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Key Takeaways
- The binomial model builds a lattice of up and down price steps and values the option backward from its expiration payoff.
- At each node, discount the risk-neutral weighted values from the next step.
- For American exercise, use the greater of immediate exercise and continuation value.
- With continuous dividend yield q, risk-neutral stock growth is exp[(r − q)Δt].
- Check convergence across step counts. Successive tree prices oscillate around the limiting value rather than improving at every added step.
What is the binomial option pricing model?
The binomial option pricing model is a method for calculating an option’s theoretical value by splitting the time to expiration into small steps in which the underlying can only move up or down. The branching builds a lattice, often drawn as a tree, of the prices the stock could reach. The model values the option at every node, working backward from expiration, which lets it test early exercise.
The approach was introduced by Cox, Ross, and Rubinstein in 1979, which is why the standard version is called the Cox-Ross-Rubinstein or CRR model. The terms binomial tree model and lattice model refer to the same idea. Its central appeal is flexibility: because it evaluates the option at each node individually, it can handle features that a single closed-form equation struggles with.
The most important of those features is early exercise. American-style options can be exercised at any time before expiration, and the binomial model can check at every node whether exercising immediately is worth more than holding. That makes it a standard engine for American-style equity options, where the Black-Scholes formula, which assumes exercise only at expiration, misses the early-exercise right.
How does the binomial model work?
The construction has two passes. The first pass builds the tree forward. Starting from today’s price, each step multiplies the current price by an up factor u or a down factor d. In the CRR form, u = e^(sigma x sqrt(dt)) and d = 1 / u, where sigma is the volatility and dt is the length of one step in years. After several steps the tree fans out into a range of possible prices at expiration.
The second pass works backward from the expiration payoffs. For a call, each final value is max(stock − strike, 0); for a put, it is max(strike − stock, 0). At each earlier node, discount the weighted next-step values at r. With continuous dividend yield q, use p = [exp((r − q)Δt) − d]/(u − d). The example has no dividends, so q = 0.
The probability p reproduces carry-adjusted stock growth inside the tree, which lets the replication argument price the option from today’s stock price and rate alone, whatever the stock’s expected return.
For a European option the backward pass simply discounts continuation values to today. For an American option the model adds one comparison at every node: it takes the larger of the continuation value and the intrinsic value of exercising right then. Whenever immediate exercise wins, the node value is reset to the exercise payoff, and that higher value flows into the discounting of earlier nodes. This node-by-node test is what lets the lattice price the early-exercise right that a closed-form European formula misses.
Options are quoted per share, so a modeled value of $7.48 per share corresponds to $748 for one standard contract that covers 100 shares.
Formula
Continuation = exp(−rΔt) × [p × Vup + (1 − p) × Vdown]; p = [exp((r − q)Δt) − d]/(u − d). American node value = max(exercise value, continuation).- In the CRR tree, u = exp(σ√Δt) and d = 1/u. σ is decimal volatility and Δt is years per step.
- r is the continuously compounded rate and q is continuous dividend yield. The worked example uses q = 0.
- p is a valid weight only while d < exp((r − q)Δt) < u, so shorten the step if the carry falls outside the up and down moves.
- Begin with expiration payoffs, then roll backward through the permitted exercise dates.
- A discrete cash dividend drops the stock by the dividend at its ex-dividend date in the tree, which is where an American call’s exercise test bites.
Worked Example
Value a one-year American put with stock and strike at $100, no dividends, a 6% continuously compounded rate, and two six-month steps. Use u = 1.20 and d = 1/1.20.
- Per-step growth e^(0.06 × 0.5) = 1.0305, so p = (1.0305 − 0.8333) / (1.20 − 0.8333) = 0.538 and 1 − p = 0.462.
- At the down node (stock $83.33) the hold value discounts to $13.71, while immediate exercise is worth $100 − $83.33 = $16.67.
- Because $16.67 exceeds $13.71, exercising early is optimal at that node, and its value is set to $16.67.
- Rolling back to today gives an American put worth $7.48 per share ($748 per contract), versus $6.15 for the otherwise identical European put.
The lattice checks exercise at every node, so it captures the $1.33 of value that early-exercise rights add here. Black-Scholes cannot see that, because it assumes the put can only be exercised at expiration.
How traders use the binomial model
Most traders do not build lattices by hand. The model runs inside the pricing engines that populate an option chain, the Greeks shown on a trading platform, and the theoretical values in an options profit calculator. Knowing what the engine does tells a trader which input moves each number, starting with volatility and the exercise test.
The clearest practical use is judging early-exercise situations. When a stock is about to pay a dividend, an in-the-money American call can be worth more exercised than held, and a binomial engine flags that by showing the exercise value winning at the relevant nodes. A trader holding a short call sees the same logic as assignment risk. For deep in-the-money American puts, the model similarly shows when capturing intrinsic value now beats waiting.
The model also clarifies what changes a theoretical price. Volatility feeds directly into the up and down factors, so the volatility input sets the modeled value. Comparing a binomial value against the market premium is really a comparison of the model’s volatility input against the market’s implied volatility. That framing keeps attention on the one input traders can most reasonably debate.
Because the lattice is transparent, it is also a useful benchmark. Running the same contract as a European option and as an American option isolates the value of the early-exercise right, and increasing the step count shows how quickly the estimate stabilizes. Those checks build intuition for when a quick closed-form estimate is good enough and when the lattice is doing real work.
How to choose inputs and check binomial tree convergence
Use the exact exercise style, dividends, maturity, and volatility input for the selected contract. The tree checks permitted exercise at each node and calculates value backward from the terminal payoffs.
Test convergence across multiple step counts. For a European call with the stock at $100, a $90 strike, a 5% rate, 20% volatility and one year to expiration, three CRR steps give $16.42 and four give $17.03, while the matching Black-Scholes value is $16.70. Add steps until successive prices agree to the cent you trade at.
Price with the risk-neutral weights, and express a directional view through the position you choose. For path-dependent or more complex contracts, choose a numerical method that represents the required payoff features. Compare the model value with executable premiums, then bring the chosen position into the scenario grid.
Limitations of the binomial model
- Set volatility for the contract being priced. The CRR tree applies one volatility to every step, so match its input to that strike and expiration’s implied volatility.
- Add steps until the price settles. Tree prices oscillate as steps are added: the convergence call moves from $16.42 at three steps to $17.03 at four around a $16.70 Black-Scholes value, and a coarse tree misplaces the early-exercise boundary too.
- Put the directional view in the position. The 0.538 up-weight in the example is a pricing weight that reproduces the 6% carry.
- Mark the ex-dividend date in the tree for a discrete cash dividend, and reach for Monte Carlo simulation or finite-difference methods on path-dependent or high-dimensional payoffs.
Bottom line
The binomial model values an option backward through a tree of up and down moves and tests exercise at every permitted node, which makes it a standard tool for American-style options. Use it to measure the value of early exercise and to compare dividend scenarios, after confirming that the price has stabilized across step counts.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute: Valuing a Derivative Using a One-Period Binomial Model
- CFA Institute: Valuation of Contingent Claims
- OIC: The Black-Scholes Formula
How entries are sourced, calculated and updated: editorial methodology.