Asymmetric Convexity

Optionality from Thales of Miletus to Modern R&D Portfolios

In Book I, Chapter 11 of the Politics, compiled at the Lyceum in Athens around 330 BCE, Aristotle preserved the earliest recorded financial option contract in history. Citizens of the Ionian port city of Miletus had mocked the natural philosopher Thales for his poverty, arguing that his astronomical calculations of eclipses and solstices were useless because they produced no wealth. According to Aristotle's account, during the winter months around 580 BCE—six months before the summer olive harvest—Thales walked through the agricultural districts of Miletus and the neighboring island of Chios and approached the owners of every stone rotary olive press in the region.

Because it was midwinter and no one knew whether the coming summer would bring drought or bumper crops, the press owners readily accepted small cash deposits—known in Greek commercial law as arrha—to reserve the exclusive right to rent their presses at a fixed rate when harvest arrived. Thales did not buy the presses, nor did he commit to paying the full summer rental fee if the harvest failed. He paid a tiny upfront premium to lock in a fixed ceiling on his rental cost while capping his maximum loss at the value of his winter deposits.

When the summer of 580 BCE produced an abundant olive crop across Ionia, every grower in Miletus and Chios needed to crush their fruit simultaneously before it rotted. Holding exclusive rental rights to every press in both territories, Thales sublet the machines at whatever spot price the growers were willing to pay, amassing a fortune in a single season. Aristotle attributed the coup to Thales predicting the weather through astronomy, yet Thales required no meteorological foresight for the trade to work. He had engineered a contract with an asymmetric payoff: his downside was strictly bounded to a few silver drachmas of winter earnest money, while his upside scaled non-linearly with the harvest.

In 1906, the Danish mathematician and telephone engineer Johan Jensen published a paper in Acta Mathematica that formalized the structural advantage Thales had exploited. A function $f(x)$ is convex over an interval if a straight line connecting any two points on its graph lies on or above the curve itself ($f''(x) > 0$), so the slope accelerates upward as $x$ increases. For any random variable $X$ subject to uncertainty and any convex function $f(x)$, Jensen proved the inequality:

When a payoff function curves upward, the average payoff across volatile outcomes exceeds the payoff of the average outcome. Consider a European call option on an asset with strike price $K = $100$ purchased for a premium of $c = $5$, whose net payoff at expiration is $\max(S_T - K, 0) - c$. If the market stays flat at its average price of $100, the option expires worthless and the holder loses $5: $f(\mathbb{E}[S_T]) = -$5$. Now introduce volatility: suppose there is a 50 percent chance the asset crashes to $50 and a 50 percent chance it surges to $150. The average asset price is still $\mathbb{E}[S_T] = $100$. Yet in the crash scenario, the option holder's loss cannot exceed the $5 premium, whereas in the surge scenario, the option nets $$150 - $100 - $5 = +$45$. The expected payoff under volatility leaps to $\mathbb{E}[f(S_T)] = 0.5(-$5) + 0.5(+$45) = +$20$. Because the left tail is amputated at $-$5$ while the right tail remains open, rising variance directly increases expected value.

Throughout the mid-twentieth century, Mervin Kelly, who served as Director of Research and then President of Bell Telephone Laboratories from 1936 to 1959, structured his Murray Hill laboratory in New Jersey as a portfolio of convex options. Rather than forcing physicists to optimize vacuum-tube repeaters for linear 5 percent efficiency gains, Kelly allocated bounded annual budgets to small teams exploring solid-state physics, information theory, and radio astronomy. Most exploratory projects failed at the fixed cost of a few researcher-years. Yet when John Bardeen, Walter Brattain, and William Shockley demonstrated point-contact germanium amplification on December 23, 1947, that single right-tail outcome—the transistor—generated trillions of dollars in downstream value.

The same arithmetic governed early biotechnology firms such as Amgen, founded in Thousand Oaks, California, in April 1980 under CEO George Rathmann. Rather than betting its entire $19 million Series A capital on one molecule, Rathmann divided the laboratory into parallel, low-budget teams pursuing synthetic indigo, poultry growth hormones, hepatitis B vaccines, and human erythropoietin. Five of those early programs yielded zero revenue and were shut down once their bounded budgets expired. In October 1983, however, geneticist Fu-Kuen Lin cloned the human erythropoietin gene after screening 1.5 million bacteriophage plaques. Approved by the Food and Drug Administration as Epogen on June 1, 1989, that single surviving call option generated more than $25 billion in cumulative sales, paying for every failed experiment in the company's history a thousand times over.