I suggest that students at George Mason University ask their finance professors how negative interest rate impact their use of the famous Black-Scholes model.

*(Bloomberg) — 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 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.

All is not lost. There is the SABR Model that is a stochastic volatility model.

The **SABR** model describes a single forward , such as a LIBOR forward rate, a forward swap rate, or a forward stock price. This is one of the standards in market used by market participants to quote volatilities. The volatility of the forward is described by a parameter . **SABR** is a dynamic model in which both and are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations:

with the prescribed time zero (currently observed) values and . Here, and are two correlated Wiener processes with correlation coefficient :

The constant parameters satisfy the conditions . is a volatility-like parameter for the volatility. is the instantaneous correlation between the underlying and its volatility. thus controls the height of the ATM implied volatility level. The correlation controls the slope of the implied skew and controls its curvature.

The above dynamics is a stochastic version of the **CEV** model with the *skewness* parameter : in fact, it reduces to the **CEV** model if The parameter is often referred to as the *volvol*, and its meaning is that of the lognormal volatility of the volatility parameter .

A **SABR** model extension for Negative interest rates that has gained popularity in recent years is the **shifted SABR model**, where the shifted forward rate is assumed to follow a SABR process

for some positive shift . Since shifts are included in a market quotes, and there is an intuitive soft boundary for how negative rates can become, shifted SABR has become market best practice to accommodate negative rates.

The **SABR** model can also be modified to cover Negative interest rates by:

for and a **free** boundary condition for . Its exact solution for the zero correlation as well as an efficient approximation for a general case are available.^{[2]}

An obvious drawback of this approach is the a priori assumption of potential highly negative interest rates via the free boundary.