The story of Brownian motion began in 1827 with a Scottish botanist by the name of Robert Brown. Brown was investigating the fertilisation process in a new flowering plant when he noticed a ‘rapid oscillatory motion’ (pictured below) in the particles in pollen grains that were suspended in water. Other researchers had noticed this movement before, but Brown was the first to study it. This was a peculiar observation that Brown originally attributed to a characteristic of male sex cells of plants. However, when noticing that the same characteristic was present in plants that had been dead for over a century, it was obvious that his conclusion was not correct.

Eventually, the overwhelming amount of evidence proving Einstein’s statistical approach to molecular motion forced his theory to be excepted. This theory was then used in many other areas of science such as the behaviour of electromagnetic radiation, which was then adopted into the financial world due to its accurate description of random processes, such as the stock market. This is where Brownian motion’s story in finance begins and is what will be discussed further on this page.

Words or phrases that are underlined in this section will take you to more in-depth definitions of the financial terms.

Two mathematicians, Fischer Black and Myron Scholes, came up with a model to describe the price of assets in the stock market. The equation for this model is similar to the heat equation and is built on Bachelier’s Brownian motion model but includes extra factors relating it to the stock market. Black and Scholes’ model holds financial assumptions, such as no transaction costs, no limits on short-selling (where an investor will buy stock shares at a certain price, sell them, and then buy them back at a lower price) and that it is possible to lend and borrow money on a fixed, risk-free, and known interest rate. This is known as arbitrage pricing theory; it follows the same mathematics as proposed by Bachelier. Its main assumption is that the market statistically behaves like Brownian motion where the drift and rate of volatility are constant. The drift is the movement of the mean price of the stock (related physically to the mean position of a particle) and the rate of volatility is the standard deviation, average distance from the mean, of the price of a stock (also related to the average distance from the mean position of a particle at a certain time). To find out more on the mathematics behind the equation, scroll to the bottom of this page.

The statistical approach of Bachelier’s Brownian motion is not without its limitations. The Bell curve of probability shows the likelihood of a stock being that price at a given time. As can be seen from the bell curve shown next to the Bachelier model section, the likelihood that the stock price goes massively up or down is extremely unlikely. However, this does not align with our experience of the stock market, given the many financial crashes and booms over recent years, something that Bachelier’s model does not predict well. For example, stock in a company called Cisco Systems has undergone ten ‘5-sigma’ events in the last twenty years (sigma being the symbol for standard deviation, the average spread, so a 5-sigma event is when the stock price moves 5 times the average spread away from the mean price). However, Bachelier’s model predicts this price to have undergone 0.003 ‘5-sigma’ events in the same space of time. This distribution, along with the sigma variations is shown below.

This section contains information on the equations that determine the prices of the stock. These are the same equations that are used when determining the value of a price or call option. This section requires previous mathematical knowledge so it may be reasonable to skip this section. It is present for readers that wish to understand the background behind the model even further.