Binomial Option Pricing: How Trees Value Options

Key Takeaways

• Binomial option pricing builds a discrete lattice using up-factor u=exp(σ√dt) and risk-neutral probability p, then discounts backward from expiration.
• As steps increase toward infinity, the binomial price converges to the Black-Scholes price for European options.
• A common mistake: confusing the risk-neutral probability p with the real-world probability, p is a no-arbitrage construct, not a forecast.
• The model handles American early exercise by comparing continuation value to immediate exercise at every node, the key advantage over closed-form BS.

What It Is

Cox, Ross, and Rubinstein introduced the binomial model in their 1979 Journal of Financial Economics paper "Option Pricing: A Simplified Approach." At each node of the tree, the underlying can move up by a factor u or down by a factor d. Repeated branching generates a lattice of possible prices. Each terminal node has a known payoff, and the option's present value is the discounted expectation of those payoffs under a risk-neutral probability.

The headline result: as the number of steps grows, the binomial price converges to the Black-Scholes price. The tree is a discrete approximation of the same continuous-time process.

The Intuition

Black-Scholes is elegant but opaque. It tells you the answer without showing you why. A binomial tree shows you the machinery. At every step the option holder faces a choice between holding or exercising, and the value at each node is whichever is larger. You can see exactly where early exercise kicks in on an American put, exactly how path-dependent features behave, and exactly what the delta is at any node by comparing the two children.

The tradeoff is computational cost. Black-Scholes is one line. A 1,000-step tree is a million-node recursion. Modern computers make that cost trivial, which is why binomial trees are standard in practitioner software.

How It Works

Choose a number of steps n over the option's life T, giving a time step dt = T/n. Set up and down factors and the risk-neutral probability using the CRR parameterization:

u = exp(sigma * sqrt(dt))
d = 1 / u
p = (exp(r * dt) - d) / (u - d)

Where sigma is volatility, r is the risk-free rate, and p is the probability of an up move in a risk-neutral world. Note that p is not the real-world probability. It is the probability under which discounted asset prices are martingales, which is the condition that prevents arbitrage.

Build the tree forward. At step i, price j ups up to S * u^j * d^(i-j). At expiration, compute the payoff at every terminal node:

Call payoff = max(S_T - K, 0)
Put  payoff = max(K - S_T, 0)

Work backward. At each earlier node, the option value is the discounted expected value of its two children:

V_node = exp(-r * dt) * (p * V_up + (1 - p) * V_down)

For American options, compare that continuation value to the immediate exercise value at each node and take the larger. That one extra line is why trees handle American exercise naturally where Black-Scholes cannot.

Worked Example

Price an American put on a stock at 100 with strike 100, three months to expiry, 30 percent volatility, 5 percent rate, and three steps. Then dt = 0.25/3 = 0.0833, u = exp(0.30 * sqrt(0.0833)) = 1.0905, d = 0.9170, and p = (exp(0.05 * 0.0833) - 0.9170) / (1.0905 - 0.9170) = 0.5014.

At expiration the stock can reach 130, 109, 92, or 77. Put payoffs: 0, 0, 8, 23. Roll back one step, discounting by exp(-0.05 * 0.0833) = 0.9958:

V = 0.9958 * (0.5014 * V_up + 0.4986 * V_down)

At the node where the stock is 92, continuation value is roughly 15.4 while immediate exercise is worth 8, so continuation wins. At the node where the stock is 77, continuation is about 18.7 but immediate exercise pays 23, so early exercise is optimal and we set the node value to 23. Rolling that correction all the way back to the root produces a higher American put price than Black-Scholes would give for the same inputs. That extra value is the early exercise premium.

Common Mistakes

1. Using too few steps. A three-step tree is good for illustration, not valuation. Convergence to Black-Scholes is slow, on the order of 1/n. Most practitioner trees use 100 to 1,000 steps for vanilla options, more for path-dependent structures.

2. Confusing the risk-neutral probability with the real-world probability. p is a pricing device, not a forecast. It is whatever value makes the discounted underlying a martingale. Plugging in a real-world estimate of the up probability and discounting at the risk-free rate produces an arbitrage-able price.

3. Ignoring the dividend correction. Discrete dividends shift the tree because the underlying drops by the dividend amount on ex-date. Failing to adjust produces prices that are wrong, especially for call options where early exercise the day before a dividend can be optimal.

4. Mishandling American early exercise on calls. On non-dividend stocks it is never optimal to exercise an American call early. Many students still apply the max(intrinsic, continuation) rule mechanically without checking whether it ever binds. It does not bind for calls without dividends.

5. Hardcoding CRR parameters on low-volatility assets. When sigma * sqrt(dt) is small, rounding and numerical error matter. The Jarrow-Rudd and Leisen-Reimer parameterizations converge faster than CRR and are preferred for production code.

Frequently Asked Questions

Q: What is binomial option pricing in simple terms? The binomial model splits an option's life into small time steps where the underlying can go up or down by known amounts. It values the option by working backward from expiration payoffs using risk-neutral probabilities and discounting at the risk-free rate.

Q: How does binomial pricing affect investment decisions? It produces a price for American options, especially deep ITM puts, that Black-Scholes undervalues because it cannot model early exercise. Practitioners rely on binomial trees when they need to know whether and when early exercise is optimal.

Q: What is a real-world example of binomial option pricing in action? Pricing an American put on a 100-strike stock with 30% vol: at one node where the stock is 77, immediate exercise pays 23 while continuation value is 18.7, so the tree flags early exercise as optimal, raising the price above what Black-Scholes would give.

Q: How many steps should a binomial tree use? At least 100 to 200 steps for practical accuracy on vanilla options. Convergence to Black-Scholes is proportional to 1/n, so a 3-step tree gives you an illustration, not a tradeable price.

Q: How is binomial option pricing different from Black-Scholes? Black-Scholes is a continuous-time closed-form formula that only handles European exercise. The binomial model is a discrete approximation that can handle American exercise, dividends, and any path-dependent feature at the cost of more computation.

Sources

1. Cox, J., Ross, S., Rubinstein, M. (1979). "Option Pricing: A Simplified Approach." Journal of Financial Economics, 7(3), 229-263. https://www.sciencedirect.com/science/article/abs/pii/0304405X79900151
2. CERGE-EI. "The Cox-Ross-Rubinstein Option Pricing Model." http://home.cerge-ei.gz/petrz/fm/f400n10.pdf
3. Brenndoerfer, M. "Binomial Tree Option Pricing: American Options and the CRR Model." https://mbrenndoerfer.com/writing/binomial-tree-option-pricing-cox-ross-rubinstein
4. Gundersen, G. "Binomial Options-Pricing Model." https://gregorygundersen.com/blog/2023/06/03/binomial-options-pricing-model/

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.