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.

  • $\begingroup$ They don't have pdf because they have a pmf. $\endgroup$ – confused Jul 23 '20 at 12:37

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.

  • $\begingroup$ You say all 3 have a pdf. And which of them have cdf? Stock returns usually aren't normal log returns, stocks have nonnormal log returns. $\endgroup$ – develarist Jul 20 '20 at 8:58
  • 1
    $\begingroup$ @develarist As you said any random variable has a cdf. I just meant we commonly assume all 3 have a pdf. Most of the common distributions like normal, t, gamma all have a pdf., normality is not important here. But you can find counterexamples mentioned in the previous post like some mixture models for which the pdf does not exist. $\endgroup$ – fesman Jul 20 '20 at 13:06
  • $\begingroup$ @develarist I also don't see why one of these would be more likely to have a pdf than the others. Typically if you assume returns don't have a pdf, then prices won't either. $\endgroup$ – fesman Jul 20 '20 at 13:10
  • $\begingroup$ here is a separate question, quant.stackexchange.com/questions/55797 $\endgroup$ – develarist Jul 20 '20 at 13:24

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