When one of the co-founders of FinMetrics worked on his PhD thesis, in particular on the modelling of the term structure of interest rates, a key requirement was not to allow rates to become negative. Indeed, during the Master of Banking and Finance we had learned about models like the Cox, Ingersoll & Ross (1985) one-factor model, said to be an evolution of the Vasicek one, since it used a square root diffusion process in order to avoid having interest rates become negative.

Let's be honest. In the 90's, if we would have said interest rates could become negative, we would probably have been seen as theoreticians who don't understand how the true world works.

One of our first corporate treasury clients even requested we make sure to add a condition to our simulations, when not using bounded models, to floor the rate at least at 0, ... ten years before they would actually become so.

## A quick review of interest rate dynamics

So let's see some longer but still recent history of empirical evidence about the dynamics of interest rates, using a simple source, the U.S. Treasury Par Curve Yield rates, which flirted with 0 for a while, while some central banks went to negative territory: the Danish National Bank, the European Central Bank (ECB), the Bank of Japan (BOJ), the Swiss National Bank (SNB), and the Swedish Riksbank.

The dynamics shown by this graph recall the three main type of movements present in those dynamics: parallel shifts, changes in the slope (twist) and modification of the hump (bowing). By the way, PCA (principal component analysis) has shown to work pretty well with summarising the volatility of the term structure into these three recognisable factors, providing evidence of the weight of these first three in the dynamics (see Frye (1997)): 83.1% for the parallel shifts, 10% for the twist and only 2.8% for the bowing.

This is in fact a nice result. It shows that, using only duration as a risk measure for interest rates, we are close to hedge 85% of potential movements in the term structure.

We can also see that short term rates seem to be more volatile than long term ones which seem to act as long term revisions of the target.

It is even interesting to see the shape of last term structure shown, that of May 5, 2023. It is globally downward sloping with two bows and a difference of more than 2% between the highest and the lowest.

## Interest rate derivatives

Although there is a myriad of interest rate models, in discrete and continuous time, one-factor or multi-factor, before-HJM or current-HJM- era, still a vast majority of participants in the market have been happy with the so-called "market model" for call and puts on interest rates. Globally speaking, it means applying Black & Scholes (B&Sch) to interest rates as if they would be prices. This is quite peculiar as we tend normally to model the relative changes in prices 𝛥P/P measured in logs, as normally distributed, but rates' dynamics are analysed through their absolute changes in r, i.e. 𝛥r. And using B&Sch for rates as the underlying, we implicitly assumes that 𝛥r/r, measured in log returns, are normally distributed.

But suddenly, the short-term rate of some central banks moved into negative territory, which made the use of B&Sch's shortcut formula useless this time since the log function doesn't accept negative values.

Since financial engineering is not short of "cooking" solutions, one idea devised was to use B&Sch on shifted values of r before using the B&Sch formula. Reuters (now Refinitiv) and Bloomberg even started proposing implied volatility matrices based on this "shifted" B&Sch formula.

Ultimately, Louis Bachelier was remembered, him who assumed prices were normally distributed (not returns), a problem for prices, but not for interest rates that can now become negative.

## Short memory

Interestingly, our surprise for negative rates turned quickly into a normal acceptance of low - if not negative - interest rates, during more than twelve years. And it seems we have to re-learn now to live with higher interest rates.

For these last twelve years, it has been relatively difficult to motivate an audience to learn the myriad of interest-rate derivatives developed in the 80s and the 90s. Some treasury officers had even a hard time in convincing their management that it was necessary to keep an eye on the interest-rate risk tolerance and the corresponding hedge policy. Many of those who today re-understand they should hedge now again....sorry should have been hedged.

Interest-rate swaps are back in town, or have always been for those with a longer view on their financing risk than ten years...important when you issue debt over twenty years.

We advise looking at the publication and research journey of an incredible person, Jamil Baz, a wonder quant closely connected to the rise, valuation and use of interest rate swaps in the 80s.

Frye, Jon (1997). "Principals of risk: Finding value-at-risk through factor-based interest rate scenarios." NationsBanc-CRT, April (1997).

Baz, Jamil, and George Chacko (2004).

*Financial derivatives: pricing, applications, and mathematics*. Cambridge University Press, 2004.Pirotte, Hugues (2004-2023). "Interest rate risk management", Université libre de Bruxelles, (Evolving) Teaching note.

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