The Trillion Dollar Equation

A single equation spawned four multi-trillion dollar industries and fundamentally transformed how people approach risk in financial markets.

• Jim Simons created the Medallion Investment Fund in 1988, delivering 66% annual returns for 30 years • $100 invested in 1988 would be worth $8.4 billion today • Simons became the richest mathematician of all time

Isaac Newton, despite his mathematical genius, lost a third of his £30,000 wealth investing in the South Sea Company in 1720 because he couldn't predict human behavior.

The most successful market traders were not veteran traders but physicists, scientists, and mathematicians who could model market dynamics mathematically.

The Greek philosopher Thales of Miletus executed the first known call option around 600 BC by paying olive press owners to secure the right to rent their presses at a fixed price during the harvest season.

• A call option gives the right to buy something at a later date for a set strike price • A put option gives the right to sell something at a later date for the strike price • Options provide limited downside risk (you lose only the premium paid) but potentially large gains

• Limits downside losses to the premium paid • Provides leverage: a $10 option on $100 stock can yield 200% returns • Can be used as insurance to hedge against price movements

For hundreds of years, traders had no mathematical way to price options. They simply bargained to agree on a price, creating chaos on trading floors.

Louis Bachelier, working at the Paris Stock Exchange, realized that stock prices are influenced by countless unpredictable factors, making them impossible to forecast with certainty.

Bachelier proposed that at any point in time, stock prices are equally likely to go up or down, and therefore follow a random walk like a coin flip, with expected future prices forming a normal distribution.

• Bachelier rediscovered the same equation Joseph Fourier used to describe heat radiation in 1822 • He called his discovery the 'radiation of probabilities' • His work laid the foundation for pricing options using probability calculations

When Bachelier finished his PhD, he had solved a centuries-old problem and beat Einstein to the random walk concept, but neither physicists nor traders noticed his work.

Scottish botanist Robert Brown observed in 1827 that microscopic particles suspended in water moved randomly, a phenomenon later called Brownian motion, but the cause remained unexplained for 80 years.

In 1905, Einstein hypothesized that Brownian motion results from trillions of molecules hitting particles from every direction, with random imbalances causing observable movement.

• Einstein's derivation assumed particles move equally likely in any direction, just like stock prices • Particle location follows a normal distribution that widens over time • This explains diffusion even in still water

By solving the Brownian motion mystery, Einstein provided definitive evidence that atoms and molecules exist, unaware that Bachelier had discovered the same mathematics five years earlier.

• Ed Thorpe, a physics graduate, invented card counting by keeping mental track of played cards • He bet larger amounts when odds favored him and smaller amounts when they didn't • Casinos countered by adding more decks, reducing the advantage of card counting

Thorpe transferred his card-counting skills to the stock market, calling it the biggest casino on Earth, and started a hedge fund that achieved 20% annual returns for 20 years.

Thorpe pioneered dynamic hedging, where an option seller owns stock to offset losses. If stock price rises, losses from the option are covered by stock gains, allowing risk-free profits with minimal fluctuation risk.

Thorpe created a more accurate option pricing model than Bachelier's, accounting for stock price drift (the tendency of stocks to increase or decrease over time based on company performance).

In 1973, Fischer Black and Myron Scholes published an equation for option pricing, with Robert Merton independently publishing a version based on stochastic calculus, fundamentally changing the financial industry.

They argued that a risk-free portfolio of options and stocks should return only the risk-free rate (US Treasury bond returns). This means fair option pricing balances risk between buyers and sellers equally.

• The Chicago Board Options Exchange was founded the same year • The formula was adopted as the benchmark for Wall Street within a couple of years • Options market volume has doubled roughly every five years since

• Exchange-traded options market (multi-trillion dollars) • Credit default swaps market • OTC derivatives market • Securitized debt market

Airlines use options to hedge against oil price increases by pricing options to buy something tracking oil prices, which pay off when oil costs rise and compensate for higher fuel expenses.

• With $1 of cash, you can buy $1 of stock • With $1 of cash, you can buy options affecting $10-20 of stock • This natural leverage in derivatives amplifies both gains and losses

Reddit traders on r/wallstreetbets used options leverage to drive up GameStop stock prices, forcing hedge fund short sellers to lose money quickly by combining stock purchases with options that multiplied market impact.

Options enable hedging for companies, governments, and individual investors. They provide liquidity during normal times, but can exacerbate market crashes during periods of stress when all securities move in the same direction.

Global derivatives markets are on the order of several hundred trillion dollars, representing multiples of the underlying securities they are based on.

Options allow a single underlying asset to be transformed into 5, 10, 20, or 50 different versions with varying risk-reward profiles that appeal to investors with different preferences.

• During normal times: derivatives provide significant liquidity and stability • During market stress: all securities can move together in the same direction • In crashes: derivatives markets can exacerbate market dislocations

If we ever discover all patterns in the stock market, knowing what they are will eliminate them. We would finally achieve a perfectly efficient market where all price movements are truly random.

• Jim Simons was a world-class mathematician whose work on Riemann geometry influenced knot theory, quantum field theory, and quantum computing • Chern-Simons theory laid mathematical foundation for string theory • He received the Oswald Veblen Prize in geometry in 1976

When founding Renaissance Technologies in 1978, Simons' strategy was to use machine learning to find patterns in the stock market, believing complexity prevents certainty but patterns provide opportunities.

• Simons hired physicists, astronomers, mathematicians, and statisticians with PhDs and proven research records • He explicitly avoided people with finance backgrounds • He sought scientists who had done science well in their respective fields

• Used hidden Markov models and data-driven strategies • Became the highest-returning investment fund of all time • Challenged the efficient market hypothesis itself

In 1988, Bradford Cornell published research showing that the efficient market hypothesis is false and that predictabilities exist in the stock market, based on the Medallion Fund's extraordinary performance.

It is possible to beat the market if you have the right models, proper training, adequate resources, sufficient computational power, and advanced mathematical techniques.

Physicists and mathematicians discovered patterns in stock markets and uncovered the randomness underlying them, providing new insights into risk and opening entirely new markets.

• Determined accurate prices for derivatives • Eliminated market inefficiencies • Demonstrated that mathematical modeling could revolutionize finance