How a tree of up and down moves turns into a price
The binomial model replaces the continuous random walk of the underlying with a sequence of simple coin-flips. Over each short time step Δt the price either rises by a factor u or falls by a factor d, and chaining those steps together produces a recombining lattice of possible prices at expiry. You start at the right-hand edge of that tree, where the option’s payoff is just its intrinsic value, then roll backwards: each node’s worth is the discounted risk-neutral average of the two nodes it feeds into. Carry that one step at a time all the way back to the root and you have today’s fair value — the same no-arbitrage logic behind the Black-Scholes calculator, just built discretely instead of in closed form.
The binomial (CRR) lattice
Δt = T / N
u = eσ√Δt, d = 1 / u
p = ( e(r−q)Δt − d ) / ( u − d )
Value = e−rΔt · [ p·Vup + (1−p)·Vdown ]
where T = years to expiry, N = number of steps,σ = volatility, r = risk-free rate, q = dividend yield, and p is the risk-neutral up-probability. For an American option, each node also takes the larger of this held value and the immediate exercise (intrinsic) value.
Worked example
Price an American put on a $100.00 stock with a $105.00 strike, 0.5 years to expiry, 25% volatility, and a 4% risk-free rate on a 50-step CRR lattice:
| Step | Amount |
|---|---|
| Spot price (S)an American put with 0.5 years to expiry | $100.00 |
| Strike price (K)25% volatility, 4% risk-free rate, 0% dividend yield | $105.00 |
| Lattice setupup factor u = 1.0253, down factor d = 0.9753, risk-neutral p = 0.5018 per Δt = 0.0100 yr step | 50 steps |
| European value from the latticeclosed-form Black-Scholes says $8.70 — the 50-step tree lands within $0.03 | $8.67 |
| + Early-exercise premiumthe value of being able to exercise before expiry — exactly what Black-Scholes cannot price | $0.31 |
| = American put valueeach node takes the larger of holding on and exercising now | $8.98 |
Computed with this calculator's default settings — open the tool above and you'll see the same numbers, then change any input to reprice your own contract.
A binomial option model that shows its lattice
- You can see it converge. Raise the number of steps and watch the lattice price oscillate in toward the Black-Scholes value, so the link between the discrete and continuous models is visible, not just asserted.
- It prices American early exercise. At every node the model compares holding against exercising now, capturing the early-exercise premium that a closed-form formula cannot.
- The lattice parameters are shown. Δt, the up factor u, the down factor d, and the risk-neutral probability p are printed in full, because those numbers are how you actually learn how the tree is built.
- Same tested core as Black-Scholes. The European results reuse the same pricing and normal-distribution code as the Black-Scholes page, so the two calculators agree where the theory says they must.
Frequently asked questions
What is the binomial option pricing model and how does it work?
The binomial model — most commonly the Cox-Ross-Rubinstein (CRR) version — prices an option by building a tree of possible future prices for the underlying. At each small time step the price either moves up by a factor u or down by a factor d, branching into a lattice of possible outcomes. You first work out the option’s payoff at every node at expiry, then roll backwards through the tree: each earlier node’s value is the discounted, risk-neutral expectation of the two nodes it leads to. Repeating that step by step back to today gives the option’s fair value.
What are u, d, and p?
They are the three parameters that define each step of the lattice. The up factor is u = e^(σ√Δt), the size of an upward move over one time step of length Δt, and the down factor is its reciprocal, d = 1/u, which keeps the tree recombining and centered. The risk-neutral probability p = ( e^((r−q)Δt) − d ) / ( u − d ) is not a real-world forecast; it is the probability that makes the discounted expected price grow at the risk-free rate, which is what no-arbitrage pricing requires. This calculator prints u, d, and p so you can see exactly how the lattice is built.
How does the binomial model relate to Black-Scholes?
They are two views of the same no-arbitrage pricing idea. The binomial model is discrete — it chops time into N steps — while Black-Scholes is its continuous-time limit. As you increase the number of steps, the CRR tree price for a European option converges to the closed-form Black-Scholes price, oscillating in and then settling down. This calculator lets you watch that convergence directly, and shares the same tested pricing core as the Black-Scholes calculator so the two agree where they should.
What is the difference between European and American options, and why does binomial handle American better?
A European option can only be exercised at expiry, while an American option can be exercised at any time up to expiry. Black-Scholes assumes European exercise and has no clean closed form for the early-exercise feature. The binomial model handles it naturally: as it rolls backwards through the tree, at each node it simply takes the larger of the “hold” value (the discounted expectation) and the “exercise now” intrinsic value. That node-by-node comparison is exactly what early exercise requires, which is why lattice methods are the standard tool for American-style options.
How many steps should I use?
More steps make the tree a finer approximation of continuous price movement, so accuracy improves as N grows — but the work grows too, roughly with the square of the number of steps. In practice the price stabilizes quickly: a few hundred steps is plenty for most options to land within a cent of the limiting value. Very small step counts (say, under ten) are useful for seeing the mechanics of the lattice, but should not be trusted as precise prices.
Disclaimer: This calculator is foreducation and illustration only. The binomial model is a model built on simplifying assumptions (constant volatility and rates, recombining up/down moves, frictionless markets); its output is not a tradeable quote and real option prices will differ. Nothing here is investment, tax, or trading advice.