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Brownian Motion and the Random Walk

Updated: Nov 20

Disclaimer:

Please note that the opinions expressed by the author in this article do not constitute financial advice and are solely for educational purposes only. When buying shares, the value of your investment may go down as well as up and you may get back less than you invest.


In this article I will provide an informal description of Brownian motion. I will discuss how random walk theory was developed, whether markets are efficient, and touch on Brownian motion’s importance to mathematical finance.


What is Brownian Motion?

Brownian motion was first observed by Robert Brown, a Scottish botanist in 1827. Using a microscope, he observed the erratic motion of pollen grains suspended in water. One can only imagine the excitement that Brown must have felt. What could be the source of this movement? Was this divine or did some physiological explanation exist? Brown could prove neither. Subsequent scientists touched at the correct theory, but it was not until 1900 when Bachelier published Theorie de la Speculation that a mathematical explanation for Brown’s observation was first formalised. It was not until 1905 when Einstein (and Smoluchowski independently from each other) described the phenomenon in terms of thermodynamics that a physical interpretation of the process was first unearthed. These men found an explanation for the seemingly inexplicable observation that Brown made over seventy years prior.


The answer to the mystery: pollen grains were being bombarded by microscopic water molecules. The explanation was ground-breaking, but the mathematics that would be developed was even more powerful.


Can we Predict the Future?

Have you ever looked at a price chart for a given asset and wondered; what caused the asset to move with that specific trajectory? You may be able to identify various macroeconomic events that could easily be ascribed to have had a causal effect on the asset’s price. The Port of Shanghai closed, a worse than anticipated quarterly earnings report was announced, insolvency, credit crunch. But the nature of causation is a deeply philosophical problem. Can we ever isolate an event from the noise? And even if we could, how would we be able to say definitively that there was some causal connection between an isolated event and the movement of an asset’s price on a chart? Ever since David Hume meditated on the nature of causation in his A Treatise of Human Nature philosophers have tried to resolve problem.


But we’re interested in making informed investment decisions and, instinctively, we know that an asset’s path (or walk) is determined by the market. The vivaciousness of buyers and sellers, coming together, and their participation in the market generates the demand and, in turn, the price.


But are Price Movements Chaotic?

Returning to our discussion on Brownian motion (BM), it’s important to stress that at the heart of BM is the notion of chaos. It is the idea that we cannot know the trajectory of the pollen grains because the number of water molecules and the seemingly infinite permutations of the ways they can interact with each other make the process unforecastable. Hence, BM gave rise to our first formalisation of a stochastic (Greek for random) process. The seemingly random movement of pollen grains germinated a cascade of intellectual intrigue and endeavour, as it provided a framework for academics to answer a question that had been articulated decades before.


Stock exchanges have existed for centuries. The oldest being the Amsterdam Stock Exchange (1602). The New York Stock Exchange (NYSE) was founded in 1792 along with the London Stock Exchange (LSE) in 1801 (to name just a few). But during the 19th Century the scale and ubiquity of exchanges exploded. With this growth people began to wonder if stock prices moved with some underlying trend, or if price movements were simply random. Data sets were observed and analysis performed. But it was not until Louis Bachelier presented his PhD thesis in 1900, that the notion of a random walk took hold in the financial community. He advocated the idea that stock price movements were random, and they follow a random walk.


The random walk states that at each time interval the price movement is random, meaning that with each tick-tock of the stock market clock the probability that the price moves up or down is unknown. This notion might seem beautiful or ghastly depending on your views of financial markets.


The attractiveness of the theory is that it is organic. It is a simple description of an vastly complex, dynamic system, which we can incorporate into models. But is this how the world actually works? Well, despite the seeming elegance (and the simultaneous insanity) of the notion that price movements are random, an academic debate has raged for the past 60 years and no consensus has been reached.


Academics Against the Practitioners

There exists a contradiction between the belief held by professional stock market analysts on the one hand and academic statisticians and economists on the other.


Analysts hold the belief that there exist underlying trends that can be obtained through a thorough analysis of all available information. By analysing all available information, they can then make informed investment decisions. For them, trends exist; it is simply a matter of reading them correctly. Their grand understanding of the market is that most investors have imperfect knowledge of the available facts (earnings reports, the broader economic outlook, exchange liquidity) and the trend only develops once this information disperses through the market. Being able to acquire knowledge early creates an opportunity to profit because as people adjust their assessment of a particular asset, they will buy and sell accordingly and the price will also adjust.


