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Douglas T. Breeden coauthored the extraction of state-contingent prices from option pricing with Robert H. Litzenberger. For sufficiently regular call prices across strikes, their second strike derivative identifies a discounted state-price density. The construction makes an entire option-price curve informative about the market's pricing of future outcomes, rather than only one payoff.^[ref]

For $Z$ with standard normal distribution, $m\in \mathbb{R}$, and $a>0$, integrate $(m+az)\varphi (z)$ over $z>-m/a$. The identity ${\varphi }^{\prime }(z)=-z\varphi (z)$ gives the displayed expectation of the positive part. This is useful for option pricing when a financial payoff is the positive part of a normal random variable, including a log-call option. At $a=0$ the formula is interpreted by its limit, $m^{+}$.

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]

Myron Scholes, with Fischer Black, developed the Black-Scholes model and its option pricing equation. Their 1973 paper connects a contingent claim's value to the cost of a dynamically adjusted replicating portfolio, under explicit assumptions about trading and the underlying price process.^[ref]

The mathematical insight is that a carefully chosen hedge can cancel the instantaneous random component of the claim's value. No-arbitrage then constrains the remaining locally riskless return, producing the Black-Scholes equation together with a final payoff condition. The result is a conditional conclusion within a model: it does not require an ability to predict the actual stock-price path.^[ref]

Institutional profile: www.gsb.stanford.edu/faculty-research/faculty/myron-s-scholes

Robert H. Litzenberger studied state-contingent claim prices with Douglas T. Breeden. Their call-spread argument relates derivatives of call option prices with respect to strike to state prices. Under suitable smoothness, the second derivative yields a discounted density, providing a bridge between observable option pricing and the pricing of narrow ranges of future asset values.^[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]