A quantum walk down Wall Street

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F.INANZ AND Physics have long been productive bedfellows. When he wasn’t writing the laws of mechanics and gravity, Isaac Newton ran the Royal Mint, making coins harder to forge and forging counterfeiters to the gallows. The quantitative instruments developed by Louis Bachelier in 1900 to study the French stock market were taken up by Albert Einstein to prove the existence of atoms. Norbert Wiener formalized it in a mathematical framework that forms the core of today’s financial models.

But the financial world has taken other great ideas from 20th century physics more slowly. This may not come as a surprise as they are generally bizarre. Shoot an electron beam through two slits on a screen and they will traverse both of them at the same time, traveling as a wave but arriving as particles. Concentrate enough energy in one region of space and matter and antimatter appear out of nowhere. Bring the right two particles together and they’ll pop back in.

All of this seems worlds apart from the day-to-day reality of traders entering buy and sell orders on their keyboard. But on closer inspection, the financial world bears a striking resemblance to the quantum world. A ray of light may appear continuous, but it is actually a stream of discrete packets of energy called photons. Cash flows come in similarly different chunks. Like the position of a particle, the true price of an asset cannot be determined without a measurement – a transaction – that in turn changes it. In both areas, uncertainty or risk is best understood not as a peripheral source of error, but as a fundamental characteristic of the system.

Such similarities have spawned a niche area of ​​research known as quantum finance. In an upcoming book, Money, Magic, and How to Dismantle a Financial Bomb, David Orrell, one of his leading proponents, examines the landscape. Mr. Orrell argues that modeling markets with the quantum mechanics mathematical toolbox could lead to a better understanding of those markets.

Classic financial models are based on the mathematical idea of ​​the random walk. You start by breaking time down into a series of steps, and then imagine that the value of a risky asset like a stock can go up or down a small amount with each step. A probability is assigned to each jump. After many steps, the probability distribution for the price of the asset looks like a bell curve centered around a point determined by the cumulative relative probabilities of its upward and downward movement.

A quantum walk works differently. Rather than going up or down with every step, the price of the asset evolves as an “overlap” of the two that is never pinned down unless measured in a transaction. With each step, the various possible paths overlap like waves, sometimes amplifying and sometimes canceling each other out. This interference creates a very different probability distribution for the final price of the asset than that produced by the classical model. The bell curve is replaced by a series of peaks and valleys.

By and large, the classic random walk is a better way of describing how asset prices move. But the quantum walk better explains how investors, when buying call options, feel about their moves that confer the right to buy an asset at a certain “strike price” on a future date. A call option is generally much cheaper than the underlying asset, but pays off when the price of the asset goes up. The main scenarios that buyers have in mind are not smooth price fluctuations, but rather a big move up (which they want to profit from) or a big decrease (which they want to limit their exposure to).

The potential return is particularly juicy for options with an exercise price that is significantly higher than the current price. However, investors are more likely to buy those with a strike close to the market price of the asset. The prices of such options are very closely matched to those predicted by an algorithm based on the classic random walk (in part because this is the model that most traders accept). But a quantum walk that assigns a higher value to such options than the classic model explains the preference of buyers for them.

Such ideas may still sound abstract. But they will soon be physically embodied on the stock exchanges, regardless of whether the theory is adopted or not. Quantum computers, which replace the usual zeros and ones with superimpositions of the two, are approaching market maturity and promise faster calculations. Any bank that wants to keep its lead has to embrace it. Their hardware meanwhile makes it easier to run quantum walk models than classic ones. Finances will catch up one way or another.

This article appeared in the Finance & Economics section of the print edition under the heading “Schrödinger’s Markets”


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