Do financial return series have a probability mass function (pmf)? 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:

*

*financial time series are finite in the number of historical observations they possess, and

*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)?
 A: 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.
A: 
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.
