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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)?

develarist
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  • 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 – develarist Aug 20 '20 at 06:20
  • 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. – Richard Hardy Aug 20 '20 at 07:26
  • 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). – Richard Hardy Aug 23 '20 at 06:43
  • Applies to prices but returns arent in dollara and cents. Do returns belong to a countable set – develarist Aug 23 '20 at 07:15
  • 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. – Richard Hardy Aug 23 '20 at 07:50
  • 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 16:45

2 Answers2

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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.

Richard Hardy
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  • 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? – Sycorax Aug 12 '20 at 16:31
  • I agree with the arguments you make overall, I'm just confused by the $*$ text is all. – Sycorax Aug 12 '20 at 16:38
  • @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. – Richard Hardy Aug 12 '20 at 17:07
  • Ok, then I think we're on the same page. We both agree that prices can take on values in the rational numbers, rationals are countable, and countable sets have pmfs. And we know that $\mathbb{Q}\cup\mathbb{Q^c}$ gives us $\mathbb{R}$, so it seems that the main thing that we need to show is that prices **can't** be irrational, which I suppose amounts to the observation that no broker tracks fractions of shares smaller than a certain positive number; that is, share's aren't tracked to infinite precision. Likewise for dollars. – Sycorax Aug 12 '20 at 17:19
  • @Sycorax, I think prices are natural numbers, because the stock exchange would not allow quoting a price that is not a whole number of dollars and cents. That makes it even simpler for us. Period. Now purely hypothetically, if decimals (or some other fractions) of cents were allowed in addition to whole number, we would have rational numbers. That is not difficult to imagine. But I find it very hard to imagine how one could obtain a price as an irrational number. For that, we need something more complicated than a fraction (which is a ratio of two whole numbers). – Richard Hardy Aug 12 '20 at 17:24
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    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. – Chris Haug Aug 12 '20 at 23:57
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    @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! – Richard Hardy Aug 13 '20 at 04:57
  • The fact that a variable is recorded as a multiple of some small, indivisible value like \$0.01 does not require us to model that variable as discrete, nor does it even mean we should conceptualize it as discrete. It's certainly not a count! It is important and useful to distinguish the objects we use in a model from their physical manifestations as observations. In particular, it's rarely helpful to model or even think of log returns as "discrete phenomena" except for the purpose of learning about asset pricing (*.e.g.*, the binomial asset pricing model). – whuber Sep 30 '20 at 14:28
  • @whuber, I guess you have downvoted my answer, and I would like to understand why. My answer and your comment agree entirely that the underlying physical phenomenon and the model can differ in some aspects such as one being discrete while the other being continuous. Yet you seem to maintain stock prices and returns are continuous. Are you defining them as some underlying processes that may either have different values from the ones we are seeing or additional values we are not seeing (in time points between our observations)? – Richard Hardy Sep 30 '20 at 17:55
  • @whuber, otherwise the prices and returns are what we observe rather than some latent processes. Then I make an argument that they are discrete. This is because they can only take a countable number of possible values. With which part of my reasoning do you disagree? – Richard Hardy Sep 30 '20 at 17:57
  • Most of it, I fear. One point of departure is Stephens' original paper on measurement types. He based his typology on the groups of transformations that preserve the necessary properties of the measurement. A second point of departure is the concept of a difference between a model and the data. From both of these perspectives, most of the evidence you adduce to conclude stock returns are necessarily discrete is just not relevant. Ultimately, what matters is the adequacy and tractability of the resulting model, so I could be persuaded to your point if you could exhibit a useful discrete model. – whuber Sep 30 '20 at 18:17
  • @whuber, I am not familiar with Stephen's paper, but I guess it contradicts the claim that a variable with a countable number of possible values is discrete. This is suprising for me, but since you are citing it, there might be some good arguments for the position. I think I do appreciate the difference between an underlying phenomenon, data generated by it and a model, even if I might have given a different impression. I do not however find the argument of usefulness relevant to the discussion. Variables are what they are, their type cannot change because of usefulness or lack thereof. – Richard Hardy Sep 30 '20 at 18:35
  • @whuber, where we agree, however, is that we can choose a model based on its usefulness. If I find a continuous model to be more convenient than a discrete one, I might choose it regardless of the type of the underlying phenomenon and the data generated by it. (It might be worth mentioning that the encoding of the data will be model dependent, where model concerns the mapping from a random phenomenon to a random variable). Anyway, thanks for your explanation. – Richard Hardy Sep 30 '20 at 18:39
  • Re: "their type cannot change." I view that as a fundamental misconception, in part because in modeling a statistical problem we have many choices we can make about the objects in that model and how they correspond to the world. Don't limit yourself by such suppositions. BTW, there are several senses of the term "discrete" potentially in play in this conversations, which makes me concerned about possible misunderstandings. The mathematical definition of a discrete random variable is not *apropos* in this context. – whuber Sep 30 '20 at 21:57
  • @whuber, I shall probably start a new thread where I could learn more about your position, but I would like to clarify some preliminaries first. I like to start from the underlying random phenomenon before introducing random variables. E.g. I maintain a coin throw or a dice roll are discrete phenomena as they have a countable number of possible outcomes. Would you agree? 1 billion coin throws where we record the number of heads is still a discrete phenomenon, for the same reason. Would you agree? Taking a logarithm of the latter number produces is still discrete for me; is it for you? – Richard Hardy Oct 01 '20 at 07:39
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/113601/discussion-between-richard-hardy-and-whuber). – Richard Hardy Oct 01 '20 at 07:39
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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.

Trusky
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