Binomial Option Pricing Model
Tree-based option valuation model that prices contracts by working backward through possible up and down price paths.
The binomial option pricing model values an option by building a tree of possible underlying prices and then working backward from expiration to today.
The model is intuitive because each step allows the underlying asset to move up or down. With enough steps, the tree can approximate continuous price behavior while still handling features such as early exercise.
Binomial Tree Intuition
The diagram shows a two-step tree. The underlying price branches upward or downward at each period, and the option value is calculated backward from the final payoff nodes.
At each step, the underlying price can move:
or:
where \(u\) is the up factor and \(d\) is the down factor.
Risk-Neutral Probability
In a simple one-period model, the risk-neutral probability is often written as:
The option value is then the discounted risk-neutral expected payoff:
where \(C_u\) and \(C_d\) are the option values at the up and down nodes.
Backward Induction
The binomial model works backward:
- Build the possible underlying prices at expiration.
- Compute option payoffs at terminal nodes.
- Discount expected values backward one step at a time.
- For American-style options, compare continuation value with exercise value at each node.
That early-exercise check is one of the main reasons the binomial model remains useful.
Example
Suppose a stock is 100, the strike is 100, the one-period up factor is 1.10, the down factor is 0.90, and the risk-free growth factor for the period is 1.02.
The risk-neutral probability is:
For a call, terminal payoffs are 10 in the up state and 0 in the down state. The current theoretical value is the discounted expected payoff, before transaction costs and model limitations.
Binomial vs. Black-Scholes
| Feature | Binomial model | Black-Scholes model |
|---|---|---|
| Time structure | Discrete steps | Continuous-time closed form |
| Early exercise | Can model directly | Basic version cannot |
| Dividends | Can handle discrete dividends | Needs adjustments |
| Intuition | Visual tree | Formula-based |
| Computation | More steps improve precision | Faster closed-form calculation |
Public Source Checks
- Cboe’s Options Calculator is useful for seeing how model inputs affect theoretical option prices and Greeks.
- FINRA’s options overview explains option premiums, time decay, implied volatility, exercise, and assignment.
- The OCC Options Disclosure Document explains standardized option risks that still apply when a theoretical model is used.
Common Confusion
Do not confuse the risk-neutral probability with a real-world forecast probability. It is a pricing tool that makes the discounted expected payoff consistent with no-arbitrage assumptions.
Related Terms
- Black-Scholes Option Pricing Model: A closed-form model often compared with binomial trees.
- Risk-Neutral Valuation: The probability framework used in the tree.
- Option Pricing Models: The broader model family.
- Call Option: A contract often valued with tree methods.
- Put Option: A contract where early exercise can matter.
- Time Decay: Time sensitivity that appears through the tree’s time steps.