Non-normal sample from a non-normal population (option returns) does the central limit theorem hold?

I'm testing a short call option strategy and found, as expected, non-normal return distributions. It is known that option returns are not normally distributed (i.e., also the population). I take the options from a population of 5 million different options and have in the end about 6000 different options options (i.e., my sample is N>30). I am woundering whether the central limit theorem holds for non-normal populations and non-normal samples? In the end, I would like to perform one-sample one-sided t-tests for the mean.

Thank you

1 Answer

A common misconception of t-test is that the distribution of the variable you are looking at should be normal; however, the assumption actually is that the mean of that distribution should be normally distributed. This is the case for any situation in which the central limit theorem can be applied, and the central limit theorem gives you a normal distribution under the assumption you are averaging iid. random variables, and that you have a defined variance. From what I understand, this should be your real concern, to check if your distribution of option prices is heavy-tailed. If so, and the exponent is below 2, then you don't have a well-defined variance, and the mean will be Levy-stable distributed, so you couldn't use the t-test.

• Damn, I forgot about iid.. My returns actually have negative one-day lag autocorrelation. So not even CLT does hold. May 21, 2021 at 12:47
• Is the first sentence actually true in a theoretical sense (cf this Q&A). I mean practically, people uses t-test regularly for the more well-behaved non-normal distributions without problem, and as you have pointed out the OP should be more concerned about the "well-behavedness". I am concerned the first sentence may mislead future readers. May 21, 2021 at 12:51
• @user322347 you should investigate, there are TLC statements that hold even with a certain autocorrelation May 25, 2021 at 9:19