# Binomial and Trinomial Trees

> Binomial and trinomial trees price options by stepping prices up and down through time. Learn how the Cox Ross Rubinstein model works, with a worked example.

Source: https://learn.tradelabsai.com/options/binomial-and-trinomial-trees/  
Track: Options · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Binomial and Trinomial Trees", https://learn.tradelabsai.com/options/binomial-and-trinomial-trees/

A binomial tree prices an option by splitting the time to expiration into steps and assuming that at each step the underlying price can move only up or down by set amounts. Working backwards from expiration, you calculate the option's value at every point. Trees are intuitive, handle American options with early exercise naturally and converge to Black Scholes as the number of steps grows. Trinomial trees add a third "middle" move for faster, more stable convergence.

## The Cox Ross Rubinstein (CRR) model

John Cox, Stephen Ross and Mark Rubinstein published the standard binomial model in 1979. For each step of length Δt:

```
u = e^(σ × √Δt)
d = 1 / u
p = (e^(r × Δt) - d) / (u - d)
```

- **u:** up factor
- **d:** down factor
- **p:** risk neutral probability of an up move (not a real world probability)

## The pricing procedure

1. **Build the price tree** forward from today's price, multiplying by u or d at each step.
2. **At expiration,** calculate the option payoff at each final node.
3. **Step backwards:** at each earlier node, the option's value is the discounted expected value of the next two nodes:

```
V = e^(-r × Δt) × [p × V_up + (1 - p) × V_down]
```

4. **For American options,** compare this with the value of exercising immediately and keep the larger. See [American Option Pricing](https://learn.tradelabsai.com/options/american-option-pricing/).
5. **The value at the first node** is the option price.

## Worked example: a two step tree

**Example: An American put with two steps**
S = $100, K = $100, T = 1 year, two steps (Δt = 0.5), σ = 20%, r = 5%.

- u = e^(0.20 × √0.5) = e^0.1414 ≈ 1.1519; d ≈ 0.8681
- p = (e^0.025 minus 0.8681) / (1.1519 minus 0.8681) = (1.0253 minus 0.8681) / 0.2838 ≈ 0.554

Prices at expiry: 100 × 1.1519² ≈ $132.69, 100 × 1.1519 × 0.8681 = $100.00, 100 × 0.8681² ≈ $75.36.
Put payoffs: $0, $0, $24.64.

Step back to 6 months:
- Up node ($115.19): value = e^(minus 0.025) × (0.554 × 0 + 0.446 × 0) = $0.
- Down node ($86.81): continuation = 0.9753 × (0.554 × 0 + 0.446 × 24.64) ≈ $10.72. Exercise now: $100 minus $86.81 = $13.19. Early exercise is better, so the value is $13.19.

Today: 0.9753 × (0.554 × 0 + 0.446 × 13.19) ≈ $5.74.

A European put (no early exercise) would use $10.72 at the down node, giving about $4.66. With hundreds of steps, the European value converges to the Black Scholes price of about $5.57, and the American value to about $6.09.

You can build larger trees in the [Binomial Option Pricing Calculator](https://learn.tradelabsai.com/tools/binomial-calculator/).

## Why trees converge to Black Scholes

As the number of steps grows, the distribution of final prices in a binomial tree approaches a lognormal distribution, which is what Black Scholes assumes. See [Binomial and Bernoulli Distributions](https://learn.tradelabsai.com/math/binomial-distribution/) and [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/). In practice, a few hundred steps give prices accurate to a cent for most options.

## Trinomial trees

A trinomial tree lets the price move up, down or stay the same at each step. Benefits:

- **Faster convergence** and smoother results.
- **More flexibility** to match term structures of rates and volatility, which is why trinomial trees are common for interest rate models such as Hull White.

## Strengths and weaknesses

| Strengths | Weaknesses |
|---|---|
| Handles American exercise directly | Slow for many underlyings at once |
| Easy to understand and code | Results oscillate as step count changes |
| Handles dividends at specific dates | Path dependent options need extra work |
| Gives Greeks from the tree | Less suited to complex volatility dynamics |

For path dependent and multi asset options, Monte Carlo simulation is usually preferred. See [Monte Carlo Option Pricing](https://learn.tradelabsai.com/options/monte-carlo-option-pricing/).

## Frequently asked questions

### What is a binomial option pricing model?

A method that models the underlying moving up or down by set amounts at each time step, then works backwards from expiration to find the option's value.

### Why use a binomial tree instead of Black Scholes?

Trees handle American options with early exercise and discrete dividends, which the basic Black Scholes formula cannot.

### What is a trinomial tree?

A tree in which the price can move up, down or stay the same at each step, giving faster and smoother convergence than a binomial tree.

Next, learn simulation based pricing in [Monte Carlo Option Pricing](https://learn.tradelabsai.com/options/monte-carlo-option-pricing/).

## Continue learning

- Next lesson: [Monte Carlo Option Pricing](https://learn.tradelabsai.com/options/monte-carlo-option-pricing/)
- Previous lesson: [Black-76 and Bachelier Models](https://learn.tradelabsai.com/options/black-76-and-bachelier-models/)
- Related: [Black-76 and Bachelier Models](https://learn.tradelabsai.com/options/black-76-and-bachelier-models/): Black 76 prices options on futures and forwards; Bachelier assumes normal price changes and handles negative prices. Learn the formulas, uses and differences.
- Related: [American Option Pricing](https://learn.tradelabsai.com/options/american-option-pricing/): American options can be exercised early, so they need special pricing methods. Learn trees, finite differences, approximations and Longstaff Schwartz simulation.
- Related: [Binomial Option Pricing Calculator](https://learn.tradelabsai.com/tools/binomial-calculator/): Free binomial option pricing calculator using a Cox Ross Rubinstein tree. Price American or European calls and puts and see the early exercise premium.
- Related: [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/): The Black Scholes model prices European options from five inputs. Learn the formula, its assumptions, a step by step example and where the model breaks down.
- Related: [Binomial and Bernoulli Distributions](https://learn.tradelabsai.com/math/binomial-distribution/): The binomial distribution gives the probability of a number of wins in a set of trades. Learn the formula, trading examples and its link to option trees.
- Related: [Monte Carlo Option Pricing](https://learn.tradelabsai.com/options/monte-carlo-option-pricing/): Monte Carlo pricing simulates many random price paths and averages the discounted payoffs. Learn the method, a Python sketch, accuracy, variance reduction and uses.