Foreword
There are many models of how stocks are priced, dealing with different aspects of pricing, and different pricing models. Intuitively, if 2 things are the same, then they should trade with the same price. But empirically, we know that they don’t always do that (see here and here). Why not?
As usual, a reminder that I am not a financial professional by training — I am a software engineer by training, and by trade. The following is based on my personal understanding, which is gained through self-study and working in finance for a few years.
If you find anything that you feel is incorrect, please feel free to leave a comment, and discuss your thoughts.
A = B = C
Let’s say we have 3 different securities, which may be stocks, futures, options, or whatever, A, B and C, which are essentially the same thing, modulo some constants (such as interest rates, trading fees, etc.). For example, SPY, ES, VOO, IVV, etc. The conventional wisdom is that the price of A, B and C should be the same — again, modulo some constants — since these are constants, we can model them out and just assume each of A, B and C are trading sans these conditions.
In practice, we know that SPY, ES, VOO and IVV don’t trade exactly in lockstep. Yes, they trade pretty closely, and divergences don’t last for very long. But in some short-ish timeframe (which may be on the order of seconds or even milliseconds), their prices can diverge.
Arbitrage
Assuming A, B and C are convertible, i.e. we can convert any of A, B or C into any of A, B or C, then there is arbitrage — if the price of A and B, say, get out of sync, someone can short the more expensive one, buy the cheaper one, and convert from the cheaper one to the more expensive one to close the short, thus making a “risk free” arbitrage profit.
The Efficient Market Hypothesis likes to assume that all these happen instantly, so there are no risk free profits to be made. But Wall St’s legions of market makers, liquidity providers and ETF arbitragers say otherwise. The fact that these companies are massively profitable suggests then that price dislocations can happen, and these companies close the price dislocations, getting us closer to (but never really reaching!) a perfectly efficient market, while making a profit for their troubles.
Not so perfect information
Now, in the perfect world, everyone knows all publicly available information at all times (i.e. information dissemination is instantaneous). In practice, that is clearly not true and can never be true — physics teaches us that nothing can move faster than the speed of light, including information. Since light moves at a finite speed, entities further from the source of information need necessarily get the information at a later time than entities closer to the source of information.
So, let’s take that a bit further, and assume that not everybody knows about the fact that A = B = C. Let’s say some segment of the market knows that A = B, another segment know that B = C and a third segment knows about A = C. Now these segments can overlap, but they are not the same, i.e. some people may know more than one of the 3 sub-equations, but not everyone knows all the sub-equations. Further, we use “know” here loosely — it’s possible that an entity actually does understand the 3 sub-equations, but for whatever reasons, decide that they only want to trade one or two of the 3 sub-equations. For example, an entity may only be a market maker in A and B. So while they can easily trade the A = B equation, they may not be able to trade the B = C or A = C equations as effectively, and so they don’t.
A moment in time
Let’s say some large entity, E, decides, for whatever reasons, they want to buy our security, and in large quantity. E can choose to buy A, B or C, and they fully understand the A = B = C relationship. However, E is a practical, real world entity, and they are not a market maker — they are a regular investor, and more interested in the productive earnings of the security, than the temporary price dislocations. So, for practical purposes, E will only buy one of the 3 of A, B or C, as it makes bookkeeping easier.
Now, even if E does the smart thing, and buy the cheapest of the 3 right now (and for the sake of argument, let’s say that’s A), if E decides at some future date to add on their position, and in order to maintain the “only one symbol on my books” rule, they will, at that future time, continue buying A. This may be true, even if A is not be the cheapest of the 3 at that point in time. In effect, at some arbitrary point in time, E may, for perfectly rational, though not financial, reasons, decide to buy A, even if A is not the cheapest.
Next, let’s say E is expecting to deploy large sums of money. We know that a trade is always between a buyer and a seller. We have a seller in E, but who’s selling? Unless there is another entity (or set of entities) that are willing to sell at least as much as E is buying at the current market price of A, E‘s buying will necessarily push the price of A up, at least in the short term. And since at any particular point of time, it is impossible to guarantee that you can always trade such that someone else is always willing to take the other side of the trade from you at the current market price, especially if you are trading in large sizes, we come to the conclusion that at least in the short term, E‘s trades will necessarily push the price of A up.
As the price of A increases, arbitragers will start to do their work by shorting A and buying B or C to convert into A to close their short. This will push the price of A down, while simultaneously pulling the prices of B and C up. However, since it’s generally not possible that the number of entities in the segments arbitraging A = B and A = C to be exactly the same, one of B or C will move up faster than the other. Let’s say for our discussion that B moves up faster than C, so we arrive at A > B > C, again, in the short term.
Since B and C diverged, our last segment of arbitragers will step in, shorting B and buying C. This has the effect of pushing C up faster, but notice how it work against the efforts of the A = B arbitragers!
Big picture
Stepping away from the instantaneous snapshots of the prices of A, B and C caused by E‘s trades, we arrive at a well known scenario described in mathematics as a “converging recurrence relation”. E‘s trades immediately pushes up the price of A, and the efforts of the arbitragers, over time, tries to spread that “information” (here the price of the security represented by A, B and C), so that all of A, B and C all reflect the same price.
Notice that we did not talk about fundamentals! It maybe that E is too optimistic, and pushes the price of A (and eventually B and C) up too high. But that doesn’t matter to the arbitragers — they are simply making risk free profits by arbitraging the equation A = B = C. So, in the short term, it’s perfectly possible that the prices of A, B and C are dislocated from fundamentals.
More interestingly, for the prices of A, B and C to converge to a final single, identical value, will require many rounds of arbitragers shorting/buying different pairs of the 3, but since the relationship is convergent, we can assume that over time, the absolute magnitude of dislocations will reduce, and the prices will eventually settle down (assuming no one else is trading these symbols other than the arbitragers).
Most interestingly, it should be noted that we cannot actually predict the final stable price of A, B and C! Without knowing the relative numbers of arbitragers in each of the 3 sub-equations, and without knowing their relative trading speeds and aggressiveness, it’s generally not possible to figure out whether the A = B arbitragers will be more successful pulling up the price of B to meet the price of A, or that the B = C arbitragers will be more successful pushing DOWN the value of B, which eventually results in A and C catching down to B.
Liquidity
One thing to note, though, is the need for liquidity. For the arbitragers to work, there needs to be external sellers and buyers in each of A, B and C. Given that we know A = B = C, then objectively, even if not everyone knows the full equation, we can generally expect that the number of arbitrage buyers in A will be less than the number of arbitrage sellers in A, whenever A is overvalued compared to B and C.
For securities which are very liquid, such as the SPY, ES, VOO, IVV, etc., the arbitragers have a deep pool of external traders to trade against, so the convergence of the prices can be fairly quick, sometimes in the order of milliseconds.
But for securities which are very illiquid, such as certain ETFs, the arbitragers may not be able to find willing buyers/sellers to take the other side of their operations, and in those cases, the convergence can take a much longer time. In some cases, illiquid ETFs have been known to diverge from their underlying for days or even weeks at a time.
Final words
Clearly, the above does not perfectly describe every security. In fact, it does not perfectly describe any security (or set of securities). It is a model to think about how prices move, out of many, many different models that try to explain subtle nuances of different parts of the market.
While it is sort of true in practice, there are a number of assumptions made which are not realistic in practice — such as no external buyer/seller willing to move prices other than E.
However, hopefully a discussion of the model, even in our made up world, is a useful exercise in thinking about how prices move in practice.
Janky Thoughts 