The Binomial Tree Model for American Options
The binomial tree model, developed by Cox, Ross, and Rubinstein (CRR) in 1979, is the standard method for pricing American-style options. Unlike Black-Scholes, which assumes European exercise (only at expiration), the binomial model evaluates the option value at every node in a discrete time lattice, allowing for early exercise at any point before expiration.
The model works by dividing the time to expiration into N equal steps. At each step, the stock can move up by factor u or down by factor d. Starting from the final nodes (expiration payoffs), the model works backward through the tree, at each node comparing the exercise value to the continuation value and choosing the maximum. This backward induction naturally handles the American early exercise feature.
American put options can be optimally exercised early, especially when the stock price drops significantly below the strike. Black-Scholes cannot capture this early exercise premium. The binomial tree model explicitly checks at every node whether exercising is more valuable than holding, producing a more accurate price for American options.
Binomial Tree Parameters
- 1dt = 90/(365*4) = 0.0616 years per step
- 2u = e^(0.30 * sqrt(0.0616)) = e^(0.0745) = 1.0773
- 3d = 1/1.0773 = 0.9283
- 4p = (e^(0.05*0.0616) - 0.9283) / (1.0773 - 0.9283) = 0.4927
- 5Stock tree: $100 -> Up $107.73 or Down $92.83
- 6At expiry nodes: max(K - S, 0) for each terminal price
- 7Work backward: at each node, max(exercise, continuation)
- 8American price = $5.24 vs European BSM = $4.98
When Early Exercise Is Optimal
| Condition | Early Exercise? | Reasoning |
|---|---|---|
| Deep ITM put, stock far below strike | Often optimal | Interest earned on strike proceeds exceeds time value |
| Near expiry, moderately ITM | Usually optimal | Time value is small, capture intrinsic value now |
| High interest rates | More likely | Higher time value of receiving cash early |
| No dividends expected | More likely for puts | No reason to wait for stock to drop on ex-date |
| High dividends expected | Less likely for puts | Expected dividend drops help put holders |
Binomial Tree Accuracy
The accuracy of the binomial tree increases with the number of steps. A 4-step tree is useful for visualization and understanding the concept, but practical pricing requires 50-200 steps. With enough steps, the binomial model converges to the Black-Scholes price for European options, confirming the mathematical consistency between the two models. For American options, the binomial price converges to the true American option value.
- 4-10 steps: Good for educational visualization and understanding the concept. Accuracy within 1-5% of converged value.
- 50-100 steps: Suitable for practical trading decisions. Accuracy within 0.1% of converged value.
- 200+ steps: Near-exact pricing. Used by professional risk management systems and market makers.
- Odd vs. even steps: For ATM options, odd numbers of steps tend to converge faster to the correct price.
- Richardson extrapolation: Combining results from different step counts can dramatically improve accuracy with fewer steps.
Comparing American and European Option Values
An American option is always worth at least as much as its European counterpart because it has all the same rights plus the additional right of early exercise. For calls on non-dividend-paying stocks, early exercise is never optimal (the American call equals the European call). For puts, and for calls on dividend-paying stocks, early exercise can be optimal, making the American option more valuable. The difference between American and European prices is called the early exercise premium.