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What is the central limit theorem?

In finance, it’s often useful to estimate the characteristics of a larger entity by taking a representative sample of its constituent parts. If you wanted to predict the performance of an asset class or an index, for instance, you don’t have time to analyse every single stock individually. If, however, you take many representative samples from the statistical population you’re interested in, you can get a pretty good idea of the larger entity’s overall performance. This approach is often used by investors to assess risk and volatility, analyse returns or construct a portfolio. The reason it works so well is the ‘central limit theorem’ (CLT), a core tenet of probability theory. The central limit theorem is the principle that, within a population that has a finite level of variance, and with a large enough sample size, the distribution of sample means from that population will form a normal distribution. From this distribution, you can approximate the mean and standard deviation of the overall population.

How does the central limit theorem work?

Say, for example, you want to find out the mean height of everyone in your town. A valid method would be the following: take a sample of ten people’s heights (this is your sample), find the mean of these heights, and plot it on a graph, where the x-axis represents height and the y-axis represents frequency. Then repeat this sample over and over and plot all the means on your graph. You will soon see that the means form a bell-curve shape or a ‘normal distribution’. Now, draw a line down the centre of this normal distribution. According to CLT, this line represents the mean of the overall statistical population. It can therefore be assumed to be a close approximation of the mean height of everyone in your town.

The central limit theorem states that the average of all the sample means can be assumed to be approximately equal to the mean of the population from which you drew the samples if you have a large enough sample size. Furthermore, it goes on to say that the larger the sample size and the greater the number of samples, the more closely the distribution of sample means will resemble a normal distribution. This makes it a powerful tool for predicting the likelihood of a given outcome within a large statistical population. There are a few conditions required for CLT to be effective: the sample size must be large enough to make the approximation valid (a minimum sample size of 30–50 is generally considered sufficient); the sample size must be identical for every sample; and the population must have a finite level of variance.

When is the central limit theorem relevant in options trading?

Using the same principle we saw in the height-measuring example, another way you might use CLT is if you were looking to find the mean return for a given index. Although it would be very time-consuming to gather data for the entire population (i.e., all the stocks in the index), you could find a good approximation of the mean returns by working with CLT. In this example, you would take a sample of thirty random stocks from the target index and find the mean return, then plot this on a graph. You would then repeat this for another random sample of thirty stocks, and another and another. Take care to remove previously selected stocks as this could lead to bias. If you plotted all the mean returns on a graph, they would form a normal distribution, giving you a good indication of the mean returns for the totality of the index.

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