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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 financial applications do the CDF versus PDF, and vice versa, come in handy?

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.

Richard Hardy
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develarist
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  • The link says that if the distribution is discrete, it doesn't have pdf. He didn't say anything special because pdf is defined for the continuous random variables anyway. And prices usually are continuous so I don't see why it wouldn't have pdf. The only concern is if the price is negative then, log return will not be determined well. – hbadger19042 Jul 19 '20 at 12:33
  • should i be asking which of the 3 listed random variables are discrete then? (not sure what discrete means here) and let's assume stock prices can't be negative, since they never are. log returns and cumulative returns, on the other hand, can be negative – develarist Jul 19 '20 at 12:35
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    To think of it, prices are actually discrete because exchange usually has ticks. So it might depend on the time span you consider. You may ask the domain knowledge better in quantitative finance than here. – hbadger19042 Jul 19 '20 at 12:39
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    The "cumulative" in the "C" of "CDF" and in "cumulative returns" are related only through a basic mathematical idea of adding things up in sequence. There's no deeper relationship than that. The question of having a PDF is unanswerable until you specify a probability model: in some cases the model will have a PDF and in others it won't. – whuber Jul 19 '20 at 20:37
  • @develarist, I was going through my old answers and noticed this one has not been accepted. Let me know if you need further clarification. Thanks. – Richard Hardy Sep 29 '20 at 16:41

1 Answers1

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Real world phenomena do not imply probability models in a strict mathematical sense. An analyst can choose a probability model for the phenomenon of interest based on multiple considerations. In particular, declaring financial phenomena such as prices, log-returns and cumulative returns to be random variables is not unambiguous mathematically. Thus the question Which of the random variables listed above... cannot be answered without qualifications.

Stock prices as well as log-returns understood as random phenomena are discrete; see "Do financial return series have a probability mass function (pmf)?". Cumulative returns are cumulative sums of log-returns, and thus they are discrete, too. Discrete phenomena are naturally and most accurately modeled by discrete random variables, though it is not impossible to approximate them with continuous random variables. But if we go with the natural choice, the corresponding discrete random variables will have PMFs and CDFs but not PDFs; PDFs are reserved for continuous random variables.

Handiness of PDFs vs. CDFs in finance merits a separate question.

Richard Hardy
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  • Many careful writers are happy to use the term _probability density function_ for discrete variables too, so in a wide sense, and they do so rigorously; it's just a question of the underlying measure. Peter Whittle's _Probability via Expectation_ is a fine example. – Nick Cox Aug 22 '20 at 14:16
  • @NickCox, thank you, I will keep it in mind. In this answer I just follow the distinction made by the OP and found in much of the less technical litterature. – Richard Hardy Aug 22 '20 at 14:19
  • I can't see that the OP is defining anything clearly. Your point, and mine too, is that the answer does depend in part on what a PDF is (is defined to be). We have to factor in also the term _probability distribution function_ which might mean density function and which might mean (cumulative) distribution function, although in this case I guess the latter is ruled out. – Nick Cox Aug 22 '20 at 14:27
  • @NickCox, I was imprecise; the distinction I referred to was actually made in OPs earlier post which I linked to. But there does not seem to be any misunderstanding here, or does there? – Richard Hardy Aug 22 '20 at 14:32
  • Thanks for underlining the importance of the linked thread. I don't think we misunderstand each other. The OP's line of questioning is often puzzling to me. My main concern is that other readers do realise that there is some variation in terminology, surprising or unfortunate though that may be. – Nick Cox Aug 22 '20 at 14:53