The Pricing of Options and Corporate Liabilities
Fischer Black (University of Chicago) and Myron Scholes (MIT), Journal of Political Economy 81(3), May–June 1973, pp. 637–654. Received November 1970, final version May 1972. Read here from the SFU-hosted copy of the JSTOR scan (stable URL); the Princeton mirror of the same scan carries no text layer and does not extract.
The paper that gave this spoke’s missing layer — option-pricing — a closed-form answer, and the one whose argument shape the rest of the corpus keeps failing to imitate.
The argument
The premise is stated in the abstract as a refusal rather than a model: “If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks.” Everything follows from that.
Black and Scholes assume seven “ideal conditions”: a known, constant short-term interest rate; a stock price following a continuous random walk with variance proportional to the square of the price, so terminal prices are lognormal and the variance rate of return is constant; no dividends; a European option, exercisable only at maturity; no transaction costs; unlimited borrowing at the short rate; and no penalty for short selling.
Under those conditions a hedged position — one share long against 1/w₁ options short, where
w₁ is the partial derivative of the option value with respect to the stock price — has a value that
does not depend on the stock price. Adjusted continuously, its return “is completely independent of
the change in the value of the stock”; adjusted discretely, the residual risk is uncorrelated with
the market and diversifiable. A position with no risk must earn the short-term rate, or “speculators
would try to profit by borrowing large amounts of money to create such hedged positions, and would in
the process force the returns down to the short term interest rate.” Imposing that equality gives a
differential equation for the option value, which with the boundary condition at maturity has exactly
one solution. It turns out to be the heat-transfer equation of physics, solved in Churchill (1963).
The solution is the formula that carries their names:
w(x, t) = x·N(d₁) − c·e^{r(t−t*)}·N(d₂)with x the stock price, c the exercise price, t* the maturity date, r the short rate, v²
the variance rate of return, and N(·) the cumulative normal distribution.
The claim that matters most here
“Note that the expected return on the stock does not appear in equation (13).”
The option’s value is independent of how fast anyone expects the stock to rise. Two investors who disagree completely about a company’s prospects must, if they accept the assumptions, agree on what its options are worth. That is what makes the result usable: the hardest input in finance — expected return — is eliminated rather than estimated, and what remains (price, strike, time, rate, volatility) is either observable or the single unknown the market is really quoting.
Two smaller results in the same passage. Maturity enters the formula only multiplied by r or v²,
so lengthening the option has the same effect as raising both. And x·w₁/w is always greater than
one: an option is always more volatile than its stock, which is the leverage that makes the
instrument attractive and dangerous in the same breath.
Why it belongs in this wiki and not only in a textbook
The corpus’s standing complaint is that every performance number it holds comes from a simulator its author wrote (backtesting, backtest-overfitting). This paper is the counter-example on method, and it is 1973: the authors derive a price from a no-arbitrage condition, state the seven conditions under which the derivation holds, and then report where their own formula fails.
Their empirical tests (Black and Scholes 1972, on a large body of call-option data) found that market prices “deviate in certain systematic ways from the values predicted by the formula.” Buyers consistently pay more than the formula says; writers receive about what it says; the gap is larger for options on low-risk stocks, so “the market appears to underestimate the effect of differences in variance rate.” And the authors close the loop themselves: given the transaction costs in that market — all of them effectively borne by buyers — “this systematic misestimation of value does not imply profit opportunities for a speculator.”
Set that beside superior-skills‘s headline of six trades at a 100% win rate, or fincept-terminal shipping factor discovery and RL with no word on validation. Black and Scholes found a real, replicated deviation between their model and the market, and reported that it was not tradeable after costs. Nothing in this corpus’s modern sources reports a negative result of any kind about its own method. That is the gap chan-algorithmic-trading argues about from the simplicity side and deflated-sharpe-ratio measures from the statistical side, shown here as a matter of what an author chooses to publish.
The assumption list is the other transferable part. Every one of the seven is false in some market — rates move, variance is not constant, stocks pay dividends, short selling is penalized, costs are real — and the paper’s usefulness comes from having them written down where a reader can check them against their own instrument. No backtest in this wiki ships that list.