Binomial option pricing model
The binomial option pricing model values an option by building a tree of possible underlying prices over discrete steps and working backward from expiry, discounting risk-neutral expected values at each node. Because each node can compare holding with exercising, it prices American-style options, which the closed-form Black-Scholes formula cannot.
Senzoukria · Glossary · Updated September 2026
At a glance
- Up and down factors (CRR)
- u = e^(σ√Δt), d = 1/u
- Risk-neutral probability
- p = (e^((r−q)Δt) − d)/(u − d)
- Main use
- American exercise, discrete dividends
How the tree works
Split the option's life into n steps of length Δt. In the Cox-Ross-Rubinstein version the price moves up by u = e^(σ√Δt) or down by d = 1/u at each step. At expiry the option's payoff is known at every terminal node. Moving backward, each node's value is the discounted risk-neutral average of its two children, e^(−rΔt)·[p·V_up + (1 − p)·V_down]. For an American option the node value is the larger of that continuation value and the immediate exercise value.
Worked example: two steps
Spot 100, strike 100 put, six months, σ = 30%, r = 5%, two steps of three months. Then u = 1.1618, d = 0.8607 and p = 0.5043. Terminal prices are 134.99, 100 and 74.08, giving put payoffs of 0, 0 and 25.92. One step back, the down node at 86.07 has a continuation value of 12.69 but an immediate exercise value of 13.93, so an American holder exercises there. Rolling back gives 6.82 for the American put against 6.21 for the European one on this coarse tree.
| Steps | American put | European put |
|---|---|---|
| 2 | 6.82 | 6.21 |
| 500 | 7.39 | 7.16 |
| Black-Scholes (European) | — | 7.17 |
Strengths and limits
- Handles early exercise, discrete dividends and some path features that closed forms cannot.
- Converges to Black-Scholes for European options as steps increase, with a small oscillation from one step count to the next.
- Greeks are read from neighbouring nodes, which makes gamma noisy on coarse trees.
- It still assumes one constant volatility along the tree unless extended; it does not by itself explain the smile.
Relevance for index traders and Senzoukria
SPX options are European, so the closed form applies; SPY and QQQ options are American, and a tree or an equivalent method captures their early-exercise premium. For the near-the-money, short-dated contracts that dominate gamma exposure, the difference between European and American gamma is usually small. Senzoukria does not run a binomial model: its locally computed greeks use closed-form Black-Scholes, and provider greeks come from the provider's own model, which the Calculation assumptions panel names by provider rather than guessing.
Related
In the same section
- Black-76
- Big trades
- Bid-ask spread
- Futures block trade
- Bid-ask bounce
- Block trade
- Bid × ask volume
- Book depth
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Frequently asked questions
- When is early exercise of an American option worth anything?
- For a call on a non-dividend-paying underlying, practically never, because selling the call keeps its time value. It can be optimal just before an ex-dividend date for deep in-the-money calls, and for deep in-the-money puts when interest earned on the strike exceeds the remaining time value.
- How many steps are enough?
- It depends on the required precision and the maturity. Prices often stabilise to the cent within a few hundred steps for monthly options; greeks need more care. Averaging results from two consecutive step counts reduces the oscillation.