Guest Post: Market Price Is An Illusion
Bo Peng submits a theoretical thought experiment on the discontinuity of security prices in a world in which two bids one millisecond apart are an eternity away from each other thanks to supercomputers in charge of markets, and thus the price discovery function can go anywhere from zero to infinity in the space of nanoseconds. Are the now daily flash crashes nothing more than an actual representation of the fractal nature of markets, as popularized by the now late genius Benoit Mandelbrot, and as was validated on Zero Hedge some time ago? If so, prepare for a market in which melt downs and melt ups become a daily, hourly and millisecondly fact of life.
Submitted by Bo Peng
Market Price Is An Illusion
You need to carry some rocks and the only thing you have is a screen sieve. But it's a fine screen. So as long as it's strong enough, the bottom is as good as a solid one. Millions of years pass, the screen sieve has always worked and gradually people forget about the holes on the bottom. But as the rocks get weathered over time, they become smaller and smaller, eventually smaller than the screen holes. Now, all of a sudden, "It's broken!" people gasp.
It's always broken. It's just that how it's broken hasn't been a problem before.
This is, in short, the origin of recent flash crashes with alarming frequencies and reach. The screen holes are the time interval between quotes. The size of rocks is the reaction time of market participants (traders) executing trades.
During the vacuum between two consecutive quotes, the notion of "market price" is undefined. It could be anything from 0 to infinity. But this is quite ugly and inconvenient. So we assume, driven by our evolutionarily honed instinct of linear interpolation, there still MUST be a market price during the vacuum and it SHOULD be somewhere close to the last quote. I call this the "Continuous Price Assumption"
This seems quite logical, and has always worked ok during the days of human traders. Human traders, or particularly human market makers, have limited reaction speed, say in the 10's of milliseconds range. Quotes pour in from all sources, faster than human traders can keep up, leaving them no time to see the vacuum. Even if conceptually they are aware of the discontinuous nature, they have little facility to exploit it. Rocks are big. Screen sieve has a rock solid bottom.
Enter GHz computers. The time between two quotes 1 millisecond apart is eternity. During this time the CPUs can scan all internet connections, check for content updates, display a few frames in three video windows, overlay their audio, refresh all the window frames with updated smeared version of the wallpaper for no logical reason, handle 271 senseless events, and then take a good nap, but not forgetting to update the "System Idle Process" CPU utilization percentage during the nap, which should be logically forbidden. It can scan all the ECNs and put out quotes, even execute trades, multiple times during the void. The rocks have become sand. The fundamental flaw in Continuous Price Assumption is no longer just theoretical, it has resulted in the invalidation of the very notion of market price.
This, I think, deserves a line by itself: The notion of market price is never what we assumed it to be. It's not continuous. It jumps all the time. And except for the singular points of quotes appearing, which has a mathematic measure of precisely 0 (a set of points on a presumed line), it doesn't even exist.
And I think this deserves repeating: The probability of hitting a quote is mathematically zero.
The market should never have worked at all. It has worked only because somebody can make money by letting you hit a quote; they artificially extend the lifespan of the last quote, to keep the appearance of a functioning market and to make money in return for this vital service. If they deem it unfavorable to extend the lifespan of the last quote, they will offer a new quote that's favorable to them. It's always been this way. Nothing has changed except the player's reaction speed.
The approach of treating "price curve" as fractals is not new. But it's been mostly a fancy word to be thrown around geek cocktail parties in hope of catching some girl's attention for a fractal second. Now, with high frequency trading playing a significant role in various markets, the fractal nature of "price curve" has become a very real feature with huge practical implications.
For those of you not familiar with fractals, it's a curve that's discontinuous everywhere, with its value undefined everywhere. The best you can do is to come up with a notion of range and probability of its value at any given point, but the range and probability distribution depend on the specific problem at hand and generally can vary greatly in time (or whatever the X-axis is) for the same system.
Back in June I wrote "The Market Is Never Right" and a commenter joked I should come up with a Peng Uncertainty Principle. Now here it is.
The Peng Uncertainty Principle
The Market Price cannot be determined unless with a trade, which will inevitably change the state of the presumed market and is by definition always in the past. To be precise, Market Price cannot be defined except in the posteriori sense.
The very notion of "current market price" is an illusion. It doesn't exist. It's a total farce, always has been, always will be.
Let this sink in for a moment.
The genie is out of the bottle. Love it or hate it, HFT is here to stay and it is the future. In the meanwhile, the flash crash must be stopped. With the insight of the Peng Uncertainty Principle, we can see that HFT itself is not the root cause. It's just a new tool that happens to reveal a fundamental flaw that's always been in the system. The right solution is not to banish HFT. It lies in attacking the root cause.
For the market to maintain a minimal measure of sanity, the inherent, unbound fractal nature of the price curve must be artificially eliminated.
Sounds impressive? Well here's a more readable version. Forbid quotes x% away from the last trade or NBBO within y milliseconds.
Why milliseconds? Why not femtoseconds? It is admittedly arbitrary. But the point is that pursuit of speed should not become a goal by itself. There's a point beyond which the pretence of price discovery no longer has value to the general population of market participants. If it's really essential for a market to move 10% in one millisecond in order to assure its functioning, then that market should not be functioning.