Negative interest rates have quite literally broken one of the pillars of modern finance.

As economists and central bankers weigh the pros and cons of sub-zero rates and their impact on the world, traders have been contending with a rather more mundane, but fundamental issue: How to price risk on trillions of dollars of financial instruments like interest-rate swaps when their complex mathematical models simply don’t work with negative numbers.

Out are certain variations of the Black-Scholes model, the framework that allowed derivatives to flourish in the past four decades. In are a hodgepodge of approximations and workarounds, including one dating to the 19th century.

Granted, the current state of affairs is more a nuisance than a serious problem. And it’s one that has been largely confined to Europe and Japan. But with sub-zero interest rates becoming a long-term economic feature and the number of negative-yielding bonds reaching $15 trillion, it’s an issue more and more traders, particularly in the U.S., are trying to wrap their heads around.

“I was quite surprised that I’ve started getting questions from U.S. clients wondering, ‘What’s the impact of negative rates? What are the mathematics?’” said Sphia Salim, a London-based rate strategist at Bank of America.

The issues are most apparent in the market for interest-rate swaps. (This market allows professional investors to lock in interest rates and lets speculators bet on whether rates on bonds or loans will rise or fall.) That’s because the Black 76 model, the main tool to price options for interest-rate derivatives, and its variants are so-called log-normal forward models.

For those who aren’t math nerds, it can essentially be boiled down to this: the formula breaks because it requires users to calculate a logarithm, and a logarithm of a negative number is undefined, or meaningless.

One option has been to dust off a framework that was first proposed nearly a 120 years ago. Known as the Bachelier model, it’s named after the French mathematician Louis Bachelier, who laid out his approach in his 1900 thesis “Theory of Speculation.” The model is best known for solving the math behind a theory from physics known as Brownian motion (some five years before Albert Einstein did the same in his revolutionary work on thermodynamics), and applying it to finance, according to a 2016 paper by Ian Thomson.

For market participants like Banco Di Credito P. Azzoaglio SpA’s Simone Ligato, its appeal is that it simply works with negative numbers -- at least for vanilla instruments like caps, floors and swaptions.

It comes with a few trade-offs. In a 2017 paper co-written by Ligato, the authors concluded that while Bachelier was “coherent enough” to replace Black 76, certain discrepancies make apples-to-apples valuation comparisons difficult. That’s led to some awkward conversation with clients.

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