Binomial Option Pricing Calculator
Price American and European options on a Cox-Ross-Rubinstein tree, with the early exercise premium stated separately.
Black-Scholes answers one question and answers it in closed form: what is an option worth if the only day it can be exercised is its last. A great many listed options can be exercised on any day before that, and for those the closed form is a lower bound rather than a price. A binomial option pricing calculator exists for the wider case, and this one is an American option calculator before it is anything else: it prices the right to exercise early and states what that right is worth on its own.
Formula: u = e^(sigma x sqrt(dt)), p = (e^((r-q) x dt) - d) / (u - d)
Your figures
American options may be exercised at any time; European options only at expiry.
The price at which the option may be exercised.
Calendar days. A tree with no time left has no branches, so this must be above zero.
Annualised, as a percentage. Volatility is what makes the tree branch, so this must be above zero.
Annualised. Roughly the yield on a short government bond.
Optional — annualised continuous yield on the underlying.
How many periods the option's life is divided into. More steps converge on the closed form; 500 is the ceiling.
Results
American put price on the tree
US$5.98
per share, over 100 steps
- Early exercise premiumWhat the right to exercise before expiry is worth, per share, on this tree.
- US$0.2362
- American price on this tree
- US$5.98
- European price on the same treeBoth styles are priced in one pass, so the difference between them is the exercise right alone.
- US$5.74
- Black-Scholes priceThe closed form for the same inputs, which prices European exercise only.
- US$5.76
- Tree less Black-ScholesDiscretisation error plus any early exercise value. It narrows as steps are added.
- +US$0.2188
- Intrinsic valueWhat the option is worth if exercised right now.
- US$0.00
- Up factor and down factorThe multiples applied to the underlying at each step. Their product is 1, which is what makes the tree recombine.
- 1.017711 · 0.982597
- Risk-neutral probability of an up moveThe weight that removes drift from the price. Not a real-world probability.
- 50.26%
- Steps used
- 100 · 1.800 days each
What this means
A 100-step Cox-Ross-Rubinstein tree values this American put at US$5.98 per share. Each step covers 1.800 days, over which the underlying is taken to multiply by 1.017711 or by 0.982597, weighted by a risk-neutral probability of 50.26%. The same tree prices the European twin at US$5.74, so on these inputs the right to exercise before expiry is worth US$0.2362 per share. Black-Scholes gives US$5.76 for the same inputs; the gap to the tree is part exercise right and part the lattice's own approximation. These are model values under stated assumptions, not a forecast of what the option will trade at.
- The tree is an approximation that converges on the closed form as steps are added, oscillating slightly as it does. Running the same inputs at two step counts shows how much of the difference to Black-Scholes is the exercise right and how much is the lattice.
- Dividends are modelled as a continuous yield. A share paying discrete cash dividends is only approximated by that, and the approximation is weakest around an ex-dividend date — which is exactly when an American call is most likely to be exercised early.
- Assumes one constant volatility and one constant rate for the whole life of the option. Real chains price a different implied volatility at every strike and expiry.
American put price on the tree: US$5.98
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Continue with these figures
About the Binomial Option Pricing calculation
The binomial tree answers the broader question by refusing to leap straight to expiry. Binomial tree option pricing divides the option's life into a run of short steps, lets the underlying move up or down by a fixed multiple at each one, and then walks backwards from the final payoffs to today. Because it stops at every node on the way, it can ask at each of them whether the holder would do better to take the money now — which is a question a closed form has no place to pose.
As a Cox-Ross-Rubinstein calculator it builds that lattice from the underlying price, the strike, the days remaining, one volatility, the risk-free rate and any dividend yield, over as many steps as you ask for. It prices the American and the European option on the same tree in a single backward pass, so the difference between the two is the value of the exercise right alone, uncontaminated by any difference in the grid they were priced on.
The Black-Scholes price for the same inputs sits beside them, along with the up factor, the down factor and the risk-neutral probability the tree is constructed from. Nothing is fetched and no venue's terms are assumed: the volatility, the rate, the dividend yield and the step count are all fields you fill in.
