion and Monte Carlo valuation for three underlying assets. Basket weights are normalized automatically.
Formula Used
Terminal asset model
\( S_i(T)=S_i(0)\times e^{[(r-q_i)-0.5\sigma_i^2]T+\sigma_i\sqrt{T}Z_i} \)
Correlation handling
A 3×3 correlation matrix is decomposed with Cholesky factorization so independent normal shocks become correlated shocks.
Typical payoff definitions
- Call on max: max(max(S1(T),S2(T),S3(T)) − K, 0)
- Put on min: max(K − min(S1(T),S2(T),S3(T)), 0)
- Basket call: max(w1S1(T) + w2S2(T) + w3S3(T) − K, 0)
- Spread call: max(max(S(T)) − min(S(T)) − K, 0)
Present value estimate
Option Price = \( e^{-rT}\times \text{Average Simulated Payoff} \)
Greeks
Delta, gamma, vega, rho, and one-day theta are estimated with finite-difference bump-and-revalue methods using common random numbers.
How to Use This Calculator
- Choose the rainbow payoff that matches your structure.
- Enter current prices for all three assets.
- Provide strike, maturity, risk-free rate, and dividend yields.
- Enter each asset volatility as a decimal, such as 0.25.
- Set basket weights when using basket payoffs.
- Enter pairwise correlations for the three assets.
- Select a simulation count for the desired precision.
- Press the pricing button to see results, Greeks, chart, and export options.
Example Data Table
| Example Parameter | Sample Value | Meaning |
|---|---|---|
| Payoff Type | Call on Maximum Asset | Uses the best-performing terminal asset. |
| S1, S2, S3 | 100, 95, 105 | Current prices of the three underlyings. |
| Strike | 100 | Contract strike level. |
| Volatilities | 0.22, 0.25, 0.20 | Annualized asset volatility assumptions. |
| Dividend Yields | 0.01, 0.01, 0.01 | Continuous income yields. |
| Correlations | 0.45, 0.35, 0.40 | Pairwise asset dependence. |
| Risk-Free Rate | 0.05 | Discounting and drift input. |
| Time to Maturity | 1.00 year | European expiry horizon. |
| Simulations | 6000 | Monte Carlo path count. |
Use the sample values above to test the page quickly. Exact results depend on the chosen payoff and simulation count.
Frequently Asked Questions
1) What is a rainbow option?
A rainbow option depends on multiple underlying assets instead of one. Its payoff may be linked to the maximum, minimum, basket value, or spread across the assets.
2) Why is correlation important here?
Correlation changes how often assets move together. That directly affects best-of, worst-of, basket, and spread payoffs, so even small correlation changes can materially alter price.
3) Why use Monte Carlo simulation?
Many rainbow structures have no simple closed-form solution, especially with several assets and custom payoffs. Monte Carlo handles this flexibility by averaging many simulated outcomes.
4) What do the confidence interval values mean?
They show the sampling uncertainty from simulation. A narrower interval means the estimate is more stable. Increasing the number of simulations usually tightens the interval.
5) Are the Greeks exact?
No. They are numerical estimates from finite-difference revaluation. They are useful for sensitivity analysis, but they still depend on simulation noise and bump size choices.
6) Why are basket weights normalized?
Normalization keeps the basket definition consistent when weights do not sum to one. It lets you focus on relative importance across assets without manually rescaling inputs.
7) Does this support American exercise?
No. This implementation prices European-style expiration payoffs only. American or Bermudan exercise requires a different numerical method, such as least-squares Monte Carlo.
8) How many simulations should I use?
Start with 5,000 to 10,000 for quick estimates. Use more when payoff tails matter, correlations are extreme, or you want tighter confidence bands and smoother Greeks.