Consider a single-period, binomial market model with a $r > 0$ interest rate (in USD per period) and a portfolio $(x, y)$ consisting of two assets: a savings/lendings account and a stock, both measured in USD.
Now, both $x$ and $y$ may be positive and negative. If $x$ is positive, the savings account holds $x$ USD; if $x$ is negative - the account holder owes the bank $x$ USD. If $y$ is positive, the account holder has $y$ stocks at his/her possession.
What does it mean for $y$ to be negative?
The only idea I have is that a negative $y$ corresponds to short selling $y$ stocks. However, in the real world, short selling a stock is accompanied by setting up an interest-accruing margin account with the broker and possibly depositing an additional collateral, and this is not reflected in the model.
I understand that the model is a simplification of the real world, but I don't think ignoring an interest accruing debt is an acceptable simplification, and, in support of this I bring the following quote from Investopedia:
Most of the time, you can hold a short for as long as you want, although interest is charged on margin accounts, so keeping a short sale open for a long time will cost more.
It also simply doesn't make any economical sense, of the sort that exists even in the most simplified models of economic interactions, that one can borrow something of value without having to pay for it.
So what does it mean for $y$ to be negative? How can I wrap my mind around it?
EDIT: Here's my proposal for an answer, let me know what you think.
The difficulty arises from the fact that the word "stock" is a misnomer: the security referred to as a "stock" does not model a real-life stock, not even in simplified form. Rather, it models some other financial instrument that has no counterpart in real life, which, together with additional financial instruments, can be used to synthesize a simplified model of a real life stock.
A much better conceptualization of what the model "stock" means is it is a variation on a savings/lending account: whereas the value of a regular savings/lending account increments deterministically with time, the value of the so-called "stock" increments randomly.
From this follows the following conclusion: instead of referring to the model "stock" as such, it would be better to call it "a random savings/lending account", as opposed to "a deterministic savings/lending account", and save the term "stock" to a different financial instrument that actually models a real-life stock, and that can be synthesized from a combination of a deterministic and a random savings/lending accounts and possibly some other financial instruments.