Calculator inputs
Example data table
Sample setup: put option, strike 95, maturity 1 year, rate 5%, dividend yield 2%, volatility 25%, and 200 tree steps.
| Spot | American value | European value | Early exercise premium |
|---|---|---|---|
| 80.00 | 16.635473 | 15.674807 | 0.960666 |
| 90.00 | 10.472160 | 10.008208 | 0.463952 |
| 100.00 | 6.263791 | 6.042268 | 0.221523 |
| 110.00 | 3.576305 | 3.471291 | 0.105014 |
| 120.00 | 1.978207 | 1.929139 | 0.049068 |
Formula used
\(u = e^{\sigma\sqrt{\Delta t}}\), \(d = 1/u\)
\(p = \dfrac{e^{(r-q)\Delta t} - d}{u - d}\)
\(C = e^{-r\Delta t}\big(pV_{up} + (1-p)V_{down}\big)\)
European node value = continuation value.
American node value = max(intrinsic value, continuation value).
Early exercise premium = American option value − European option value.
This page uses a Cox-Ross-Rubinstein binomial tree. It handles calls and puts, allows continuous dividend yield, and measures how much value comes from exercise flexibility.
How to use this calculator
- Enter the current spot price and strike price.
- Set time to expiry in years, then enter rate, dividend yield, and volatility.
- Choose call or put, then select a tree depth. Higher steps usually improve stability.
- Set chart range percentages and point count for the sensitivity graph.
- Click Calculate premium to show the result above the form.
- Review the summary cards, sensitivity table, and exercise frontier snapshot.
- Use the CSV or PDF buttons to export your pricing report.
Frequently asked questions
1) What does early exercise premium measure?
It measures the extra value created by the right to exercise before expiry. The calculator finds it by subtracting the European price from the American price under identical assumptions.
2) Why is the premium often zero for some calls?
Non-dividend American calls often have little reason to exercise early, so their American and European values can match closely. Positive dividend yield can change that relationship.
3) Why do puts usually show more early exercise value?
Deep in-the-money puts can benefit from locking in intrinsic value sooner, especially when rates are positive. That makes early exercise more attractive than waiting in some nodes.
4) What tree step count should I use?
Start around 100 to 300 steps for quick work. Increase the count until the price stabilizes to your preferred decimal precision. Very low step counts can distort the estimate.
5) Why might the probability input logic fail?
Extreme rates, volatility, or very coarse time steps can push the risk-neutral probability outside its valid range. Raising the number of steps often fixes the issue.
6) Is this calculator suitable for discrete dividends?
This version uses a continuous dividend yield, which is a practical approximation for many cases. Exact discrete dividend schedules need a more specialized tree setup.
7) What does the exercise frontier table show?
It lists sample tree steps where immediate exercise dominates continuation value. The stock ranges highlight where the option holder would prefer exercise over holding.
8) Can I use these results for risk management?
Yes, as an analytical estimate. It helps compare exercise flexibility across scenarios, but production pricing and risk systems may require calibration, market conventions, and model governance.