When Financial Prices and Returns Have PDFs and CDFs

Summary

The document asks whether stock prices, log returns, and cumulative returns have probability density functions, cumulative distribution functions, or both, and when each representation is useful. The included answer explains that a cumulative distribution function exists for any random variable, while a probability density function exists only when the distribution has an appropriate density. For a continuous variable, the density is the derivative of the cumulative distribution function; a jump or other nondifferentiable behavior can prevent a PDF from existing.

The answer notes that common continuous modeling assumptions, such as normally distributed log returns, imply PDFs for returns, cumulative returns, and prices. This is a model-based statement, not a claim that empirical financial data must follow those distributions. The question’s request for graphical comparisons and guidance on choosing CDFs versus PDFs is not addressed in the included answer, nor does it explore the non-stationarity concern raised about cumulative returns.

Key ideas

• Every random variable has a cumulative distribution function, but not every variable has a probability density function.
• For a continuous distribution, the PDF is the derivative of the CDF when that derivative exists.
• A distribution with jumps may lack a PDF even though its CDF is defined.
• A continuous model for log returns can imply PDFs for returns, cumulative returns, and prices.
• The answer does not provide the requested plots or discuss applications favoring PDFs or CDFs.

Tags

# Which financial time series have a PDF and/or CDF?


# Which financial time series have a PDF and/or CDF?












Consider the following types of financial time series for a single publicly-listed stock:

- Price data

- Log returns

- 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.

## Answer by fes (score 5, accepted)

https://quant.stackexchange.com/a/55774

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.