Brownian Motion, Random Walks, and the Hot Hands Fallacy

  • Bachelier first surmised that Brownian Motion could be applied to asset prices
  • Paul Samuelson found his work many years later, and wrote his theory on random price fluctuations
  • Malkiel applied a random walk to markets, assuming that past trends cannot predict future movements
  • Humans can find patterns in anything, even if they do not exist

All of the below is purely research and should not be interpreted as speculation or opinion. This does not represent my views.

The determination of these fluctuations is subject to an infinite number of factors: it is therefore impossible to expect a mathematically exact forecast. Contradictory opinions in regard to these fluctuations are so divided that at the same instant buyers believe the market is rising and sellers that it is falling.

Source: Louis Bachelier, The Theory of Speculation

What is Brownian Motion?

Brownian Motion is the movement of small particles suspended in liquid or gas.These particles collide with one another, and upon impact, move in a random, zig-zaggy fashion. This is the central thesis of randomness and chaos.

Source: Byju

Because Brownian Motion occurs on the microscopic scale, probabilistic models are used to describe it. In this piece, we will address Brownian motion through the application of a stochastic process, which is often used to describe variables that move randomly, like the particles above.

Bachelier: Predictable Prices are Exploitable Prices

You might know of Brownian Motion from Einstein in his Annus Mirabilis papers. Or, you might know of it from Louis Bachelier, a French graduate student who published his work 5 years before Einstein, proposing that Brownian motion could be a tool in predicting price changes in the stock market.

Bachelier theorized that as soon as a price begins to become predictable, it becomes immediately exploitable. Under the belief that markets are efficient, predicted price patterns will immediately disappear, as market competition brings the price back to equilibrium. As described by Bachelier’s thesis defense committee:

The buyer believes in a probable rise, without which he would not buy, but if he buys, there is someone who sells to him and this seller believes a fall to be probable. From this it follows that the market taken as a whole considers the mathematical expectation of all transactions and all combinations of transactions to be null.

Paul Samuelson found Bachelier’s  Théorie de la spéculation almost 56 years later, and subsequently published his work on random price fluctuations, titled Proof That Properly Anticipated Prices Fluctuate Randomly. This is the premise for the Efficient Markets Hypothesis, expanded by Eugene Fama in his 1970 review, Efficient Capital Markets: A Review of Theory and Empirical Work.

The Efficient Market Hypothesis: All Price Movement is Random

The Efficient Market Hypothesis states “asset prices fully reflect all available information in the market.” Thus, when markets are well-informed and competitive, all price movement must be random, because all prices already reflect all available information.

The change in the prices of financial instruments relies on a myriad of economic variables, including geopolitical factors, nature, and macroeconomic metrics. Those that are known are already accounted for, and those that are not known, cannot be accounted for, according to EMH. Therefore, according to this theory, any forecasting methods are flawed since they can never predict the price movement.

Source: Research Gate

Thus, markets reflect “anticipated or forecastable changes” and leaves “unanticipated or unforecastable changes open to speculation, changes which must be assumed to behave randomly.” Expected returns therefore result in a martingale, which is a sequence of random variables (prices) with an expected future value is equal to the present value. The FV = PV. 

A Random Walk

EMH is based on the assumption that stock markets are random. Specifically:

After speculators have incorporated all available knowledge into their trades, one expects that the result will be prices showing unpredictable fluctuations, independent of their past history.

Markets are popularly described by the random walk hypothesis, and Brownian motion is the limiting case of a random walk (described indepth here). Burton Malkiel popularized the idea of a random walk in his aptly named book “A Random Walk Down Wall Street”.

Shown below is a random walk, prices colliding into each other, constantly moving, creating the market. The market is a stochastic process, comprised of a succession of random steps. Stock prices often follow a wavy path, fluctuating over the short-term, as shown below.

Source: Towards Data Science

The random walk assumes that past trends cannot be used to predict future movements. This ties into the EMH because the assumption is that prices cannot be predicted, as any movement is driven by unforeseen events.

Random walk brushes off both technical and fundamental analysis, stating “the theory of random walks in stock-market prices presents important challenges to both the chartist and the proponent of fundamental analysis”. According to this theory, predicting the market is futile, according to Malkiel and others.

