Race of the models
In our June pipeline, we introduced a new measure of volatility - a single volatility index. The idea is very simple - replace the entire ATM volatility surface with one best-fitting short-rate volatility number. First, it is useful for risk managers because is drastically simplifies quantitative understanding of the role of volatility in a retrospective risk assessment. In particular, it sets up a framework for measuring the Vega. Second, this method may become a fair judge of the performance of term structure models. Indeed, each short-rate model has its own volatility specification, i.e. produces its own volatility index. The best model is the one in which the index is most independent of the rate level. For the ultimate model selection, we usually complement this type of analysis with others, such as historical, day-to-day, rate behavior, and the evidence of volatility skew for traded swaptions.

For quite a long time, the Hull-White (HW) model performed very well across all metrics. With rates continuing their freefall through spring, many started wondering whether HW would remain creditable. A simple reason for this doubt is that a normal model does not preclude a plunge into the negative rate territory -- unseen in U.S. history. The only way to prevent negative rates is to reduce the basis point volatility when rates fall. Both Black-Karasinski (BK) and Squared Gaussian (SG) models accomplish this mission, albeit at a different pace.

As shown in Figure 3, a sharp decline in rates in June of 2003 (below 3%) triggered a new mode. The SG model performed visibly better with HW to follow. As for the BK model, it has remained a minor-leaguer, where it seems to have been relegated 10-15 years ago. As we see, any change in the rate level causes a mirror reflection in the BK (i.e. proportional) volatility. A strong remaining dependency on rates indicates wrong functional volatility form for this model.

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