The Black-Scholes derivation descends directly from a corrected Treynor equation

Evidence

This was a reformulation of the later, less successful, steps in Treynor’s approach to the mathematics of capital asset pricing (Treynor 1963), and the aspect of it on which Black and Treynor collaborated had involved Treynor writing an expression for the change in the value of a cash flow in a short, finite time interval At; expanding the expression using the standard calculus technique of Taylor expansion; taking expected values; dropping the terms of order AP and higher; dividing by At; and letting At tend to zero so that the finite-difference equation became a differential equation. Treynor’s original version of the latter was in error because he had left out “some terms involving second derivatives” that did not vanish, but Black and he worked out how to correct the differential equation. — Donald MacKenzie, An Engine, Not a Camera, 569

Categorization

• XREF: Establishes the intellectual genealogy from CAPM into Black-Scholes — central to the performativity thesis’s claim that models have a lineage. — connects to: Deriving the Black-Scholes equation required invoking CAPM’s riskless-rate logic. — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 581; Black’s original derivation assumed CAPM pricing of both stock and warrant — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 1266
• QUESTION: Interesting that the founding mathematical move began with an error (dropped second-derivative terms). Worth chasing how much of option theory rests on corrections of prior errors.

Broader Connections

• Deriving the Black-Scholes equation required invoking CAPM’s riskless-rate logic. #corroborates (0.85) — Donald MacKenzie, An Engine, Not a Camera, 581