Applications of Randomness

We can think of a random walk as something that a child would draw with a crayon. She puts her crayon on the paper, and scribbles all over the page. There is no rhyme or reason to her movement, and presumably, she doesn’t base the movement of the crayon on what she just drew. She just draws lines.

The chart looks like something that a child would draw, when asked to draw a line. It goes up and down, with many peaks and valleys. When you look at the USD-Euro exchange rate below you see that same movement.

Source: Robert Nau

Taking the first difference of the time series, or the daily changes, would be a better depiction of the price pattern. This allows us to predict the change that comes from the model, rather than trying to interpret the level of the series. This model works when movements are equally as likely to go up or down, a 50/50 shot in either direction.

Let’s look at the first difference of the above chart (T2 – T1) or the daily changes:

Source: Robert Nau

How can we tell that these changes are random? That involves diving into their autocorrelations, which is the “correlation between the variable and itself lagged by k periods.” If the values are truly random, these autocorrelations should be equal to ~0.

If the values are random, they shouldn’t have any correlation at all. The red bands below are the 0.05 significance level, testing whether the autocorrelations are statistically significant from zero. The data stay within the bands, supporting ~0 autocorrelation.

Source: Robert Nau

This is the random walk without drift model, assuming that each step that the market took has a mean value equal to zero. If there were drift in the model, that would mean that the step size was equal to a nonzero value.

With a random-walk-with-drift model, instead of the child drawing with the crayon tip, she would be drawing with the broken crayon that favors the right or left. However, we are assuming that our kid has a fresh crayon, straight out the box, and that we are operating with a random-walk-without-drift model.

Below is that child drawing with her fresh crayon. This the random-walk-without-drift model applied to the USD-Euro exchange rate. The forecasts follow the same path as the data, lagging behind by one period. The long-term path is a straight line because it is assumed that there is no drift in the model (new crayon) rather than either upward or downward drift (broken crayon).

Source : Robert Nau

As you can see, the confidence bands demarcated in red grow in the form of a sideways parabola. This is because the standard error of the forecast (k-step-ahead forecast) is larger than a one-step ahead forecast by the square root of k. Random walk models incorporate the square root of time into their errors. Parabolas are created by taking the square root of a variable. Combining the two creates the sideways parabola on our graph.

The random walk model is sometimes called the naïve model. Arguing that things will continue as they will always have doesn’t seem very complex. But the model has important applications to financial theory, including the Black-Scholes Model.

Conclusion: Humans Are Very Good at Finding Patterns, Even if They Are Not There

All of us have watched basketball, and witnessed the moment that a player keeps hooping. Basket after basket, the player is unstoppable. We assume that this streak is a pattern.

But really, it’s a string of random occurrences trending upwards. There is no pattern to the player making basket after basket, rather, it’s all by chance. Humans are pattern-seeking animals. Apophenia is the scientific name for this phenomenon, and it shows up in the gambler’s fallacy, Type I errors in statistics, and the previously described “hot hands” fallacy seen in sports.

We tend to bucket items into recognizable features. It’s an evolutionary characteristic that enabled us to survive. However, it does not always mean that those patterns exist.

Source: NBER

These authors state it simply. “If a player makes a field goal, he is less likely to make his next field goal attempt.” However, even that could be argued against. Perhaps that it is completely random. We look for patterns in everything. There are many variables that go into making field goals and free throws, but a common variable is psychology.

We can all fall prey to the “Hot Hands” fallacy, for both better and worse.

The better: “I am untouchable! Watch out, world.”

And the worse: “I will never keep having such great luck! Something is about to go wrong.”

Both should be weighted equally, and perhaps that’s where the horizontal line in the random walk forecast really comes from – a net zero effect, taking into account all items in a level and even fashion.

Circling back to Bachelier, Brownian Motion, and all of the above, Paul Samuelson wrote the below in his 1972 book:

“The theorem is so general that I must confess to having oscillated over the years in my own mind between regarding it as trivially obvious and regarding it as remarkably sweeping. Such perhaps is characteristic of basic results.

Source: Paul Samuelson

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