Binomial Option Pricing formula
T = days ÷ 365 dt = T ÷ n
u = e^(σ × √dt) d = 1 ÷ u
p = (e^((r − q) × dt) − d) ÷ (u − d)
terminal node j of n: S × u^(2j − n)
payoff there: call max(node − K, 0)
put max(K − node, 0)
rolling back one step, at every node:
continuation = e^(−r × dt) × [p × V(up) + (1 − p) × V(down)]
European = continuation
American = max(continuation, intrinsic at that node)
early exercise premium = American price − European price- S, K
- Underlying price now and the strike the option is exercised at.
- T
- Time to expiry in calendar years. 180 days is 180 ÷ 365.
- n
- Steps in the tree. Each covers dt years, or days ÷ n calendar days.
- u, d
- The multiples applied to the underlying on an up and a down move. Their product is exactly 1.
- p
- Risk-neutral probability of an up move — the weight that removes drift, not a forecast.
- σ (sigma)
- Annualised volatility as a decimal, held constant over the whole life of the option.
- r, q
- Annualised risk-free rate and continuous dividend yield, both as decimals.
- continuation
- What holding on is worth at a node: the discounted, probability-weighted value of the two nodes it leads to.
The choice of d = 1 ÷ u is what makes the whole thing tractable. An up move followed by a down move lands back on the price it started from, so the lattice recombines and a tree of n steps has n + 1 terminal nodes rather than 2ⁿ. At a hundred steps that is the difference between five thousand-odd nodes and a number with thirty digits in it, which is the difference between an answer that appears as you type and no answer at all.
The probability p is not an opinion about where the underlying is going. It is defined as the weight that makes the expected value of the underlying after one step equal its forward price, e^((r − q) × dt) times where it stands now, and that definition is what strips the drift — and therefore the trader's view — out of the price entirely. Two people who disagree completely about direction still get the same number from this tree, which is the point of the construction.
That construction only holds together while p sits strictly between 0 and 1. If the one-step growth were below the down move or above the up move, one branch would beat the risk-free rate with certainty and there would be an arbitrage rather than a price, so those inputs are rejected instead of being quietly priced. It happens when the gap between the rate and the dividend yield is large relative to what the volatility can produce over a single step, and adding steps or raising the volatility both pull p back towards a half.
Both exercise styles are computed on the same tree in the same pass, and that is deliberate rather than incidental. A lattice is an approximation, so its price carries a small discretisation error; subtracting Black-Scholes from the American tree would fold that error into a figure meant to isolate the exercise right. Subtracting the European tree from the American one cancels it, because both carry the same error. What is left is the premium for early exercise and nothing else.
Where that premium comes from differs by option type, which is why the same tree is worth running on both. An American put earns it from interest: exercising early turns the strike into cash that can then earn the risk-free rate for the rest of the option's life, so the premium grows with the rate and with the time remaining. An American call on a share paying nothing earns no premium at all — exercising early throws away time value and pays the strike sooner than necessary — and only a dividend large enough to outweigh that makes early exercise on a call worth anything.
Accuracy improves with steps, but not smoothly. The tree's terminal nodes either straddle the strike neatly or they do not, and that alternates with the parity of n, so the price oscillates around the true value as steps are added rather than creeping towards it. Running the same inputs at two nearby step counts is the cheapest way to see how much of the gap to Black-Scholes is the lattice and how much is real.
Binomial Option Pricing: a worked example
An American put five points in the money: underlying at 95, strike 100, 180 days to expiry, 25% volatility, a 5% risk-free rate, no dividend, priced over 100 steps.
- T, in years180 ÷ 365 = 0.493151 years
- dt0.493151 ÷ 100 = 0.004932 years, or 1.8 days a step
- u, the up factore^(0.25 × √0.004932) = e^0.017556 = 1.017711
- d, the down factor1 ÷ 1.017711 = 0.982597
- p, the up probability(1.000247 − 0.982597) ÷ (1.017711 − 0.982597) = 0.5026
- European on the tree8.1093
- American on the tree8.4807
- Early exercise premium8.4807 − 8.1093 = 0.3714
- Black-Scholes8.0942
- Tree less Black-Scholes8.4807 − 8.0942 = 0.3865
- Intrinsic value nowmax(100 − 95, 0) = 5.00
8.4807 per share for the American put against 8.1093 for its European twin, so the right to exercise before expiry is worth 0.3714 — around 4.4% of the option's value, and none of it visible in the closed form. The remaining 0.0151 between the European tree and Black-Scholes is the lattice's own approximation at a hundred steps; raising the step count shrinks that figure while leaving the 0.3714 essentially where it is, which is what separates the two.
