What is the binomial option pricing model?
The binomial model breaks the time to expiration into steps. In each step the stock can move to one of two prices, up or down, which creates a tree of possible paths. The version most people use was published by John Cox, Stephen Ross and Mark Rubinstein in 1979, and it is often called the CRR model.
The model is popular for two reasons. It is intuitive, because you can follow every calculation on paper, and it handles American options, because it can compare exercising with holding at every node. As the number of steps grows, the binomial price of a European option converges to the Black Scholes price.
The tree also makes the logic of option pricing visible. At each node you can see the stock price, the option value and the hedge ratio, which is why the model is widely used to teach replication and risk neutral pricing.
What is the binomial option pricing formula?
Everything rests on a one step formula. Over a step of length Δt, the stock either rises to S × u or falls to S × d, and the option is worth Vu or Vd in those two states. Today's value is the discounted risk neutral average of the two.
In the formulas below, r is the risk free rate, q is the continuous dividend yield and sigma is the volatility. The probability p is not a forecast. It is the probability that makes the stock's expected return equal the risk free rate minus the dividend yield, which lets the model price options without knowing anyone's real expectations. The same value comes from a hedge: holding (Vu minus Vd) / (S × u minus S × d) shares, financed at the risk free rate, replicates the option over the step.
The formulas for the Cox Ross Rubinstein tree are:
- Option value = e^(minus r × Δt) × (p × Vu + (1 minus p) × Vd)
- u = e^(sigma × square root of Δt), the up factor
- d = 1 / u, the down factor
- p = (e^((r minus q) × Δt) minus d) / (u minus d), the risk neutral probability of an up move
What does a one step and two step binomial example look like?
Start with a single step to see the mechanics. Use illustrative inputs: a stock at 100, a strike of 100, six months to expiration (T = 0.5), a risk free rate of 4 percent, volatility of 25 percent and no dividend. With one six month step, u = e^(0.25 × square root of 0.5) = 1.1934 and d = 0.8380, so the stock ends at 119.34 or 83.80. A call struck at 100 pays 19.34 or 0, p is about 0.513, and the call is worth e^(minus 0.02) × 0.513 × 19.34, about 9.72 per share.
Now use two steps, so Δt = 0.25. Then u = e^(0.25 × 0.5) = 1.1331, d = 1 / 1.1331 = 0.8825, and p = (e^(0.01) minus 0.8825) / (1.1331 minus 0.8825), which is about 0.509. Each step is discounted by e^(minus 0.01), about 0.9900. The stock tree reaches 113.31 and 88.25 after one step, and 128.40, 100.00 and 77.88 at expiration.
Working backward means starting at the final nodes, where the option is worth its payoff, and applying the one step formula node by node until you reach today. For an American option, each earlier node also compares that result with immediate exercise. Here is the American put:
- At expiration the put pays max(K minus S, 0), which is 0, 0 and 22.12 at the three final nodes.
- At the 113.31 node both future values are 0, so the put is worth 0 there.
- At the 88.25 node, holding is worth 0.9900 × (0.509 × 0 + 0.491 × 22.12) = 10.76, but exercising pays 100 minus 88.25 = 11.75. The American put takes 11.75, so early exercise is optimal at this node.
- At the start, the value is 0.9900 × (0.509 × 0 + 0.491 × 11.75) = 5.71 per share.
How many steps does a binomial tree need?
In the example, a European put cannot exercise at the 88.25 node, so it uses 10.76 there and comes out at 5.23. The two step tree therefore puts the early exercise premium near 0.48. But two steps are far too few for accuracy: the Black Scholes value of the same European put is about 6.03. With several hundred steps, the tree's European put converges to that value and the American put settles near 6.2.
Accuracy does not improve smoothly. Prices zigzag as you add steps, because the strike falls at a different spot relative to the nodes when the step count switches between odd and even. Averaging the results for N and N+1 steps cancels much of that oscillation, and CalcOptions uses exactly that smoothing. The cost is speed, since the work grows with the square of the number of steps. Other refinements, such as the Leisen Reimer tree and Richardson extrapolation, attack the same problem.
How does the tree handle early exercise and dividends?
For an American option, every node takes the larger of two values: the continuation value from the one step formula, and the exercise value, which is S minus K for a call or K minus S for a put. The nodes where exercise wins trace out the early exercise boundary, the stock price below which an American put should be exercised, or above which a call should be exercised ahead of a dividend.
A continuous dividend yield enters through the drift term r minus q in p, which is how CalcOptions handles dividends. Discrete cash dividends on specific dates need extra machinery, such as subtracting the present value of upcoming dividends from the stock price or building a tree that does not recombine. Because CalcOptions uses a continuous yield, check the timing of any large dividend yourself.
When should you use a binomial tree instead of a closed form approximation?
The binomial tree is the benchmark. Given enough steps it converges to the true American value, and it is flexible enough for discrete dividends and unusual exercise rules. Its weakness is speed, which matters when you price a whole option chain or solve for implied volatility over and over.
That is why CalcOptions uses the Bjerksund and Stensland (2002) approximation by default and offers the smoothed CRR tree and Barone Adesi and Whaley as alternatives. For most listed options the three methods agree closely, and the differences are usually small. When they do differ noticeably, typically for long dated, deep in the money puts, the tree is the more trustworthy reference.
Trees are also a good way to sanity check other numbers. If an approximation and a smoothed tree disagree by more than you expect, look at the inputs first, especially the dividend yield and the time to expiration, before trusting either price.
Frequently asked questions
Is the binomial model better than Black Scholes?
The binomial model is better for American options because it can value early exercise, which Black Scholes cannot. For European options the two agree, since the binomial price converges to the Black Scholes price as steps increase. Black Scholes is faster when early exercise does not matter.
How many steps should a binomial tree use?
Use enough steps that adding more no longer changes the price in a meaningful way. Prices oscillate between odd and even step counts, so averaging N and N+1 steps, as CalcOptions does, reaches a stable answer with fewer steps. Two or three steps are useful for learning but far too coarse for trading.
What is the risk neutral probability in the binomial model?
The risk neutral probability is p = (e^((r minus q) × Δt) minus d) / (u minus d), the up probability that makes the stock earn the risk free rate minus the dividend yield on average. It is a pricing tool, not a forecast of what the stock will do. Using it lets the model price options without estimating the stock's real expected return.
Try it in a calculator
Long Put
A long put is the purchase of a single put option, which gives you the right to sell 100 shares at the strike price on or before expiration.
Short Put
A short put, also called a naked or uncovered put, is the sale of a put option backed by margin rather than the full cash needed to buy the shares.
Protective Put
A protective put is a position that owns 100 shares of stock and buys one put option to set a floor under them.
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.