Binomial Option Pricing Calculator With a Binomial Tree

How to use the binomial option pricing calculator with a binomial tree

1. Choose call or put, and American or European exercise.
2. Enter the stock price, the strike and days to expiration.
3. Enter implied volatility, the risk free rate and the dividend yield as percentages.
4. Set the number of steps. More steps give a more accurate price.
5. Read the option value, u, d, p and the step length in the results panel.
6. Study the drawn binomial tree, which shows up to 6 steps with the same inputs.

How the binomial option pricing model works

The binomial model splits the time to expiration into N equal steps. In each step the stock either moves up by a factor u or down by a factor d, so after N steps the possible stock prices form a tree. At every final node the option's value is known: it is the intrinsic value. The model then works backward, one step at a time, to today.

At each earlier node, the value of holding the option is the discounted, probability weighted average of its two possible values one step later, using the risk neutral probability p. For an American option the model also checks whether exercising at that node is worth more than holding, and keeps the larger of the two. That early exercise check is what makes the binomial tree the standard reference model for American options.

Binomial option pricing formula (Cox Ross Rubinstein)

The calculator uses the Cox Ross Rubinstein parameters, published in 1979. Δt is the length of one step in years, σ is implied volatility, r is the risk free rate and q is the dividend yield. The up and down factors make the tree match the stock's volatility, and p makes the expected stock return equal the risk free rate minus the dividend yield. Node (j, i) is the node after j steps with i down moves. The formulas are:

Δt = T ÷ N, with T = days ÷ 365
u = e^(σ × √Δt)
d = 1 ÷ u
p = [e^((r minus q) × Δt) minus d] ÷ (u minus d)
Stock price at node (j, i) = S × u^(j minus i) × d^i
At expiration: V = max(0, S minus K) for a call, max(0, K minus S) for a put
Holding value = e^(minus r × Δt) × [p × V(up) + (1 minus p) × V(down)]
American node value = max(holding value, exercise value)
European node value = holding value

Worked example: American put, 100 steps

The calculator loads with an American put on a $100 stock with a $100 strike, 45 days to expiration, implied volatility of 30%, a 4.1% risk free rate, no dividend yield and 100 steps. Each step is 0.45 days long, with u = 1.01059, d = 0.98952 and p = 0.49977. The American put is worth $3.9794 against a Black Scholes value of $3.9425, so the right to exercise early adds about 3.7 cents. Switch to European exercise and the tree returns $3.9425, matching Black Scholes to four decimals.

The drawn tree uses 6 steps of 7.5 days each, with u = 1.04394, d = 0.95791, p = 0.49905 and a discount factor of 0.99916 per step. Its root value is $3.8511, below the 100 step price because a coarse tree tracks the stock's path less precisely. Early exercise is optimal at six nodes, all on the low side of the tree: $87.90 after 3 steps, $91.76 and $84.20 after 4 steps, and $95.79, $87.90 and $80.65 after 5 steps.

Take the lowest node after 5 steps, where the stock is $80.65. One step later the stock is either $84.20, where the put is worth $15.80, or $77.26, where it is worth $22.74. The calculation at that node runs:

• Holding value = 0.99916 × (0.49905 × 15.8033 + 0.50095 × 22.7422) = 19.2631
• Exercise value = 100 minus 80.6527 = 19.3473
• Exercising beats holding by about 8 cents, so the node is marked for early exercise and takes the value 19.35

How many steps does a binomial tree need?

In this binomial tree calculator, more steps give a more accurate price, because the tree's stock prices get closer to a continuous path. The error shrinks roughly in proportion to 1 ÷ N, but not smoothly: prices oscillate as N switches between odd and even. The calculator damps that wobble by averaging an N step tree and an N plus 1 step tree, and it accepts up to 2,000 steps.

In the default example the 100 step American put is $3.9794, and the 500 step tree used by the option pricing calculator gives $3.9780, a difference of about a tenth of a cent. The drawn tree is capped at 6 steps so every node stays readable, while the headline price always uses the full step count.

Binomial tree vs Black Scholes

A European binomial tree converges to the Black Scholes price as the number of steps grows, and with these inputs 100 steps already match it to four decimals. The difference is that a tree can check for early exercise at every node, which Black Scholes cannot. That makes it the natural way to price American puts and American calls on dividend paying stocks, and to see where the early exercise boundary lies.

The tradeoff is speed. A tree with N steps evaluates about N² ÷ 2 nodes, so it is slower than closed form approximations such as Bjerksund and Stensland or Barone Adesi and Whaley. For a single option the difference is negligible, and the option pricing calculator shows the tree next to the faster models so you can compare them.

Frequently asked questions

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Educational tool, not financial advice. Calculations are theoretical and depend on your inputs; real fills, fees, early assignment and dividends can change results. Options involve risk and are not suitable for every investor.