How Brownian Motion Built Modern Finance

Tommy Lin

London, UK

Stock Market Display Attribution: Wix

The foundation of modern finance - derivatives markets, algorithmic trading, risk management - surprisingly rests on equations originally developed to describe pollen grains moving in water, a concept seemingly unrelated to finance.

In 1900, thea French mathematician called Louis Bachelier wrote his doctoral thesis where he argued that stock prices move like particles in Brownian motion, following what we now call a continuous random walk. At that time, nobody paid attention to his thesis. His work was dismissed as too applied for mathematicians and too mathematical for economists.

Five years later, in 1905, Einstein independently developed almost identical mathematics, but on the aspect of physics, not finance. He proved that the random unpredictable motion of particles was caused by their bombardment from invisible, moving atoms which make up a fluid, thus providing evidence for the existence of atoms. Einstein consequently earned lasting fame, winning the Nobel Prize in 1921. Meanwhile, Bachelier did not reach anything close to that level of recognition;. fFor most of his life, he remained in the shadows. Yet, Bachelier had discovered something profound about both physical particles and financial markets. Today, he is widely credited as the ‘father of mathematical finance.’

The Paradox of Predictability

What makes Brownian motion so interesting is that it is not “random” in the simple sense of the word. The parallel between Brownian motion and financial markets runs deeper than Bachelier or Einstein might have imagined. Both systems exhibit an interesting paradox: deterministic behaviour at the micro-scale produces unpredictable outcomes at the macro-scale.

In Brownian motion, individual pollen grains are constantly bombarded by water molecules from all directions. According to classical physics, each collision obeys Newton's laws. In theory, with complete knowledge of every molecule's position, velocity and mass, one could precisely predict the pollen's trajectory, demonstrating that Brownian motion is indeed deterministic.

Yet in practice, the argument starts to fail. A single glass of water contains over 8 × 10²⁴ molecules;. The sheer number of collisions makes prediction impossible without infinite computational power. We therefore model Brownian motion as a stochastic (random) process - not because it violates physical laws, but because its complexity renders it effectively unpredictable, a property found in all non-deterministic systems.

This is why, on a macroscopic level, we treat Brownian motion as a stochastic process despite being deterministic in principle. More specifically, Brownian motion is a continuous random walk.

Financial markets behave very similarly. At the micro-level, individual trading decisions appear deterministic: buy orders match sell orders, supply meets demand, prices adjust accordingly. Yet when tens of millions of these interactions occur simultaneously - each trader reacting to price movements created by other traders, with small perturbations like large orders or rumours triggering feedback loops - the system becomes chaotic and unpredictable.

Whether markets are truly deterministic or fundamentally random remains debated but, for practical purposes, it doesn't matter either way. Financial markets are modelled as stochastic processes, the same mathematical framework used for Brownian motion.

The Random Walk

Bachelier's breakthrough was recognising that stock prices follow a random walk - a mathematical model where each change is independent of its history and occurs randomly.

Imagine flipping a fair coin repeatedly: if it lands on heads, you step right (+1), and if tails you step left (-1). Your position after each flip is the cumulative sum of all your steps. The key point to understand is that each flip is independent - knowing you flipped three heads in a row tells you nothing about the next flip. The coin has no memory. This independence makes the system unpredictable.

The counterintuitive mathematical property of random walks is that after n steps, your typical distance from the starting point is √n steps, not n steps. After 100 steps, you're typically √100 = 10 steps away, not 100.

Formally, the mean squared displacement (MSD) grows linearly with n, meaning the typical distance from the starting point grows like √n. This “square root of time” behaviour is a key feature of Brownian motion.

This √n scaling applies directly to stock prices.

If a stock moves ±2% daily, then over a 5-day trading week, the typical price change is:

±2% × √5 ≈ ±4.5%

Over a year (approximately 252 trading days):

±2% × √252 ≈ ±31.7%

However, stock prices differ from standard Brownian motion in one critical way: they cannot become negative. A company's share price might fall to zero, but never below or else itthat violates the limited liability rule. This is why financial models use geometric Brownian motion, where prices multiply by random factors (eg 0.96) rather than adding random increments.

