A simple explanation of Ergodicity in finance: Part I

Recently, the world of finance has become increasingly interested in an esoteric property of random processes. Proponents claim that re-examining economics through the lens of ergodicity can fundamentally change our understanding of the domain. Over the past century, mathematical methods and statistical analysis have become completely prevalent in the world of finance. But have we missed something at the foundational level?

I wrote this blog for 2 reasons. First, to give a novice (like me) an intuitive understanding of ergodicity. Secondly, to shed light on how this property can redefine finance.

An intuitive description of ergodicity

In simple terms, an ergodic process is a random (stochastic) process that has the same ensemble and time average. Let’s break this down.

Take a random process. To find its expected value, you would typically take an average of many identical processes over a certain period. This is the average most people are familiar with. On the other hand, the time average considers the average value of one process over time.

The figure below depicts this well,

Let’s look at a couple of examples to help make this point more clear,

Example 1: Imagine a game of Russian roulette. Say there’s one bullet loaded in the gun, and if you choose to play the game you win $1m. How do we evaluate this gamble?

Say that you run an experiment where 6 people play this identical game, but only once each. If I ran the experiment a large number of times, what I would find is that 83% of those who played become millionaires, and 17% received a one-way ticket upstairs.

Then you run another experiment, where the same person plays the game 6 times. Suddenly the game looks a lot less enticing.

Example 2: Here’s a more serious example. You play a game where you toss a coin, where if the coin is heads you win 50% and if its tails you lose 40%. Say you start off with £100, what is the average outcome of this game?

If we had to find the ensemble average we would say have 50% chance of make £50 and a 40% chance of losing £40.

E[X] = 0.5*50 + 0.5*-40 = +5

Great! You might be tempted to take this positive expected value bet. But’s let have a look how this may play out over time.

Say you play the game 2 times, with one heads and one tail. Whichever outcome happens first, you end up £90. Suddenly this gamble looks less attractive.

What about if you play it 4 times, with 2 heads and 2 tails? You end up with £81. By now it should be clear that this gamble gets progressively worse for the player over time.

We can now see that the behaviour over time of a random process is not the same as behaviour over identical processes at a single point in time.

Below is a more formal definition of an ergodic process, where the LHS represents the time average and the RHS represents the ensemble average.

Ergodicity and finance:

Having read the above, three key questions come to mind,

• Does trading and investing resemble a repeated gamble?
• Are the random processes we observe in finance ergodic?
• And if they aren’t, what are the consequences?

The questions get progressively harder. Let’s have a look at first two questions, and the third question will be addressed in a later blog.

Does trading and investing resemble a repeated gamble?

Yes. Any participant in the financial markets will realise that both trading and investing (if you care to make the distinction) are not one-shot games, but rather repeated gambles with payoffs.

Some may argue, what about someone who just buys and holds an investment? First, almost no one invests their entire savings in one go. The typical buy-and-hold investor, will invest over multiple periods as his savings increase. Secondly, one can easily contend that ‘not selling’ is just as much of a gamble as ‘buying’.

If investing resembles a repeated gamble, our analytical toolkit should be adapted to this instead of one-shot games.

Are the random processes we observe in finance ergodic?

Anyone who’s done quant finance 101 will recognise the below equation,

That equation says that the change in price over time is determined by it’s drift rate and it’s variance.

The term of interest for us here is dWt, the term that represents the well-known Wiener process. Is a Wiener process ergodic? No, a random walk is considered to be weakly non-ergodic — defined as a process with a distribution which becomes either infinitely broad or infinitely narrow as time goes by.

Below is a good example of a random walk (amongst other things) that I stole from Ole Peters’ blog,

Most mathematical models of stock returns assume a random walk-like element, and as consequence, they are non-ergodic just by assumption! What do we observe empirically?

All one needs to do is look at a graph of the S&P 500, and you will have your answer if it’s ergodic or not. You can see the clear drift upwards, with random fluctations around it. In fact, upon inspection, you will find that the vast majority of economic variables that we use also display this non-ergodic property!

S&P 500

To conclude, we are reminded of Nassim Taleb’s quote ‘ never cross a river that’s on average 4 feet deep’. In the next part, we examine the consequences of the trading equivalent of crossing seemingly shallow water bodies.

Sources:

This blog borrows heavily from the below blog and lecture notes,

1) Ole Peter’s blog and lecture notes: https://ergodicityeconomics.com/