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Stock returns, computed from stock prices as $r_t = \ln (p_{t}) - \ln (p_{t-1})$, are real-valued and unbounded giving the impression that they are continuous random variables. But aren't they actually discrete random variables given that:

  1. financial time series are finite in the number of historical observations they possess, and
  2. they do fluctuate within a feasible range of real values (percentage up and down ticks) known (inferred) beforehand from the source price data? (i.e. a real value of 5.1 would not appear as an observation in a daily-frequency time series because that would mean the stock jumped 610% in one day)

If so, does that mean they have probability mass functions (pmf) and not probability density functions (pdf)?

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  • $\begingroup$ i think there are alot of finance textbooks that say returns are continuous random variables. even prices i find hard to think of as discrete because values in dollars and cents are in many way also uncountable. someone else had an answer explaining all this but for some reason it's not here anymore. they especially addressed the two arguments of finite and feasible range not being requirements of a pmf $\endgroup$
    – develarist
    Commented Aug 20, 2020 at 6:20
  • $\begingroup$ Well, these textbooks are either wrong or they do not claim that. They might claim we may model returns as if being continuous, but that does not make the returns continuous in reality. The simple technical reasons are provided in my answer. The other answer you mentioned has been deleted because it was wrong, and the answerer realized that once we discussed with him in the comments. $\endgroup$
    – Richard Hardy
    Commented Aug 20, 2020 at 7:26
  • $\begingroup$ Mathematically, values in dollars and cents are countable because natural numbers are a countable set. This is a simple fact from which the rest of the argumentation is derived (using also the fact that rational numbers are a countable set, too, the proof of which goes back to the countability of natural numbers). $\endgroup$
    – Richard Hardy
    Commented Aug 23, 2020 at 6:43
  • $\begingroup$ Applies to prices but returns arent in dollara and cents. Do returns belong to a countable set $\endgroup$
    – develarist
    Commented Aug 23, 2020 at 7:15
  • $\begingroup$ Log-returns are a simple transformation of prices, and the transformation does not make them other than countable. Here is a more detailed explanation. Log-prices have the same number of possible values as prices do, as the transformation is 1 to 1. Log-returns are pairwise differences of log-prices and are countable by the same argument as why rational numbers (which are pairwise ratios of natural or whole numbers) are countable. Simple returns are analogous to log-returns in this respect. This is basic mathematics and thus easy to verify. $\endgroup$
    – Richard Hardy
    Commented Aug 23, 2020 at 7:50

2 Answers 2

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Log-returns of stock prices are discrete phenomena, and they can be modelled as discrete random variables with a probability mass function – though not for the two reasons you have listed.

Log-returns are discrete because stock prices are discrete, and log returns are differences of logs of consecutive prices, as your formula shows. Stock prices are discrete because they only take values in dollars and cents (whole numbers thereof), and they have at most a countable number of possible values. (I think there are some technical rules in stock exchanges preventing extreme movements; trading in a share is sometimes stopped if it fluctuates too wildly. If so, stock prices have a fixed, finite range within which they can move in a day. This would make the set of their possible values not only countable but also finite.)

Even though it may be natural to model log-returns as discrete random variables, it is often convenient to approximate them by continuous random variables.

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  • $\begingroup$ I understand that since stock quotes are almost always priced to the cent, these prices are countable and therefore can have a pmf. Likewise, we know that buying 3 shares for \$10 implies a price that is exactly 10/3, which is also rational. So too does buying 0.127 shares for \$10 imply that the price is a certain rational number. And rational numbers are also countable, so we can assign pmfs to all prices which are rational. But I'm having trouble parsing the $*$ text. It seems that you're saying returns are a pmf on $\mathbb{R}\setminus\mathbb{N}$? But prices can be whole dollars? $\endgroup$
    – Sycorax ♦
    Commented Aug 12, 2020 at 16:31
  • $\begingroup$ I agree with the arguments you make overall, I'm just confused by the $*$ text is all. $\endgroup$
    – Sycorax ♦
    Commented Aug 12, 2020 at 16:38
  • $\begingroup$ @Sycorax, all I was trying to say is that if prices were rational numbers (which fractions are), my argument would still hold. But I guess I can just remove the explanation if it is more confusing than helpful. $\endgroup$
    – Richard Hardy
    Commented Aug 12, 2020 at 17:07
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    $\begingroup$ On a historical note, some exchanges (e.g. US exchanges until 2001) used to quote prices in dollars and 1/16ths of dollars rather than in "decimal" terms (i.e. dollars and cents), so they effectively had fractional cents. It doesn't change the fact that discrete distributions are not usually used to model stock prices even though they are theoretically discrete, as you say. $\endgroup$
    – Chris Haug
    Commented Aug 12, 2020 at 23:57
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    $\begingroup$ @ChrisHaug, funnily enough, I knew that but I did not know the English word for 1/16, so I skipped this historical note. But thank you for sharing it! $\endgroup$
    – Richard Hardy
    Commented Aug 13, 2020 at 4:57
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If so, does that mean they have probability mass functions (pmf) and not probability density functions (pdf)?

In theory yes, in practice the mathematics become way too complicated and the results will only be theoretically better. Since finance is about money, and "time is money", it wouldn't make much sense to throw too much money into a theoretically better solution.

This is the argument I've heard both from statisticians and finance people. I'ts the underlying principle for continuous-time finance.

That being said, your point makes more sense for some asset classes where the range of discrete values is not as continuous as for stock returns, e.g. real state.

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  • $\begingroup$ so you're saying stock returns are in fact continuous variables $\endgroup$
    – develarist
    Commented Sep 29, 2020 at 19:22
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    $\begingroup$ You seem to conflate continuous time with continuity of the variables in a stochastic process. $\endgroup$
    – whuber ♦
    Commented Sep 30, 2020 at 14:29
  • $\begingroup$ Can't time be considered a variable ? $\endgroup$
    – Trusky
    Commented Oct 9, 2020 at 21:26

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