Discretization often leads to instability, particularly if not done carefully. The prototypical example is naive discretizations of stable ODEs. Of course, all we have are toy models and speculation. The only real way to know is to try it.
If I remember right, a couple of Asian equities markets have some sort of "auction on clock tick" mechanism, so we might be able to look to their experiences.
> Of course, all we have are toy models and speculation
These two statements could be construed to be in conflict, especially as I haven't the faintest idea what naive discretizations of stable ODE's is, and Google isn't of much assistance in providing an explanation that I can digest.
You lead with a fairly strong statement presented as a known fact and then later indicate it's just a model.
To me the idea of slowing things down sounds vaguely sensible, but I also know that I don't know enough to really judge. So it sounds like no one else really knows either, but the entrenched players like things the way they are, which makes sense, because the ones who have the money to be at the table running things are probably also the ones who have the money to throw at HFT and come out ahead.
It is a well known fact about dynamical systems in general.
I.e., if you ask me about the stock market, a flight control system, an electric circuit, a biological system and a computer network, I'll tell you that continuity gives you a better shot at stability than discretization in most of them.
I.e., if you don't know what you are talking about, lean towards continuity. If you do understand things, then explain the mechanics and back it up with empirics.
The hypothesis being advanced by people proposing point-in-time clearances is not necessarily that it will make the market smoother. It's that it will free up a lot of resources for other purposes.
For what it's worth - and not to distract from your real point - Exponential (or the more general Lyapunov) stability are a better explanation of instability. Stiffness is more a property of the method used to solve the ODE, rather than of the ODE itself.
Stiffness is a property of certain dynamical systems, which certain ODE solvers are better suited for. That said, "stability" is an ugly word in numerical methods, particularly for differential equations, as it could mean a number of things. I think this is the source of the confusion here.
It seems that investors trading on an exchange operating at discrete
time intervals would be at a arbitrage disadvantage if other
exchanges offered continuous trading in the same shares.
So a demand for markets that are liquid and efficient rewards
trading that is continuous in time over time-discrete trading,
and firms are competing hard to shave milliseconds from latency
with co-located hardware and faster trans-oceanic cables.
On the other hand, pricing is limited by regulation to penny
increments, with some pressure to telegraph smaller increments.
Are we likely to see a push for trading with offers and trades
denominated in milli-cents or micro-cents? Would smaller pricing
increments be welcomed by traders or regulators or neither?
Are any markets in the world trading at tiny denominations?
(NB: assumes pricing in integer quantities of tiny denominations)
I don't see a reason that "normal" investors would be at an arbitrage disadvantage if trading on an exchange that was discretized to, say, 100ms. Arbitrageurs would still have an incentive to keep the prices on the "slower" exchange in line with the "continuous" exchange's prices, trading at the discrete ticks to exploit price differences. So, price differences past an epsilon shouldn't persist for more than a few ticks, which is more than fast enough for most non-HFT investors, who don't typically make trading decisions with anything close to sub-second precision anyway.
I think a bigger question mark with my admittedly offhand proposal is whether discretizing might actually increase gaming opportunities, since the change could do non-obvious things to the strategy space. On the other hand, it might also remove existing market-gaming opportunities (which are fairly poorly understood, and axiomatically assumed not to exist by the idealized equilibrium analysis economists typically use). Alas, current game-theory solvers don't scale up anywhere close to well enough to give solid answers in either direction (real markets are a bit more complex than 4-participant games iterated over 10 timesteps...).
>I don't see a reason that "normal" investors would be at an arbitrage disadvantage if trading on an exchange that was discretized to, say, 100ms. Arbitrageurs would still have an incentive to keep the prices on the "slower" exchange in line with the "continuous" exchange's prices, trading at the discrete ticks to exploit price differences. So, price differences past an epsilon shouldn't persist for more than a few ticks, which is more than fast enough for most non-HFT investors, who don't typically make trading decisions with anything close to sub-second precision anyway.
Sure, but what you're missing is that the slow exchange's prices are always going to be worse. Not a lot worse - probably only a penny either way on some ticks, and zero on other ticks - but a little. So why would any investor with the choice ever choose to trade on the slow exchange rather than the continuous one?
"Worse" in what sense? Won't the prices deviate pretty randomly around the true price, sometimes behind a penny higher than the continuous exchange, and sometimes a penny lower, making it basically a wash?
No. If the discrete exchange makes it possible to withdraw an offer in between ticks, then the HFT guys would do that every tick - so you actually wouldn't get a price (or you'd get a very wide spread), and it would basically fail to be an exchange. So let's assume any order you have on the book stays there until the next tick's auction. In that case the HFTers are going to give a much higher spread, because if new information comes in at 3.2s and the stock is suddenly worth more than it was at 3s, but they can't withdraw their sell order until 4s, then obviously they lose money. So occasionally you'd get a better price on the discrete exchange (when the market moves further than the spread in the space of one tick) - but the HFTers would be terrified of this situation, and make their spreads wide enough that they think it's impossible. So 99% of the time, you'd get a worse price.
If I remember right, a couple of Asian equities markets have some sort of "auction on clock tick" mechanism, so we might be able to look to their experiences.