The Black-Scholes Breakthrough

The core application of Brownian motion in finance came from the pricing of options - financial contracts giving the right (but not the obligation) to buy or sell a security at a predetermined price on a specified future date.

Before 1973, over the counter (OTC) option pricing was fragmented, relying solely on intuition and guessing. How much should you pay today for the right to buy a stock at £100 in six months, when you don't know what the stock price will be? Therefore, all traders and institutions needed an objective method to price such contracts.

Economists Fischer Black, Myron Scholes, and Robert Merton realised that if stock prices follow geometric Brownian motion, the theoretical fair price of any European-style option could be derived mathematically. They assumed stock prices evolve according to the following stochastic differential equation of GBM:

dS = μS dt + σS dW

S = stock price

μ = drift (average % expected return per unit time)

σ = volatility

dt = infinitesimal time increment

dW = Brownian increment (random walk)

Here, the stock price change has a predictable drift component (μS dt) and an unpredictable component (σS dW) proportional to volatility.

Solving this required stochastic calculus because ordinary calculus assumes smooth functions, unlike the jagged paths of Brownian motion. In 1951, a Japanese mathematician called Kiyoshi Itô created Itô's Lemma, a chain rule (used for differentiating a composite function) for random processes – in this case, how the option price changes as the stock price moves along a random walk.

Black and Scholes used dynamic hedging—constantly adjusting stock holdings against the option—to cancel out the random fluctuations (∂W) from the stock's random walk. Itô’s lemma helped them analyse this. In this way, your portfolio's returns must equal that of a risk-free rate ie t-bills otherwise arbitrage would be possible. With this equality in mind, you can solve the partial differential equation (the Black-Scholes equation, derived via Itô’s lemma) - and solving it gives the famous Black-Scholes formula - from option expiration to finally get today's fair price.

Their 1973 paper "The Pricing of Options and Corporate Liabilities" is widely regarded as having transformed financial markets, as it created the mathematical foundation for modern derivatives pricing, now a trillion-dollar industry. Scholes and Merton received the Nobel Prize in Economics in 1997. Fischer Black unfortunately did not receive the Nobel Prize because he died in 1995, and Nobel Prizes are not awarded posthumously.

Limitations of Black-Scholes

Yet, the Black-Scholes model contains deep flaws. On October 19, 1987, global stock markets crashed, with the Dow Jones falling 22.6% in a single day. Under geometric Brownian motion assumptions, some estimates suggest that such an event should occur approximately once every 10⁵⁰ years—far longer than the age of the universe. Yet it still happened!

The problem was that real markets exhibit fat tails. Extreme events occur far more frequently than normal distributions predict. Markets do not follow smooth random walks; they jump, crash, and behave in ways that most standard models cannot accommodate.

Researchers have developed more sophisticated models—stochastic volatility, jump-diffusion processes, fat-tailed distributions—but none solve this fundamental problem. In essence, even state-of-the-artstate- of the art models fail to capture the complexity of markets and each have their unique weaknesses.

The Unmodelable Variable

It sounds audacious, perhaps absurd knowing that modern finance heavily relies on equations describing pollen in water. The irony runs deeper, where these models were meant to quantify uncertainty, yet overconfidence in them created new risks for funds which relied on these models.

Statistician George Box wrote: "All models are wrong, but some are useful." Although Black-Scholes is empirically wrong about human behaviour, extreme events, and market psychology, it remains useful for pricing options. They are also much simpler than other models out there.

The idea that markets cannot be modelled raises an interesting point: markets, after all, exist to serve our need for the efficient exchange of resources. Yet we fail to consider modelling our own behaviour, arguably more complex than the social mechanism of markets itself.

Ultimately, the equation for market dynamics requires a variable without an equation: us.