How to use the Binomial Option Pricing calculator
- Choose call or put, then the exercise style. American allows exercise at any point; European only at expiry, and the tree prices both regardless so the comparison is always on screen.
- Enter the underlying price and the strike, then the calendar days remaining. Weekends count, because time value decays across them.
- Enter one annualised volatility for the whole life of the option. If you only have a traded price, the implied volatility calculator converts one into the other.
- Enter the risk-free rate and, for a share or index that pays one, the continuous dividend yield. On a call the yield is what creates any early exercise value at all.
- Set the step count. A hundred is enough for ordinary equity inputs; run the same figures at 101 and at 200 to see how much the answer is still moving.
- Read the early exercise premium first. It is the figure the tree exists to produce, and it is the part of the price the closed form beside it cannot see.
Limitations of the Binomial Option Pricing calculator
Every calculation makes assumptions. These are the ones that matter here.
- The tree is an approximation that converges on the closed form as steps are added, and it oscillates rather than settling as it does. A price quoted to four decimals is not accurate to four decimals at a low step count.
- Dividends are modelled as a smooth continuous yield. A share paying discrete cash dividends is only roughly represented by that, and the approximation is at its weakest around an ex-dividend date — which is exactly when an American call is most likely to be exercised early.
- One volatility applies to every node and every step. Real chains price a different implied volatility at each strike and expiry, and no single figure entered here reproduces that surface.
- The rate is constant for the option's whole life, which matters most on the long-dated options where the early exercise premium on a put is largest.
- Returns are log-normal on this lattice, so the underlying moves by small multiplicative steps and never gaps. Overnight jumps around earnings and news are the case the model is structurally unable to represent.
- The American price is the value of exercising optimally at every node. A holder who exercises early when the tree says hold, or holds when it says exercise, does not realise it.
- A model value under stated assumptions, not a quote. Bid-ask spread, commission, borrowing cost on a short position and assignment fees are all outside it.
- The step count is capped at 500, because the work grows with the square of it and this runs on every keystroke.
Frequently asked questions about Binomial Option Pricing
Why is the American price never below the European price?
Because at every node the American holder takes the greater of continuing and exercising, and the European holder can only continue. Having a choice cannot be worth less than not having one, so the difference is either positive or zero — never negative. Any tiny negative figure a lattice produces is floating-point noise, and it is suppressed here rather than reported as a premium.
When does an American call carry an early exercise premium?
Only when the underlying pays a dividend large enough to outweigh the time value thrown away by exercising early. Exercising a call means paying the strike sooner than you have to and surrendering the remaining optionality, so with no dividend it is never worth doing and the two prices coincide exactly. That is why this calculator opens on a put.
How many steps does binomial tree option pricing need?
A hundred is enough for ordinary equity inputs to sit within a cent or so of the converged value, and the errors that remain are smaller than the bid-ask spread on most chains. If precision matters, run the same inputs at two nearby step counts: the size of the wobble between them is a fair estimate of how much lattice error is still in the number.
Why does the tree not exactly match Black-Scholes?
Two separate reasons, and they are worth telling apart. The lattice takes discrete steps where the closed form takes a continuous limit, which leaves a small approximation error that shrinks as steps are added. On an American option there is also genuine early exercise value that Black-Scholes does not price at all, and that part does not shrink with more steps.
Is the risk-neutral probability the chance the option finishes in the money?
No. It is the weight that makes the discounted expected value of the underlying equal its forward price, which is a device for removing drift from the arithmetic rather than a statement about the world. It sits close to a half for most inputs regardless of how anyone expects the underlying to behave, and it should not be read as a forecast of anything.
Why was my input rejected as arbitrage-violating?
Because the numbers entered put the one-step growth outside the range the up and down moves span, which makes the risk-neutral probability fall outside 0 to 1 and means one branch would beat the risk-free rate with certainty. It usually follows a large gap between the rate and the dividend yield set against a small volatility. Raising the volatility, adding steps or narrowing that gap all fix it.
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