Calculator
Example Data Table
| Scenario | Stock | Strike | Rate | Dividend | Volatility | Maturity | Steps |
|---|---|---|---|---|---|---|---|
| European Call | 100 | 100 | 5% | 1% | 25% | 1.00 | 50 |
| European Put | 95 | 100 | 4% | 0% | 20% | 0.75 | 60 |
| American Put | 80 | 85 | 6% | 2% | 30% | 1.50 | 75 |
Formula Used
This calculator applies a recombining trinomial tree. Each step allows the stock to move up, stay in the middle, or move down. The model discounts expected future option values back through the tree until the root node is reached.
dt = T / N
dx = sigma * sqrt(3 * dt)
u = e^(dx)
d = e^(-dx)
m = 1
nu = r - q - 0.5 * sigma^2
Pu = 0.5 * [ (sigma^2 * dt + nu^2 * dt^2) / dx^2 + (nu * dt / dx) ]
Pd = 0.5 * [ (sigma^2 * dt + nu^2 * dt^2) / dx^2 - (nu * dt / dx) ]
Pm = 1 - [ (sigma^2 * dt + nu^2 * dt^2) / dx^2 ]
Option(n,j) = e^(-r * dt) * [ Pu * V(n+1,j+1) + Pm * V(n+1,j) + Pd * V(n+1,j-1) ]
For American style:
Option(n,j) = max( intrinsic value, continuation value )
Delta, gamma, and theta are approximated from nearby tree levels, using finite differences around the root.
How to Use This Calculator
- Enter the current stock price and strike price.
- Provide the annual risk-free rate and dividend yield in percentages.
- Enter annualized volatility and time to maturity in years.
- Choose the number of tree steps for accuracy and speed balance.
- Select call or put, then choose European or American exercise.
- Click Price Option to generate the result above the form.
- Review the metrics, chart, and probability outputs.
- Use CSV or PDF export for reporting or documentation.
FAQs
1. What does a trinomial model do?
It values an option by allowing three possible price paths at each step: up, middle, and down. This often creates smoother convergence than a simple binomial tree.
2. When should I use more tree steps?
Use more steps when you want higher numerical precision, especially for longer maturities or larger volatilities. More steps usually improve stability but increase computation time.
3. Why compare with Black-Scholes?
For European options, Black-Scholes is a common benchmark. Comparing both values helps you see whether the tree is converging reasonably under the chosen number of steps.
4. Why can American and European prices differ?
American options may be exercised before expiry. That added flexibility can increase the option value, especially for puts or dividend-sensitive contracts.
5. What does dividend yield change?
Dividend yield reduces expected future stock growth under risk-neutral pricing. That usually lowers call values and can support higher put values.
6. Why were probabilities adjusted sometimes?
Certain extreme inputs may produce invalid raw probabilities. The calculator normalizes them into valid bounds and shows a warning so your result remains usable.
7. Are the Greeks exact?
No. They are finite-difference estimates taken from nearby tree values. They are useful for analysis, but they are still numerical approximations.
8. Is this calculator suitable for reporting?
Yes. It includes a result table, chart, sample data section, and CSV or PDF export tools, making it useful for presentations, notes, and internal reviews.