Economist

Harry Markowitz developed modern portfolio theory, making the tradeoff between expected return and variability a mathematical allocation problem. His 1952 Portfolio Selection by Harry Markowitz emphasizes that the variability of a combination depends on relationships between its components, not just on each component considered alone.^[ref]

For an original toy example, let two return random variables $X,Y$ each have variance one. Equal weighting givesWith zero covariance the result is $1/2$; with covariance one it is one. The calculation explains why counting assets alone does not determine diversification. Expected values, covariance estimates and allocation constraints supply distinct inputs to the optimization.

The 1990 economics prize recognized his theory of portfolio choice. The awarding academy's account identifies both the 1952 paper and his expanded 1959 treatment.^[ref]

Robert Litterman developed the Black-Litterman model with Fischer Black. Their 1992 paper makes equilibrium expected returns a neutral starting point for portfolio modeling, then incorporates views that can differ in their confidence. The contribution addresses how uncertain inputs enter modern portfolio theory, rather than treating every proposed expected return as equally reliable.^[ref]

An elementary Bayesian inference analogy illustrates the role of uncertainty. For an unknown quantity $\mu$ with normal distribution prior $N(m_0,s_0^2)$ and an independent observation error giving $y\mid \mu \sim N(\mu ,s_y^2)$, Gaussian conditional expectation gives the posterior meanThe more uncertain input receives less weight. This scalar example explains a modeling principle; the full Black-Litterman model treats related expected returns jointly, with a covariance matrix describing their uncertainty.

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]

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

Steven L. Heston introduced the Heston model, a stochastic volatility model in which the asset price and its instantaneous variance evolve together. The variance follows a square-root diffusion, while correlation between the price and variance innovations allows asymmetric option-price patterns.^[ref]

The model is useful because richer uncertainty does not remove all analytical structure. An affine expression for the characteristic function of the log price allows Fourier inversion to recover the probabilities used in European option valuation. The resulting pricing formula includes numerical integration; “closed-form” here does not mean that every price is an elementary expression without a numerical step.^[ref]

Institutional profile: www.rhsmith.umd.edu/directory/steve-heston

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

Fischer Black developed the Black-Scholes model with Myron Scholes. Their 1973 analysis shows how continuous adjustment of a replicating portfolio can eliminate the local exposure to the underlying random price change, leading to a no-arbitrage equation for the value of a contingent claim.^[ref]

This approach makes the hedge part of the mathematical explanation of the price. It also makes the assumptions visible: ideal continuous adjustment and the specified price dynamics are ingredients of the derivation, so changing the trading rules or dynamics can change the valuation problem. The Black-Scholes equation is therefore a useful starting point for studying both a tractable model and the effects of relaxing its assumptions.^[ref]

An economist who developed foundations of game theory and the Von Neumann-Morgenstern utility theorem with John von Neumann.

Researcher whose work includes Peer effects, teacher incentives, and the impact of tracking: evidence from a randomized evaluation in Kenya by Esther Duflo and others.

Economist studying interventions to reduce violence and improve youth outcomes.

Economist whose research is represented by School readiness and later achievement by Greg J. Duncan and others.