What Is Brownian Motion?
By: Farhad Malik
Tags: Finance, Data Science, Fintechexplained, Mathematics, Statistics

What Is Brownian Motion?

Explaining Wiener Process — A Must-Know Forecasting Concept In Quantitative Finance In Laymen Terms

Brownian motion is a must-know concept. They are heavily used in a number of fields such as in modeling stock markets, in physics, biology, chemistry, quantum computing to name a few. Additionally, the process forms the foundation of most simulation models.

Brownian motion is also gaining popularity in data science forecasting projects and forms the underlying foundation of Monte Carlo simulation.

Hence Brownian motion is a topic that all of us should be familiar with.

1. Article Aim

This article aims to provide a basic understanding of standard Brownian Motion. Furthermore, it will explain the basic properties of the concept.

Lastly, I will demonstrate how we can build a Brownian motion using Excel.

I will be aiming to explain the concept in layman terms.

Random paths

2. What Is Brownian Motion?

Before I begin, it is important to understand what Brownian motion is.

Brownian motion is a physical process. Albert Einstein explained the phenomenon in 1905 which was first discovered by Robert Brown in 1827. In a nutshell, it is the random movement of particles in a fluid. This indeterministic movement is due to the collision of the particles with the molecules in the fluid.

Let’s go over it once more.

If we analyse gas (smoke) particles immersed in a liquid under a microscope, we will notice that the particles of gas have a random walk.

They move in an unpredictable manner.

Photo by César Couto on Unsplash

This is due to the collisions of the particles which make the gas particles move in a random or stochastic fashion.

This random motion of a gas particle, in this instance, is known as Brownian motion. It was discovered by an English botanist Robert Brown in 1827. Brown was analysing how pollen of a plant moves in the water.

We often come across the term Wiener process and the Bachelier process in quantitative finance. For the sake of simplicity, we can consider the Wiener process as the mathematical name of the Brownian motion process. These concepts have been used to model randomness in financial markets.

Wiener process is a continuous-time stochastic process

Let’s imagine I flip a dice and start noting the result down. The result is going to be a sequence of random numbers without any pattern. Can we predict its next value? What about the way a stock price moves? It is dependent on a number of factors such as macro to microeconomic factors to news to the potential mood of the dealers in the market amongst many factors. In short, the movement is stochastic and random in nature.

Brownian motion is denoted as W(t) where t refers to the time points.

Photo by Nik Shuliahin on Unsplash

3. What Are The Properties Of Brownian Motion?

Brownian motion is not a concept that can be explained by using a mathematical expression (formula).

Brownian motion is defined by the properties and how it acts.

Let’s refer to the Brownian motion over time t as W(t). Therefore W(1) is the value of the variable at time 1.

y axis is time

This is the realisation of a Brownian motion path. It is random. We cannot absolutely pin down the next value of the stock. I will explain the properties of Brownian motion now.

3.1 W(0) = 0

It means that at time 0, the value of the Brownian motion path is 0.

3.2 W(t) ~ N(0, t)

This property might seem complicated at first but this is the key property to understand.

All it is indicating is that the underlying probability distribution of the Brownian motion over time t is normally distributed with mean 0 and variance t.

Mean is an extremely important property of distribution because it tells you where the center of the distribution is.

This is also known as the expected value of Brownian motion.

A note on N(0, t):

N(mean, variance): N indicates that the process is normally distributed. The first parameter is the mean and the second parameter is the variance. Therefore N(0,t) means Brownian motion over time t is normally distributed with mean 0 and variance t.

But First, What Does Probability Distribution Mean?

If we want to predict a variable accurately then the first task we need to perform is to understand the underlying behaviour of our target variable.

What we need to do first is to determine the possible outcomes of the target variable and if the underlying outcomes are discrete (distinct values) or continuous (infinite values). For the sake of simplicity, if we are estimating the behaviour of a dice then the first step is to know that it can take any value from 1 to 6 (discrete).

Then the next step would be to start assigning probabilities to the events (values). Consequently, if a value cannot occur then it is assigned a probability of 0%.

The higher the probability, the more likely it is for the event to occur.

As an instance, we can start repeating an experiment for a large number of times and start noting the values we retrieve for the variable.

Now what we can do is to group the values into categories/buckets. And for each bucket, we can start recording the number of times the variable had the value of the bucket. For example, let’s assume we throw a dice 10000 times. As there are 6 possible values that a dice can take, we can create 6 buckets to represent the six possible values. And after a dice is thrown each time, we can start recording the number of occurrences for each value and start assigning it to a bucket.

We can plot the chart and it will form a curve. This curve is known as a probability distribution curve and the likelihood of the target variable getting a value is the probability distribution of the variable.

Once we understand how the values are distributed then we can start estimating the probabilities of the events, even by the means of using formulas (known as probability distribution functions). As a result, we can start understanding its behaviour better. The probability distribution is dependent on the moments of the sample such as mean, standard deviation, skewness, and kurtosis.

If you add all of the probabilities then it will sum up to 100%.

