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A binomial price tree lists the possible stock prices in the Cox--Ross--Rubinstein model. Starting at $S_0$, each step multiplies the price by $U$ or $D$, with $0<D<U$. After $n$ steps and $k$ up moves,Constant multiplicative steps make the tree recombine: an up move followed by a down move reaches the same price as a down move followed by an up move. Thus level $n$ has $n+1$ prices despite $2^n$ possible paths.

For $S_0=100$, $U=1.1$, $D=0.9$, the second-level prices are $121,99,81$. With risk-free multiplier $R=1$, the no-arbitrage up probability is $q=(R-D)/(U-D)=1/2$. A European call with strike 100 pays $21,0,0$ at that level, so its model value is $R^{-2}{q}^2\cdot 21=5.25$. This probability is determined by replication and discounting, not a forecast of the physical chance of an up move.^[ref]

For a time step $h>0$, choose the undiscounted stock multipliers and money-market multiplier asFor sufficiently small $h$, $D_h<R_h<U_h$ and the risk-neutral up probability is $q_h=(R_h-D_h)/(U_h-D_h)$. A one-step log-stock increment takes values $\pm \sigma \sqrt{h}$; expanding $q_h$ gives mean $(\rho -{\sigma }^2/2)h+O(h^2)$ and variance ${\sigma }^{2}h+O(h^2)$. The Donsker invariance principle, in its triangular-array form, gives the risk-neutral log-price limitThus prices converge to the geometric Brownian motion used in the Black-Scholes model.^[ref]

The parent model's discounted multipliers are instead $u_h=U_h/R_h$ and $d_h=D_h/R_h$. Their log-price drift loses the $\rho$ term, as required for the discounted price to be a martingale. Convergence of claim prices also needs the payoff functional to be continuous at almost every limiting path, together with Uniform integrability when payoffs are unbounded. A process limit alone does not justify taking expectations of every payoff.

The Cox--Ross--Rubinstein model is a discrete-time frictionless market with a positive stock price $S_n$ and money-market account $B_n=R^n$, where $R>0$. In each step the stock is multiplied by $U$ or $D$, with $0<D<U$ and both moves having positive probability. The usual model has independent moves, permits borrowing, lending and short sales, and ignores transaction costs and dividends. It gives a finite binomial tree for pricing claims by self-financing replication.^[ref]

The market is arbitrage-free exactly when $D<R<U$. Indeed the risk-neutral up probability is then $q=(R-D)/(U-D)\in (0,1)$, making the discounted stock a martingale. If $R\le D$, borrowing to buy the stock gives a nonnegative one-step gain and a positive gain in at least one possible state; if $R\ge U$, the reversed position does so. Both possibilities contradict absence of arbitrage.

Discounting by the money-market account gives ${\tilde{S}}_n=S_n/B_n$. Its up and down multipliers are $u=U/R$ and $d=D/R$, so the same inequality becomes $d<1<u$. Keeping these discounted multipliers separate from $U,D$ prevents a missing discount in a payoff valuation formula. The model is useful because two terminal states and two traded assets let one solve replication by two linear equations, without estimating a physical expected stock return.

John C. Cox developed mathematical methods for option pricing and risk-neutral pricing. With Stephen A. Ross and Mark Rubinstein, he introduced the Cox--Ross--Rubinstein model, whose binomial price tree gives a particularly transparent example of a replicating portfolio.^[ref]

At each node, the two possible next-step payments determine the holdings in a stock and a riskless asset. Repeating this calculation by backward induction values the final payment through the tree. For an American option, the calculation also compares continuation value with the value of immediate exercise. This turns the distinction between European and American exercise rules into a concrete computational decision.^[ref]

Institutional profile: mitsloan.mit.edu/faculty/directory/john-c-cox

Mark Rubinstein worked on option pricing, including the Cox--Ross--Rubinstein model developed with John C. Cox and Stephen A. Ross. Their 1979 paper constructs an accessible discrete-time model in which successive up and down moves form a binomial price tree. Replicating a contingent payment with traded assets determines its value under the model's no-arbitrage assumptions.^[ref]

The construction makes the relation between hedging and valuation explicit. Its risk-neutral probabilities are weights consistent with the model's prices, rather than predictions of the actual frequency of an upward market move. Refining the time steps also connects the discrete model to the Black-Scholes model.^[ref]

Let $S_t$ be the undiscounted stock, $B_t=R^t$ the money-market account, and $Q$ the risk-neutral measure of the Cox--Ross--Rubinstein model. A terminal payoff $G$ in currency units has arbitrage-free valueEquivalently, the discounted claim $H=G/B_T$ has discounted value ${\tilde{V}}_t=V_t/B_t={\mathbb{E}}_Q[H\mid {\mathcal{F}}_t]$. These are different unit conventions for the same valuation. For a European payoff $G=g(S_T)$ and $n=T-t$,The outside discount and the undiscounted stock multipliers must appear together. If a payoff is instead expressed using the discounted stock, its function must be transformed accordingly.^[ref]

For $S_0=100$, $U=1.2$, $D=0.8$, $R=1.05$, a one-step European call option with strike100 pays20 or0. Its risk-neutral up probability is $q=0.625$, so its value is $0.625\cdot 20/1.05\simeq 11.9048$. Holding one half-share and borrowing about38.0952 reproduces both payoffs. This replication explains why the price uses risk-neutral weights rather than the investor's physical up probability.

Stephen A. Ross developed arbitrage pricing theory, which relates expected asset returns to exposures to common sources of risk under assumptions limiting arbitrage. His mathematical approach asks how restrictions on feasible trading strategies constrain prices, rather than starting solely from an individual investor's preferences.^[ref]

With John C. Cox and Mark Rubinstein, he also developed the Cox--Ross--Rubinstein model for option pricing. The binomial price tree separates the possible stock-price paths from the valuation rule: a replicating portfolio and no-arbitrage produce risk-neutral pricing even when the actual up-move probability is not specified.^[ref]