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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]

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]

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]