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Wednesday, 22 June 2011

Trading at the Speed of Light

Time is Money

Special relativity doesn’t generally figure high on the agenda of your average trader. Or even your far from average trader, come to that. For most of us, for most of all of time, the speed of light hasn’t ever really figured as an important investment constraint.

However, this is finance and time is money. Quite literally, it seems, as the arms race between the high frequency terminators of the automated trading industry reaches the limits of the known universe. Welcome to where arbitrage meets Einstein. And wins.

Light Speed

The speed of light – 299,792,458 m/s – is a fundamental limit on the behaviour of the known universe. It’s a so called universal constant, a fact that threw nineteenth century physics into a bit of a tizzy when revealed by the results of the Michelson-Morley experiment. Prior to this scientists had assumed that light propagated through a medium, the lumiferous æther, in the same way that sound propagates through air, and that light would travel at different speeds depending on the direction of this medium. The failure to discover any difference led to Albert Michelson earning a Nobel Prize for an experiment which "failed"

This fact explains, amongst other things, why the sky at night isn’t as bright as the sky in the day: after all, if there are an infinite number of stars in the universe – an approximately true statement – why isn’t every possible part of the night sky glowing with the light of a distant sun? To answer this, and a ton of other hard to figure stuff, we needed Einstein, whose childlike approach to the kinds of impossible questions that only a child would think to ask, was superbly tuned to the nature of the weird and wonderful behaviour of the universe at its utmost limits.

Relativity Revealed

To reveal the nature of relativity consider the following thought experiment. A train passes through a railway station and while it does the guard on the train bounces a ball on its floor. To the guard, bouncing the ball, it moves in a straight line, from hand to floor and back to hand, a simple “I” movement. Say it moved 3 feet from hand to floor.

However, to the person standing on the platform, wondering why the guard was bouncing balls instead of stopping their train, the ball didn’t just move up and down but also from left to right, executing a “V” movement. It went up and down by 3 feet and 6 feet from side to side: it travelled further.

Now the distance travelled by the ball is equal to the time it takes the light to reach our eyes multiplied by the speed of that light. However, the speed of light is a universal constant so the distance travelled is directly related to the time that passed between ball leaving hand and returning. This means that the length of time experienced by the guard and the stranded passenger was different: time is relative to your frame of reference and the faster you travel the less time you experience.

At human scale this difference isn’t usually relevant, but on a universal scale it matters: the reason the night sky isn’t ablaze with light is that some of it hasn’t reached us yet. In fact most of it never will because we’re moving away from it at faster than the speed of light … an argument otherwise known as Olbers' paradox.

Higher Frequency Trading

Despite the irrelevance of relativity to most of human life it is perhaps inevitable that it’s become important at the bleeding edge of high frequency trading. As discussed in Rise of the Machines, HFT allows those organisations possessed of the most powerful computing technology and deep enough pockets to pay for preferential access to markets, to trade on the smallest and most short-term of pricing anomalies. This is an arms race, as companies rack up the pressure to attempt to gain the tiniest of advantages over their competitors: and as trading gets faster and time intervals smaller we tend towards that most fundamental of universal properties, the speed of light.

Of course, such an idea has all the hallmarks of an April Fool joke: indeed such an article was published recently:
“However, Dr. Paolo Alfisy at Koblenz was able to breach the light barrier by using microwave photons - energetic packets of light - to communicate trading data to and from an exchange platform. Photons have negative mass and are therefore able to breach the light barrier provided they are passed through a vacuum.”
Dr. Paolo Alfisy = April's Fool Day. Those funny journalists.

Trading at the Speed of Light

Astonishingly, though, this next bit is not a joke: Alex Wissner-Gross and Cameron Freer have written what’s probably the first paper on financial analysis which includes space-time diagrams: it’s called Relativistic Statistical Arbitrage and it’s an analysis of where best to physically locate your trading centres to take advantage of delays in light propagation between different exchanges. The speed of light is relevant, of course, because most communications takes place over fibre optic cables.

Now, as the diagram on page 5 of the paper shows, this does have the real-world problem that many of these locations are in the middle of the ocean or in other inhospitable locations like Siberia, Côte d'Ivoire and what looks suspiciously like the dodgy bar at the end of my road. The net effect of this strategy, assuming it could be implemented, would be to remove pricing variations between geographically dispersed locations, at least to the level that’s actually possible given the laws of physics, the topology of the planet and the bartender's willingness to check server availability in between drawing pints.

Space-Time Trading

To go beyond this and exploit relativistic effects in quantitative finance is, at the moment beyond us, although Espen Gaarder Haug has written an amusing paper on the possibilities:
“It turns out that we already today have the technology and people to conduct an experiment with measurable effects on spacetime finance. The technology in question is the space shuttle. The space shuttle has a typical velocity of about 17,300 miles per hour (27,853 kph). Let us for simplicity assume a dollar billionaire got a free ticket to travel with the space shuttle. Further assume he leaves the 1 billion dollars in a bank that pays interest equivalent to 10% annually compounding, but with compounding every thousand of a second to make the calculation more accurate...

If the billionaire spends one year on earth according to his wristwatch he will receive $100,000,000.00 in interest income, while he will receive $100,000,000.04 in interest rate income if staying in space. That is a difference of 4 cents. This is a measurable quantity of money, but of course not economically significant, especially not for someone already a billionaire. The barrier to significant profits is that we are at a very early stage of space travel”.
Which suggests Mr Haug has never seen what a billionaire is prepared to do for another 4 cents.

Interstellar Trade

Oddly enough, the first research around these issues appears to date from 1978 by Paul Krugman. In The Theory of Interstellar Trade he posits:
“How should interest charges on goods in transit be computed when the goods travel at close to the speed of light? This is a problem because the time taken in transit will appear less to an observer travelling with the goods than a stationary observer. A solution is derived from economic theory, and two useless but true theories are proved.”
Of course, finance is weirder than even economists can predict: although it’s probably best not to hold our collective breath on intergalactic trading the idea that space travel can allow high frequency traders to increase their profits is likely the best hope for the space programme. After all, the history of exploration has always been driven more by a thirst for profit than a thirst for knowledge.

Related articles: Utility, the Deux Ex Machina of Economics, Rise of the Machines, Fall of the Machines

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