# Prof. Jayanth R. Varma’s Financial Markets Blog

A blog on financial markets and their regulation

## Interest rate modelling at the zero lower bound

A long time ago, before the Libor Market Model came to dominate interest rate modelling, a lot of attention was paid to how interest rate volatility depended on the level of interest rates. If rate are moving up and down by 0.5% around a level of 3%, how much movement is to be expected when the level changes to 6%? One school of thought argued that rates would continue to fluctuate ±0.5%; this very conveniently allows the modeller to assume that rates follow the normal distribution. An opposing school argued that a fluctuation of ±0.5% around a level of 3% was actually a fluctuation of 16 of the level. Therefore when the level shifts to 6%, the fluctuation would be ±1% to preserve the same proportionality of 16 of the level. This was also convenient as modellers could assume that interest rates are log-normally distributed.

It was also possible to take a middle ground – the celebrated square root model related the fluctuations to the square root of the level. A doubling of the level from 3% to 6% would cause the fluctuation to rise by a factor of √2 from 0.5% to 0.71%. People generalized this even further by assuming that the fluctuations scaled as (level)λ where λ=0 gives the normal model, λ=1 leads to the log-normal, and λ=0.5 yields the square root model. Of course, there is no need to restrict oneself to just one of these three magic values. The natural thing for any statistician to do is to estimate λ from the data using standard maximum likelihood or other methods. Long ago, I did do such estimations for Indian interest rates.

The Libor Market Model killed this cottage industry. It was most natural to assume log normal distributions for the interest rates and then let the option implied volatility smile deal with departures from this distributional assumption. And there matters rested until the problem resurfaced when interest rates were driven down to zero after the global financial crisis. The difficulty is that zero is an inaccessible boundary point for a log normal process. A log normal process (geometric Brownian motion) can not reach zero (in any finite time) starting from any positive rate, and if you somehow started it out from zero, it could never leave zero (because the volatility becomes zero).

The regulatory push to mandate central clearing for OTC derivatives has turned this esoteric modelling issue into an important policy concern because central clearing counterparties (CCPs) have to set margins for a variety of interest rate derivatives where the modelling of volatility becomes a first order issue. A variety of different approaches are being taken. The OTCSpace blog links to a couple of practitioner oriented discussions on this subject (here and here). Among the solutions being proposed are the following:

Shift to a normal model
This would eliminate under margining at zero interest rates, but potentially create severe under margining at high rates.
Combine normal and log-normal fluctuations
The idea is that there are two sources of fluctuations in interest rates – one behaves in a “normal” and the other in a “log-normal” manner. This may be intractable for valuation purposes, but might be acceptable for risk modelling since it solves the under margining problem at both ends of the interest rate spectrum.
Interest rate plus a small constant is log normal
For example, assume that the fluctuations in interest are proportional to the level of rates +1%.

As an aside, I believe that the zero lower bound is actually a bound not on the interest rate, but on the contango on money. In other words, the zero lower bound is simply the proposition that money (being the unit of account itself) can neither be in contango nor in backwardation. The standard cost of carry model for futures pricing tells us that the contango on money is equal to the risk free interest rate PLUS the storage cost of money MINUS the convenience yield. It is this contango that is constrained to be zero.

If the convenience yield of money is larger than the storage costs (as it usually is in normal times), the contango is zero when the interest rate is positive. In an era of unlimited monetary easing, the convenience yield of money can become very small and the zero contango implies a slightly negative interest rate since the storage cost is not zero. For physical currency, the storage cost is high because of the need to guard against theft. For insured bank deposits, the bank needs to recoup deposit insurance in some form through various fees. Of course, uninsured bank deposits are not money – they are simply a form of haircut prone debt (think Cyprus). Actually, Cyprus makes one sceptical about whether even insured bank deposits are money.