There are a large number of probability distributions and the most widely used probability distribution is known as “normal distribution”.

Let’s Now Move Onto Normal Probability Distribution

If we plot the probability distribution and it forms a bell-shaped curve and the mean, mode and median of the sample are equal then the variable has a normal distribution.

This is an example of a normal distribution bell-shaped curve:

It is important to understand and estimate the probability distribution of your target variable.

Going back to Brownian motion, every increment s of Brownian motion is also normally distributed.

Therefore,

W(t) — W(s) ~ N(0, t — s)

Here, t and s are two different time points, where t > s

This is a very important property.

Let’s assume that we gather values of a Brownian motion over different times. The most important point to note is that the distribution of the values at each time point is going to be normally distributed.

Therefore, if I take all of the values at time point 1 and plot a distribution then it will follow a bell-shaped curve.

The property states that at time t = 1, the expected value of all of the values is 0. Therefore, if we increment the time from 0 to 1 then the expected value at time 1 will also be 0.

It also implies that the expected value of browian motion at any time is going to be 0. It is a constant and does not change.

The variance of the Brownian motion is going to be the time t minus time s.

Plus the probability distribution of time t = 1 is normally distributed. Hence, the mean stays the same and the variance changes. Consequently, the variance is proportional to the time t

Variance (W(t) — W(s) = t — s

3.3 W(t) Has Stationary and Independent Increments

The first point to note is that we are talking about the increments of time by a constant.

The increments are stationary. It means that when I displace the Brownian motion, from time t to t + constant and from the time s to time s + constant then the probability distribution of both of the Brownian motions will be N(0, t1-t0). In short, the unconditional joint probability distribution does not change.

The increments are independent. It means that the increments are not dependent on each other. The underlying probability distribution will always exhibit the same properties regardless of where the increments are and the two events will not impact each other.

In short, at each increment, the expected value is going to be 0 and the variance is going to be the difference between time.

Therefore, I can take two Brownian motions from W(1) to W(2) and W(2) to W(3), and I will expect that the increments will always be stationary and independent.

Therefore, regardless of where you look at the Brownian motion, the underlying probability distribution is going to be normal with mean 0 and variance as the difference in time.

3.4 Covariance (W(t), W(s)) = Minimum Of t and s

This is a very important property. What it is indicating is that if we were to pick two values W(t) and W(s) where t and s are the two time points and generate the covariance of the two Brownian motions then the covariance will be the minimum of time t and s.

Covariance is a statistical property. Essentially, it indicates how two variables co-move with each other; whether they increase or decrease together or whether they are completely independent of each other.

3.5 Brownian motion is a Markov Process

All it means is that the present and future behaviour of a Brownian motion does not depend on its past. Hence, how a stock price behaves in the past does not dictate how it will behave in today or in the future.

Past does not matter

3.6 Brownian Motion is Martingale

It indicates that past knowledge is not required to predict the future. More specifically, the expected value is not going to change and the expected value in a future time point is going to be the same as the existed value today.

Essentially, it states that if we measure a Brownian motion walk through a timepoint a to a timepoint u then 1. the process will be stochastic in nature and 2. at any timepoint t (some timepoint between a and u) , the expected value of the next timepoint u given all previous values, is going to be equal to the current expected value of timepoint t.

3.7 Brownian motion is continuous. It is not differentiable.

Brownian motion is infinitely irregular to an extent that it cannot be differentiated. It means that the Brownian motion does not have any jumps. Therefore there is no discontinuity in the process.

The Brownian motion is not differentiable because its walk is absolutely random.

3.8 Brownian motion is fractal

Brownian motion is fractal implying that the shape is infinitely irregular and this is the reason why it is referred to as fractal. In short, we cannot find the gradient at any point due to its irregular shape.

4. Let’s Generate Brownian Motion In Excel

To further explain the concept, let’s simulate Brownian motion in excel. We can do it by using the inverse comulative distirbution of standard normal distribution.

Enter Headers:

Enter Time in the cell A1 and Brownian Motion Walk 1 in cell B1

Drag cell B1 to cell B11. This will generate 10 headings:

2. Generate time points

Enter 1 in cell A2. Enter =1+A2 in cell A3.

Drag the cell A3 to A31. This will generate 30 time points.

3. Generate Brownian Motion first walk

Enter 0 in cell B2. This is our first property that W(0) = 0

Enter =B2+NORMSINV(RAND()) in cell B3. This will generate a random variable using inverse comulative distirbution of standard normal distribution.

Drag the cell B2 to B31. This will generate random numbers for each of the time. This is the first random walk.

4. Generate 10 random walks

Drag the cell B2 to K2. Then the cell B3 to K3. Lastly, drag to K31.

This will generate 10 random walks for time 1 to time step 30.

5. Plot the walks

Select cells A1 to cell K31. Click on Insert-> Scatter -> XY plot

4. Summary

This article provided a basic understanding of standard Brownian Motion. Furthermore, it explained the basic properties of the concept. Lastly, it explained how we can generate Brownian motion in Excel.

I hope it helps.