Risk-neutral probability
q = (exp(r Delta t) - d) / (u - d)The pricing weight makes the expected gross return equal the risk-free growth factor.
Derivatives
Binomial Option Pricing (CRR)
Price options on a tree of future prices by working backwards under a risk-neutral world.
Mental model
A tree prices an option by valuing its future branches and working backward under no-arbitrage.
The binomial model assumes that at each small time step the underlying price can move up or down by fixed factors, forming a recombining price tree.
Starting from maturity, the option value at each node is the discounted risk-neutral expectation of its values in the next step, with American options allowing early exercise at each node.
Explore the mechanism
How does backward induction turn terminal option payoffs into today's no-arbitrage value?
Cox–Ross–Rubinstein (CRR) binomial option pricing. The chart shows how the option price converges as the number of time steps N increases. The orange line is a high-step reference; the green point is the current N. Below we also decompose the price into intrinsic and time value.
Price (N = 100): 9.3936Limit ≈ 9.4084 at high N
Numbers
Price at N = 100: 9.3936
High-step reference (N = 400): 9.4084
Discretisation error ≈ -0.0149 (|Δ| ≈ 0.0149; ≈ -0.16% of limit)
Intrinsic value ≈ 0.0000, time value ≈ 9.3936
Moneyness S₀/K ≈ 1.000 (out of the money)
Risk-neutral p ≈ 0.503
Interpretation
The CRR lattice approximates a continuous-time model by slicing the maturity into N steps. For small N the price can sit away from the high-step reference; as N grows, the blue curve stabilises around the orange line and the vertical Δ shrinks.
The decomposition “intrinsic + time value” is how traders read this price: intrinsic reads off the payoff if you exercised now, time value is what you pay for optionality and volatility until expiry. In-the-money options with very low time value are close to behaving like forwards.
The risk-neutral probability p should lie in [0,1]; if it needs clamping, your u/d choice is outside the textbook no-arbitrage region. In practice, a binomial tree is a numerical control grid: N is like your re-hedging frequency. More steps push you towards the continuous model but increase computation; American features justify the lattice when early exercise or path structure matters.
UNDERSTAND
What to notice
The tree is equivalent to constructing a self-financing hedging portfolio (replicating strategy) at each step, so pricing and hedging are two views of the same object.
As the number of steps increases and the time step shrinks, the CRR binomial model converges to Black–Scholes under standard assumptions.
The quantitative structure
Notation
u, d- up and down multipliers per step
q- risk-neutral up probability
Delta t- step length
Backward induction
V = exp(-r Delta t) [q V_up + (1-q) V_down]Each node is the discounted risk-neutral value of its successor nodes, subject to early exercise when allowed.
This model assumes
- The one-step market is arbitrage-free, requiring d < exp(r Delta t) < u.
- Trading and rebalancing follow the tree's discrete dates.
A small example
A one-step call
Spot is 100. In one step it becomes 120 or 90. Strike is 100, the gross risk-free factor is 1.05.
- Risk-neutral probability is (1.05 - 0.90) / (1.20 - 0.90) = 0.50.
- Terminal call payoffs are 20 and 0.
- Discount the expected payoff: (0.5 * 20 + 0.5 * 0) / 1.05 = 9.52.
ResultThe one-step no-arbitrage call value is approximately 9.52.
Where it appears
- pricing American exercise features
- teaching replication and risk-neutral valuation
- valuing contracts on discrete state trees
Where the intuition breaks
What this view leaves out
- A coarse tree introduces discretization error, while complex path dependence or many state variables can make the tree prohibitively large.
Common mistakes
“Each up/down probability is a forecast of the next market move.”
Not quite.The risk-neutral probability is chosen for pricing consistency; it need not equal the real-world probability.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Binomial Option Pricing (CRR)
Binomial Option Pricing (CRR)YOU ARE HERE
Built onArbitrage and No-ArbitrageThe CRR tree prices derivatives by replication and no-arbitrage at each step.
Built onRisk-Neutral Measure (Intuition)Risk-neutral probabilities provide a compact pricing representation of the replicating binomial tree.
UsesOption Payoff Types and Path DependenceTrees handle exercise features and certain path-dependent payoff structures through backward induction.
Validated byPut–Call ParityA consistent binomial implementation should preserve put–call parity for comparable European contracts.
Related toBlack–Scholes Option PricingCRR trees and Black–Scholes are complementary option-pricing frameworks, with binomial prices converging toward the continuous-time benchmark under standard settings.
Built on · Arbitrage and No-Arbitrage
The CRR tree prices derivatives by replication and no-arbitrage at each step.
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