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A quantum walk down Wall Street

FINANCE 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 forcing counterfeiters to the gallows. The quantitative tools developed in 1900 by Louis Bachelier to study the French stockmarket were taken up by Albert Einstein to prove the existence of atoms. Norbert Wiener formalised them into a mathematical framework that remains at the heart of today’s financial models.

Yet finance has been slower to absorb other big ideas from 20th-century physics. That is perhaps unsurprising, because they are generally bizarre. Fire a beam of electrons through two slits onto a screen and they will pass through both at once, travelling as a wave but arriving as particles. Concentrate enough energy in a region of space, and matter and antimatter pop out of the void. Introduce the right two particles to each other and they pop back into it.

All this seems worlds away from the mundane reality of traders tapping out buy and sell orders on their keyboards. But on a closer look finance bears a striking resemblance to the quantum world. A beam of light might seem continuous, but is in fact a stream of discrete packets of energy called photons. Cash flows come in similarly distinct chunks. Like the position of a particle, the true price of an asset is unknowable without making a measurement—a transaction—that in turn changes it. In both fields uncertainty, or risk, is best understood not as a peripheral source of error, but as the fundamental feature of the system.

Such similarities have spawned a niche area of research known as quantum finance. In a forthcoming book, “Money, Magic, and How to Dismantle a Financial Bomb”, David Orrell, one of its leading proponents, surveys the landscape. Mr Orrell argues that modelling markets with the mathematical toolbox of quantum mechanics could lead to a better understanding of them.

Classical financial models are rooted in the mathematical idea of the random walk. They start by dividing time into a series of steps, then imagine that at each step the value of a risky asset like a stock can go up or down by a small amount. Each jump is assigned a probability. After many steps, the probability distribution for the asset’s price looks like a bell curve centred on a point determined by the cumulative relative probabilities of the moves up and the moves down.

A quantum walk works differently. Rather than going up or down at each step, the asset’s price evolves as a “superposition” of the two possibilities, never nailed down unless measured in a transaction. At each step, the various possible paths interfere like waves, sometimes amplifying each other and sometimes cancelling out. This interference creates a very different probability distribution for the asset’s final price to that generated by the classical model. The bell curve is replaced by a series of peaks and troughs.

Broadly speaking, the classical random walk is a better description of how asset prices move. But the quantum walk better explains how investors think about their movements when buying call options, which confer the right to buy an asset at a given “strike” price on a future date. A call option is generally much cheaper than its underlying asset, but gives a big pay-off if the asset’s price jumps. The scenarios foremost in the buyer’s mind are not a gentle drift in the price but a large move up (from which they want to benefit) or a big drop (to which they want to limit their exposure).

The potential return is particularly juicy for options with strikes much higher than the prevailing price. Yet investors are much more likely to buy those with strikes close to the asset’s market price. The prices of such options closely match those predicted by an algorithm based on the classical random walk (in part because that is the model most traders accept). But a quantum walk, by assigning such options a higher value than the classical model, explains buyers’ preference for them.

Such ideas may still sound abstract. But they will soon be physically embodied on trading floors, whether the theory is adopted or not. Quantum computers, which replace the usual zeros and ones with superpositions of the two, are nearing commercial viability and promise faster calculations. Any bank wishing to retain its edge will need to embrace them. Their hardware, meanwhile, makes running quantum-walk models easier than classical ones. One way or another, finance will catch up.

This article appeared in the Finance & economics section of the print edition under the headline “Schrödinger’s markets”