There are fundamental and technical analysis. Both agree on the assumption that trends exist and one can profit on them. By analysing the company, sector, and broader economy the fundamental investor hopes to understand the factors that lie behind price changes. They might hope to generate an assessment of future balance of supply and demand, general business conditions, or profit prospects. Technical analysis on the other hand focuses on charts to guide their investment decisions. They believe that a discernible trend exists now and will carry through into the future (it will continue until it doesn’t). The facts that exists now will govern the price of the asset until sometime in the future.


Investors’ delayed responses to facts generates the trend. Once the facts are disseminated and become widely known, investors will speculate, and the price of the asset will adjust accordingly. A trend that an analyst might identify could be the fact that if a company has had a successful year and their annual report shows increased earnings, then this increase will indicate a higher probability that the company’s earning will be greater the following year. A trend is the existence of a probability of a future price change being dependent on the present price change. In essence, the past informs the future.


This might seem intuitive, but for a good portion of the last century and to many academics to this day the opposite is true. To them, the price of a stock day-to-day, or month-to-month is a random process that is determined in a fashion that is no different to flipping a coin.

To them, trends are simply interpretations, that are made retrospectively by people looking to justify their investment decisions.


Back to the Random Walk

The notion of a random walk was coined by Karl Pearson in 1905. It follows from the work the Bachilier’s stochastic analysis of Brownian Motion. The random walk was subsequently popularised by Burton Malkiel in his 1973 book A Random Walk Down Wall Street. The random walk is the discrete analogue of Brownian motion. Note, that the difference between discrete and continuous is not particularly important for our discussion.


Let’s describe the random walk in more detail. Imagine that at each time [0,t] (from zero to a time, t) we flip a coin, and at each time we count the number of heads as +1 and the number of tails as -1. When we consider an infinite number of such flips (i.e. taking the limit), we would expect to arrive back where we started (at zero). This process is an example of a Martingale, meaning that the future expectation is equal to the current value. It’s also an example of a Markov process, meaning that the process has no memory. The conditional probability is only dependent on the current state and not the whole history of the process.



The Random Walk and Finance

In the case of the stock market we may view price movements as random processes, and in doing so we may come to find that asset prices follow a random walk. This would mean that the future price only depends on the current state and not any underlying trend we might interpret into the price chart. It may also be the case that we expect the future price to be equal to the current price. This is because there is no more information available to incorporate into the price of the asset.


In essence we have no way of developing a strategy to “beat the market” consistently on a risk-adjusted basis since market prices already reflect all available information (including expectations of asset’s future value) and any new information will be incorporated instantaneously. This observation leads to another important notion in finance, the Efficient Market Hypothesis.


Efficient Market Hypothesis

The Efficient Market Hypothesis (EMH) is interconnected with the notion of Brownian Motion and the Random Walk Hypothesis (RWH). Although it would be a mistake to use them interchangeable since they are only synonymous under very specific circumstances. The EMH states that in an informationally efficient market, price changes must be unforecastable. Put simply, prices fully reflect all available information.


The random walk is produced by investors buying and selling, each actively seeking greater wealth. As soon as informational advantages are available, investors act on that information and buy or sell that particular asset. Thus, the information becomes incorporated into the market price and profit opportunities are quickly eliminated.


Intuitively, this sounds compelling. However, if markets were informationally efficient then clearly the return on investment for gathering information would be zero. There would be no point in big investment banks, spending millions on research when they could be flipping a coin.


Profit opportunities (inefficiencies) must exist to compensate investors for the cost of trading and information-gathering

These inefficiencies may arise because traders trade on

1. Noise – the wrong information – delayed information

2. Unexpected liquidity – investors may acquire capital they wish to speculate

3. Investors are happy to “pay up” – pay for the privilege of executing trades immediately rather than wait for a more appropriate buying opportunity


An old economics joke highlights the seeming insanity of the EMH. An economist and their friend are taking a walk through the park when the friend spots a £100 note on the path. Just as he’s about to reach down and pick it up, the economist says “don’t bother – if it were a real £100 note someone would have already picked it up”.


Intuitively you may feel strongly one way or the other with regard to the EMH. However, economics have not come to a consensus. This is primarily because it is not a well-defined and thus refutable statement. Nevertheless, they struggle away; analysing price data; performing variance-ratio test, simple volatility-based specification test, and other statistical analysis tests to try and get some semblance on the reality of price movements.


To Conclude

Unfortunately, I have no thesis, no conclusion to present you with. However, I will say that the notion of a random walk, and its continuous analogue BM have been instrumental in pioneering new areas of finance. Brownian motion allowed for the development of Ito Calculus and the refinement of the Black Scholes (Merton) Equation for option pricing, which spawned a new frontier of finance.