The dynamic hedge ratio at the core of option pricing resisted analytic solution.

Evidence

What Scholes could not work out, however, was how to construct a zerobeta portfolio. He could see that the quantity of shares that had to be sold short must change with time and with changes in the stock price, but lie could not see how to determine that quantity. `After working on this concept, off and on, I still couldn’t figure out analytically how many shares of stock to sell short to create a zero-beta portfolio.” (Scholes 1998, p. 480) Like Black, Scholes was stymied. — Donald MacKenzie, An Engine, Not a Camera, 577

Categorization

• QUESTION: The dynamic rebalancing insight — that short position must change continuously with price and time — prefigures the eventual delta-hedging solution; worth tracking how the fixed quantity becomes the derivative.
• XREF: connects to: The binomial model reduces option pricing to a stock that only rises or falls — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 1274; The Cox-Ross-Rubinstein model made option pricing computable numerically. — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 618; Numerical backward recursion solves option problems that resist analytic solution — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 1287

Broader Connections

• The binomial model reduces option pricing to a stock that only rises or falls #responds-to (0.78) — Donald MacKenzie, An Engine, Not a Camera, 1274