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]

1. For American options, compare this with the value of exercising immediately and keep the larger. See American Option Pricing.
2. The value at the first node is the option price.

Worked example: a two step tree#

You can build larger trees in the Binomial Option Pricing 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 and 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#

For path dependent and multi asset options, Monte Carlo simulation is usually preferred. See 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.