Jul 10, 2026
Stephen Ross generalized the Black-Scholes/Merton theory from a bond, stock, and option to any collection of instruments. He showed valuation had nothing to do with probability. Positive measures having mass 1 show up, but they are not the “probability” of anything. Instead of the sophisticated machinery of Ito processes and partial differential equations he used the Hahn-Banach theorem: a point not in a convex set can be separated by a hyperplane.
This short note provides a simple and rigorous mathematical model for valuing, hedging, and managing the risk of all derivative instruments. It is based on (Ross 1978) “A Simple Approach to the Valuation of Risky Streams” where he showed
If there are no arbitrage opportunities in a market, then there must exist a (not generally unique) positive linear operator that can be used to value all marketed assets.
Ross’s “not generally unique positive linear operator” is a measure used to convert prices and cash flows to values that generalize (Graham and Dodd 1934) and (Samuelson 1965). If repurchase agreements are available in the market it corresponds to the usual stochastic discount.
Ross identified a jump in stock price as a cash flow. He was focused on the equity world where dividend payments cause stock prices to jump down by that amount after their ex-dividend date. Of course they also jump from close to open and there is no cash flow associated with that. In the fixed income world bonds are defined by their cash flows. The price of a futures contract is always zero and pays periodic cash flows based on the change in market quotes. Adding an explicit knob for cash flows to Ross’s theory results in a more realistic model.
Market instruments have both prices and cash flows. Trading strategies create synthetic market instruments where the mark-to-market corresponds to price and the amount involved in active trading corresponds to cash flows.