# What does a negative stock amount mean in a single-period, binomial market model?

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.

• Typically a single period binomial model uses a multiplication factor instead of an additive factor. Hence the stock cannot be negative. – user9403 Mar 31 '16 at 9:38
• Better terminology than "random account" would be "an underlying hedge account". – Brian B Mar 31 '16 at 12:55

Negative y means you're short the stock. This may have costs, just as being long the stock may have income from lending it to someone else. These costs can easily be incorporated as a growth adjustment to the stock.

Since it costs more to borrow than you make from lending, the growth adjustment should depend on whether your overall position is long or short. (But not whether the particular derivative being priced implies a long or short stock hedge.)

I'm answering my own question. I don't base the following answer on anything except my opinion.

The only idea I have is that a negative $y$ corresponds to short selling $y$ stocks. [...] 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.

I disagree. In my opinion it is important to differentiate between what a positive and a negative value of a stock represent. A positive value of a stock is indeed a simplified model for holding a real life stock. However, a negative value of a stock is not a simplified model of a short sale. Rather, to use OP's words,

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...

... short sale.

I will now address OP's comment that it

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.

This is a valid comment. However, the financial instrument represented by a negative stock value is, as mentioned above, not a model of a real-life financial instrument; it is rather a complete mathematical fabrication. It is indeed an instrument that would never be traded in real life markets, however, it is also important to be able to define mathematically financial instruments that are not market efficient if only to be able to rigorously define - by way of differentiation - what it means for a financial instrument to be market efficient. The instrument described by a negative stock value is just such a non-efficient instrument. However, in combination with other financial instruments, some real-life financial instruments, e.g. short sales, may be defined.

I don't have any problem with the short stock position. In real life, you borrow the stock (costing some pretty small amount), sell it in the market, and receive cash, which you can deposit to earn interest. The cost of borrowing the stock is not determined by the market rate for short term loans. It is just a fee paid to whoever owns the stock, as compensation for the inconvenience and risk of lending it out.