Binomial Options Tree Calculator (CRR)
Price European or American options with the Cox-Ross-Rubinstein binomial tree and see how early exercise changes the value.
🌳 What is the Binomial Options Tree Calculator?
The binomial options tree calculator prices options using the Cox-Ross-Rubinstein (CRR) model, a discrete-time alternative to the Black-Scholes formula published in 1979 by John Cox, Stephen Ross, and Mark Rubinstein. Instead of a single closed-form equation, the binomial method builds a tree of possible future stock prices, one small up or down move per time step, then works backward from expiry to today, computing the option's value at every node along the way.
The binomial tree's defining advantage over Black-Scholes is its ability to correctly price American-style options, which can be exercised at any point before expiry, not just at a single fixed date. At each node in the tree, the model checks whether exercising immediately is worth more than continuing to hold the option, and takes whichever is larger. This node-by-node flexibility is exactly what a single closed-form formula cannot capture, which is why professional options-pricing systems rely on binomial (or related numerical) methods specifically for American-style contracts.
A common misconception is that binomial pricing and Black-Scholes pricing are fundamentally different models. In fact, for European-style options, they converge to the exact same answer: as the number of tree steps grows toward infinity, the CRR binomial price approaches the Black-Scholes price as a mathematical limit. This calculator's convergence chart shows this directly, letting you watch the binomial price settle onto the Black-Scholes reference line as you add more steps.
This calculator is useful because it lets you toggle between European and American exercise style, compare the binomial result directly against the Black-Scholes closed-form price, and see the exact dollar value of early-exercise flexibility, the early exercise premium, all from the same five core option-pricing inputs used across this site's derivatives tools.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — European Call, Convergence Check
A $100 strike, 1-year European call on a $100 stock (5% rate, 20% volatility), priced with a 100-step tree.
Example 2 — American Put, Early Exercise Premium
The same $100 strike, 1-year put on a $100 stock (5% rate, 20% volatility, no dividend), but American-style, priced with a 100-step tree.
Example 3 — American Call with Dividends
A $140 strike, 6-month American call on a $150 stock (4% rate, 25% volatility, 3% dividend yield), priced with a 50-step tree.
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Cox-Ross-Rubinstein (CRR) binomial model?
The CRR model, published in 1979, prices options by modeling the underlying's price as a discrete tree where, at each small time step, the price moves up by a factor u or down by a factor d, then works backward from expiry to today to find the option's fair value, discounting expected payoffs at each node.
What are u, d, and p in the binomial model?
u = e^(sigma x sqrt(dt)) is the up-move factor, d = 1/u is the down-move factor (ensuring the tree recombines), and p = [e^((r-q)dt) - d] / (u - d) is the risk-neutral probability of an up-move, where dt = T/N is the length of each time step.
How does the binomial model handle American-style early exercise?
At every node in the tree, the model compares the value of holding the option (the discounted expected value from the next step) against the value of exercising immediately, and takes the larger of the two. This node-by-node comparison is exactly what a closed-form formula like Black-Scholes cannot do, since it can only value exercise at a single fixed date.
Why does the binomial price converge to the Black-Scholes price for European options?
As the number of steps N increases, the discrete binomial price distribution approaches the continuous lognormal distribution that Black-Scholes assumes, and the CRR tree is mathematically constructed so its limit as N approaches infinity is exactly the Black-Scholes formula for European options.
What is the early exercise premium?
The early exercise premium is the extra value an American option carries over its European counterpart, purely from the flexibility to exercise before expiry. This calculator computes it as the binomial (American) price minus the Black-Scholes (European) reference price.
Why is the early exercise premium zero for American calls without dividends?
Exercising a call early means giving up the remaining time value and receiving the stock immediately, but you could instead simply hold the call and keep earning interest on the unpaid strike price. Without dividends, this makes early exercise strictly worse than waiting, so American and European call prices are identical when there is no dividend.
Why do American puts almost always have a positive early exercise premium?
Exercising a put early converts the option into cash (the strike price) immediately, which can then earn interest until expiry. This time-value-of-money benefit exists even without dividends, which is why American puts are worth strictly more than European puts in nearly every realistic scenario.
How many steps should I use in the binomial tree?
More steps improve accuracy but increase computation time. For most practical purposes, 100 to 200 steps gives a price stable to 2-3 decimal places; this calculator's convergence chart shows exactly how the price stabilizes as steps increase for your specific inputs.
Does the binomial model handle dividends?
Yes, this calculator applies a continuous dividend yield q in both the risk-neutral probability formula and the terminal payoff calculation, the same continuous-yield approach used in the Black-Scholes-Merton extension. Dividends reduce both call values and increase the likelihood of early exercise being optimal for American calls.
Is the binomial model used in real trading?
Yes, especially for American-style equity options, where no simple closed-form formula exists. Many professional options-pricing systems use binomial or trinomial trees, or related numerical methods, specifically because they can properly value the early-exercise flexibility that Black-Scholes cannot.