Step 1
Know the contract
An option gives the holder the right, not the obligation, to buy or sell an asset at a fixed strike price.
- A call gives the right to buy.
- A put gives the right to sell.
- The strike price is fixed in advance.
This page explains American option pricing from the ground up. Instead of only showing the final number, it breaks the process into simple ideas: what an option is, why American options are harder to price, how tree methods work, what each formula means, and how early exercise changes the answer.
If you are new to this topic, follow the ideas below in order. The page is built so each section teaches one layer before moving to the next.
An option gives the holder the right, not the obligation, to buy or sell an asset at a fixed strike price.
American options can be exercised any time up to expiry, so we must check whether exercising early is better than waiting.
A pricing tree lists possible stock prices over time and works backward from the final payoffs to today’s price.
This is the first big concept to understand. The difference is not the payoff formula itself. The difference is when the holder is allowed to exercise.
You can exercise only at expiry. Because there is no exercise decision before maturity, the pricing problem is simpler.
You can exercise at any time up to expiry. That means every node in the tree asks: “Should I exercise now or continue?”
If immediate exercise gives more value than holding the option for later, the rational holder exercises. This is why the price of an American option is never less than the value from simply continuing forward in the tree.
Change the inputs and the entire page updates: formulas, metrics, charts, the mini teaching tree, and the explanation of what the numbers mean.
The formulas below are the heart of the model. Each one updates using your current inputs, and each symbol is explained in plain English.
max(S - K, 0). A call is valuable only when the stock price is above the strike.
max(K - S, 0). A put is valuable only when the stock price is below the strike.
Future cash flows are multiplied by e-rΔt to convert them back to present value.
This mini-tree uses a small number of steps so you can inspect the nodes one by one. Switch between binomial and trinomial trees, then click a node to see the stock price, immediate exercise payoff, continuation value, and final American value at that point.
Filled red nodes indicate places where immediate exercise beats continuation. Blue nodes indicate holding the option is better.
Select a node in the tree to inspect the pricing decision at that point.
The lines below show how both trees approach a stable answer as the number of time steps increases.
Pricing becomes more computationally expensive as the number of steps rises, especially for trees with more branches.
For puts, this boundary marks the approximate stock price below which early exercise becomes optimal. For calls without dividends, the boundary is often weak or absent.
The interpretation below is generated from your current parameter choices, so the report teaches the numbers instead of just displaying them.