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A quantum walk on Wall Street

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FINANCE AND physics has long been productive bedfellows. When not 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 stock market were taken up by Albert Einstein to prove the existence of atoms. Norbert Wiener formalized them within a mathematical framework which remains at the heart of current financial models.

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Yet finance has been slower to absorb other great ideas from 20th century physics. This is perhaps not surprising, since they are generally weird. Shoot a beam of electrons through two slits on a screen and they will pass through both at once, traveling like a wave but arriving as particles. Concentrate enough energy in one region of space, and matter and antimatter spring out of the void. Present the two good particles to each other and they reappear there.

It all seems a far cry from the mundane reality of traders typing buy and sell orders on their keyboards. But on closer inspection, finance bears a striking resemblance to the quantum world. A beam of light may appear continuous, but is actually a flow of discrete packets of energy called photons. Cash flow comes in equally distinct chunks. Like the position of a particle, the actual price of an asset is unknowable without making a measurement – a transaction – which in turn changes it. In both fields, 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 a forthcoming book, “Money, Magic, and How to Dismantle a Financial Bomb,” David Orrell, one of its main supporters, examines the landscape. Mr. Orrell argues that modeling 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 may rise or fall slightly. Each jump is assigned a probability. After many steps, the probability distribution for the asset’s price looks like a bell curve centered on a point determined by the cumulative relative probabilities of up and down movements.

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

Generally speaking, the classic random walk is a better description of how asset prices move. But quantum walk better explains how investors think about their movements when they buy call options, which confer the right to buy an asset at a given “strike” price at a future date. A call option is usually much cheaper than its underlying asset, but gives a big payoff if the asset’s price rises. The priority scenarios in the buyer’s mind are not a slight price drift but a sharp rise (from which he wishes to benefit) or a sharp fall (to which he wishes to limit his exposure).

The potential return is particularly juicy for options with strike prices much higher than the going price. Still, investors are much more likely to buy those with strike prices close to the market price of the asset. The prices of these options closely match those predicted by a classic random walk-based algorithm (in part because this is the model most traders accept). But a quantum walk, by assigning these options a higher value than the classic model, explains the buyers’ preference for them.

Such ideas can still seem abstract. But they will soon take physical form in the trading rooms, whether the theory is adopted or not. Quantum computers, which replace zeros and the usual ones with overlays of the two, are approaching commercial viability and promise faster computations. Any bank wishing to retain its advantage will have to adopt them. Their hardware, meanwhile, makes performing quantum walk models easier than conventional models. One way or another, finance will catch up.

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


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