After a detailed look at the implications of a prolonged period of historically low interest rates and a close examination of how swaptions have been trading, Andrew Davidson & Co. has introduced two new term structure models.

These models - known as the Hull White Model (HW) and the Squared Gaussian (SG), along with an enhanced version of the existing Black Karasinski (BK) model (all included in the firm's VectorsTM suite of analytical tools) - are all based on a different distribution of interest rates over time.

BK, which has been called the bread and butter for option traders since it was first introduced, assumes lognormality. The HW model assumes normality, while the SG model assumes "normal process squared".

In a release, representatives from Andrew Davidson noted that "Depending upon the prevailing level and volatility of interest rates, risk managers, traders and portfolio managers may now choose the most appropriate model for the given interest rate environment."

All in favor of HW

In the most recent publication of the company's Quantitative Perspectives, Senior Consultant Alex Levin proposed the use of the Hull White model, which assumes normality, as opposed to using the Black Karasinski model, which assumes lognormality. He argued that both the recent historically low rates and implied volatility skew for swaptions strengthen the case for rate "normalization", and, in turn, spurn lognormality.

"One argument is to take a look at historical deviations of rates and when you do so, you would not find much relationship between volatility and the level of rates," explained Levin.

He added that this supports an argument against the Black Karasinski model - which assumes that the daily volatility of rates should be proportional to the level of rates - as proportionality has not been a factor nor has there been any disputable relationship between volatility and the level of rates for the last decade.

"A weak or absent relation between absolute volatility and rate level is a sign of normality rather than lognormality," wrote Levin in the Quantitative Perspectives report. "It also prompts quoting rate uncertainty (and, therefore, option prices) in terms of absolute volatility (such as 110 basis points) rather than relative volatility (say, 20%). Recently, many brokers have begun communicating exactly in that way."

Aside from the historical behavior of rates, market perception and anticipation of future rate distribution are even stronger arguments for the use of the Hull White model.

Levin said that market participants could look at and "measure" the implied volatility skew for swaptions, which is the line that shows how volatility changes with a strike. He wrote, "If the market participants believed in lognormality, there would exist little reason for the implied volatility to change with the option's strike."

He explained that the way that swaptions trade should give an indication of what the market thinks about the future distribution of rates. It would also confirm that, according to market participants, the future distribution of rates is nowhere near lognormal.

"Looking at historical data and measuring the way swaptions trade, you can make an assumption on what market participants think about distribution," said Levin. "Research implies that the rates are perceived to be close to normally distributed. This is why you have to select a model that most appropriately reflects this indication, and the Hull White model is the most reflective of that today."

Levin also stated that the switch from the Black Karasinski model to the Hull White model would systematically shorten effective duration for most mortgage instruments (except IOs and POs). For example, he quantified a duration reduction of 0.4 years for the current-coupon agency pass-through, thereby reducing the hedging needs for mortgage originators, bankers and investors.

Negative rates also an issue

One disadvantage of the HW model is that it does not preclude rates going into negative territory. In the Quantitative Perspectives report, Levin also questioned what detrimental effects could happen if the HW model is used, considering that interest rates have never been negative in U.S. history.

The report asked, "Indeed, the odds of such an event are far from infinitesimal, but how badly can it damage the value of an MBS?"

Analysis showed that using the HW model is "rather harmless; it will not lead to sizable mispricing even in the worst, mortgage-irrelevant case," wrote Levin. So, the use of the HW model does not look as detrimental to analytics as some people might perceive it to be. However, this conclusion would need some periodic review as interest rates continue to fall.

If practitioners are interested in a model that precludes negative rates, they could use the Squared Gaussian, which makes use of lognormal analysis and considers the relationship between the level of rates and the level of volatility, albeit somewhat dampened.

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