Rubinstein (1976) The Valuation of Uncertain Income Streams and the Pricing of Options

Summary

Develops a general discrete-time asset pricing framework based on a positive stochastic discount factor $Y_t$ common to all securities, and applies it to derive the Black-Scholes option pricing formula without requiring continuous trading or risk neutrality. The key result is that bivariate lognormality of the stock return and the pricing kernel is sufficient for Black-Scholes in discrete time under risk aversion. Published in The Bell Journal of Economics 7(2): 407–425.

Key Claims

General valuation (Theorem 1): Under the single-price law of markets and nonsatiation, there exists a positive random variable $Y_t$ — the same for all securities — such that $P_0 = \sum_t [E(X_t)/R_{Ft} + \mathrm{Cov}(X_t, Y_t)/R_{Ft}]$ . Securities paying more in low- $Y_t$ states command a premium; $Y_t$ is the stochastic discount factor. $Y_t$ is unique only when markets are complete (number of securities = number of states).
CPRA pricing kernel (Theorem 4): Under constant proportional risk aversion (CPRA) with coefficient $b$ and weak aggregation, $Y_t = R_{Mt}^{-b}$ where $R_{Mt}$ is the gross market return. Corollaries: (a) the market portfolio follows a random walk if per capita consumption follows a random walk (or $b = 1$ ); (b) the term structure is unbiased under the same conditions.
Log-Capital Asset Pricing Model (Log-CAPM): For securities jointly lognormal with the market: $\ln E(r_i) = \ln r_F + b \cdot \mathrm{Cov}(\ln r_i, \ln r_M)$ . When market returns are lognormal, $b$ can be estimated: $b = [E(\ln r_M) - \ln r_F] / \mathrm{Var}(\ln r_M)$ .
Discrete-time Black-Scholes (Theorem 5): If $S_t$ and $Y_t$ are jointly lognormal and investors agree on $\mathrm{Var}(\ln R)$ , then the Black-Scholes formula holds exactly in discrete time with risk-averse investors and no continuous hedging. Bivariate lognormality replaces the continuous-trading assumption.
Stein–Rubinstein lemma: For bivariate normal $(x, y)$ and any differentiable $g$ : $\mathrm{Cov}(x, g(y)) = E[g'(y)]\,\mathrm{Cov}(x, y)$ . Used in the option pricing proof. First derived in Rubinstein's 1971 PhD dissertation; independently noted by Stein (1973).

Concepts Introduced or Extended

Entities Mentioned

Mark Rubinstein

Quotes

"To my surprise, the resulting option pricing formula is identical to the Black-Scholes (1973) formula even though only costless discrete-time trading opportunities are available so that investors cannot create a perfect hedge, investors are risk averse, and must simultaneously choose among a large number of securities."

My Take

The central contribution is robustness of Black-Scholes: continuous hedging is not the essential ingredient — bivariate lognormality of the stock and pricing kernel is. The CPRA pricing kernel result ( $Y_t = R_{Mt}^{-b}$ ) is the discrete-time antecedent of Hansen-Jagannathan bounds and modern Stochastic Discount Factor (SDF) asset pricing. The Stein–Rubinstein lemma became a workhorse of finance theory — it underlies Merton's Intertemporal Capital Asset Pricing Model (ICAPM), beta representations for non-normal returns, and the Gaussian kernel trick in option pricing proofs — though it is often attributed to Stein (1973) alone. For this wiki the paper's primary value is as foundational background for Contingent Claim Model Error (Jacquier-Jarrow 2000) and the option pricing strand more broadly.