Which financial time series have a PDF and/or CDF?

Question

Consider the following types of financial time series for a single publicly-listed stock:

1. Price data
2. Log returns
3. Cumulative returns

Each is computed from the item listed before it: log returns are based on differences of prices, and cumulative returns are cumulative products of log returns.

• Which of the random variables listed above possess a probability distribution function (PDF),
• which have a cumulative distribution function (CDF), and
• which have both a PDF and CDF?
• for what sort of financial applications is the CDF preferred over the PDF, and vice versa?

I ask because the following post says all random variables have a CDF, but not all of them have a PDF. So I wanted to see how this applies to commonly used financial data, which are prices and returns. Graphical depictions of the above datas' CDF and PDFs displayed side-by-side would help in the explanation.

I'm particularly curious about cumulative returns. Since they're cumulative, it automatically makes me think it corresponds and is represented best by a CDF, so in a way I'm wondering if cumulative returns are more useful than they're made out to be, despite being non-stationary.

Answer 1

For a continuous variable the PDF is the derivative of CDF. So returns or prices don't have a pdf if the cdf is not differentiable, e.g. it "jumps" at some point. The simplest models we use, like normally distributed log-returns, imply that returns, cumulative returns and prices all have a pdf.