Price American call and put options using the Cox-Ross-Rubinstein binomial tree model. Includes interactive tree visualization with early exercise detection and European price comparison.
| European (Black-Scholes) price | — |
| European (CRR tree, no early exercise) | — |
| American (CRR) price | — |
| Early exercise premium | — |
For more details on the documentation and methodology, follow the Python script.
The Cox-Ross-Rubinstein (CRR) binomial tree model, published in 1979 by John Cox, Stephen Ross, and Mark Rubinstein, is one of the most widely used methods for pricing options. It discretizes the continuous price process into a recombining lattice of up and down moves over \(n\) time steps, each of length \(\Delta t = T/n\). As the number of steps increases, the CRR model converges to the Black-Scholes price for European options.
Unlike the Black-Scholes model, which provides a closed-form solution only for European options, the binomial tree can price American options by checking for optimal early exercise at every node during backward induction. This makes it the traditional and most widely adopted discrete method for valuing American-style derivatives in both academic and practical settings.
At each time step, the stock price can move up by a factor \(u\) or down by a factor \(d\), with risk-neutral probabilities \(p\) and \(1-p\) respectively. The CRR parameterization is:
The exercise payoff at any node represents the immediate value from exercising the option. Option values are computed by backward induction: the continuation value is the discounted risk-neutral expectation, and for American options, the value at each node is the greater of the continuation value and the exercise payoff:
Key variables:
The early exercise premium is the additional value an American option has over an otherwise identical European option. It arises because the holder of an American option can exercise at any time before expiration, not just at maturity. The binomial tree captures this premium naturally through backward induction: at each node, the model compares the continuation value \(C_{i,j}\) (the discounted expected value of holding the option) with the immediate exercise payoff \(E_{i,j}\). Whenever the exercise payoff exceeds the continuation value, the option value at that node equals the exercise payoff, and that node is marked as an early exercise node.
For call options, early exercise tends to be optimal when the dividend yield exceeds the risk-free interest rate (\(q > r\)), because the dividends lost by holding the option outweigh the time value of waiting. For put options, early exercise can be optimal even without dividends when the option is deep enough in the money, because the interest that could be earned on the exercise proceeds exceeds the remaining time value. The charts above illustrate how the early exercise premium varies with the dividend yield while keeping all other parameters fixed, using \(n = 1000\) steps for precision. The CRR binomial tree remains the traditional and widely used approach for quantifying this premium in both academic research and industry practice.
