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.
What makes this calculator different
- 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.